Photon emission by vortex particles accelerated in a linac

This paper demonstrates that charged spinless vortex particles accelerated in linear accelerators retain their orbital angular momentum with negligible losses due to photon emission, proving that their quantum states are robust enough to reach relativistic energies in both conventional and wake-field accelerators.

The Gist

The Big Picture: Spinning Top in a Rocket

Imagine you have a tiny, spinning top (an electron) that isn't just spinning on its own axis; it's also swirling around in a spiral path, like a galaxy or a hurricane. In physics, we call this a "vortex particle" because it carries a special kind of twist called Orbital Angular Momentum (OAM).

Now, imagine you want to shoot this swirling top out of a giant cannon (a particle accelerator or "linac") to make it go incredibly fast—almost the speed of light.

The Big Question: As this swirling top gets blasted forward by the rocket engine, will it lose its spin? Will the friction of the acceleration cause it to unravel and stop swirling, turning into a boring, straight-moving particle?

The Answer: The authors of this paper say: No, it's incredibly robust. The particle keeps its swirl almost perfectly intact, even while being accelerated to relativistic speeds.

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The Problem: The "Spooky" Physics of Acceleration

In the world of quantum physics, things get weird when you accelerate particles.

• The Old Way: Usually, physicists calculate how particles behave when they are sitting still or moving at a constant speed. They use a "stationary" approach, like watching a car drive on a highway at a steady 60 mph.
• The Real World: Accelerators are like a car slamming on the gas pedal. The speed is changing constantly. This is a "non-stationary" process.
• The Fear: When a charged particle accelerates, it usually emits light (photons). Think of a car tire spinning on wet pavement and spraying water droplets. The authors were worried that as the "vortex electron" accelerates, it might spray off "twisted photons" (light that also carries a twist), effectively robbing the electron of its own swirl. If this happened too fast, the electron would lose its special quantum properties before it even reached the target.

The Solution: A "Quasi-Classical" Model

To solve this, the authors built a new mathematical model. Instead of trying to solve the impossible, messy equations of the entire universe, they used a clever shortcut called the WKB approximation (think of it as a "good enough" map).

The Analogy:
Imagine the electron is a surfer riding a massive wave (the radio wave in the accelerator).

1. The Wave: The accelerator uses radio waves to push the electron forward.
2. The Surfer: The electron is a tiny wave packet (a bundle of energy) riding that wave.
3. The Trick: Because the surfer (electron) is so tiny compared to the massive ocean wave (the accelerator field), the surfer only feels a tiny, flat, constant push at any single moment. The authors realized they could treat the electron as if it were in a simple, constant electric field, which made the math solvable.

They also had to deal with a "ghost" problem. In strict quantum math, accelerating particles can sometimes create "ghost" particles (electron-positron pairs) out of thin air. The authors' model cleverly ignores these ghosts because, in real-world accelerators, the fields aren't strong enough to create them anyway. This keeps the math clean and focused on the single electron.

The Findings: The Swirl is Safe

After running the numbers, they found some very reassuring results:

1. The "Lifetime" is Huge: They calculated how long it takes for the electron to lose its swirl due to emitting light. They found that the "lifetime" of the swirl is orders of magnitude longer than the time it takes to travel through the accelerator.

   • Analogy: If the accelerator is a 10-second sprint, the time it would take for the electron to lose its spin is like 10,000 years. It's basically impossible for it to lose its spin during the race.

2. The "Big Loss" is Rare: They looked at the different ways the electron could lose its swirl.

   • Analogy: Imagine the electron is a spinning top. It could lose a tiny bit of spin (a wobble) or a huge chunk of spin (falling over). The math shows that the electron almost never falls over. It only ever wobbles a tiny bit, and even that is very rare.
   • Most of the time, the electron emits a photon that carries almost no twist, leaving the electron's own twist untouched.

3. The Electric Field Doesn't Matter (Much): You might think a stronger push (higher electric field) would shake the swirl apart. But for realistic electron beams (which are very short and tight), the strength of the electric field barely changes the outcome. The electron is so "point-like" compared to the distance light needs to form a wave that the field strength doesn't disrupt the swirl.

Why This Matters

This paper is a green light for future experiments.

• Before this: Scientists were worried that if they tried to accelerate "vortex electrons" (twisted electrons) to high energies, they would lose their special quantum identity.
• Now: We know that we can take these twisted particles, blast them through conventional particle accelerators (like the ones at CERN or SLAC), and they will arrive at the other end still swirling.

The Takeaway:
Think of the vortex electron as a spinning coin. The authors proved that you can shoot that coin out of a cannon at 99% the speed of light, and it will still be spinning when it hits the target. This opens the door to using these "twisted" particles for new types of imaging, studying magnetic materials, and exploring the fundamental nature of the universe, all without worrying that the acceleration process will destroy the very thing that makes them special.

Technical Summary

1. Problem Statement

The paper addresses the stability of charged particles carrying Orbital Angular Momentum (OAM) (known as "vortex" or "twisted" particles) when accelerated to relativistic energies in linear accelerators (linacs).

• The Core Issue: When a charged particle with a phase vortex (OAM projection $l$) is accelerated, it may emit photons. If these emitted photons carry away angular momentum, the particle's quantum state (its vorticity) could degrade or collapse.
• Theoretical Gap: Standard Quantum Electrodynamics (QED) calculations often rely on the S-matrix formalism, which assumes stationary states and infinite time intervals. However, acceleration in a linac is a non-stationary process occurring over finite time and space. Existing models either ignore the acceleration dynamics or rely on exact solutions (like the Klein-Gordon equation in constant fields) that introduce unphysical electron-positron pair admixtures, making them unsuitable for realistic accelerator conditions where fields are far below the Schwinger critical limit.
• Key Question: Is the OAM of a vortex electron (or ion/muon) robust enough to survive the acceleration process in a conventional linac, or will radiative losses destroy the vortex state before relativistic energies are reached?

2. Methodology

The authors develop a non-stationary perturbation theory framework within Scalar QED to model this process.

• Quasi-Classical (WKB) Approach: Instead of using exact Klein-Gordon solutions (which are delocalized and contain pair admixtures), the authors employ a Wentzel–Kramers–Brillouin (WKB) approximation. This allows them to construct a localized, normalized wave packet that follows a classical relativistic hyperbolic trajectory in a longitudinal electric field ($E$) while maintaining a stable vacuum (no pair creation).
• Field Configuration:
  • Longitudinal Electric Field ($E$): Accelerates the particle along the $z$-axis.
  • Longitudinal Magnetic Field ($H$): Acts as a focusing lens, confining the particle transversely and quantizing the transverse motion into Landau states.
• State Definition:
  • Transverse: Described by scalar Landau states with quantum numbers $(n, l)$, where $l$ is the OAM projection.
  • Longitudinal: Described by a wave packet with a coherence length $\sigma$, centered on a classical trajectory $z_{cl}(t)$.
• Calculation of Emission:
  • The transition matrix element ($S_{fi}$) for photon emission is calculated to the first order.
  • The authors integrate over the acceleration time and the photon momentum, accounting for the temporal coherence of the emission process (interference between emission amplitudes at different times along the trajectory).
  • They distinguish between two regimes based on the packet length $\sigma$ relative to the radiation formation length ($L_{form}$): the narrow-packet regime ($\sigma \ll L_{form}$) and the wide-packet regime.

3. Key Contributions

1. Effective Model for Accelerated Vortex Particles: The paper establishes a rigorous effective model using WKB states to describe vortex particles in accelerating fields, avoiding the mathematical and physical pitfalls of exact solutions (pair admixtures) while retaining accuracy for realistic accelerator parameters.
2. Quantification of OAM Lifetime: The authors derive the lifetime ($\tau$) of the vortex state against OAM-changing radiative transitions. They define this as the characteristic time over which the particle preserves its OAM projection $l$ against photon emission.
3. Regime Analysis (Narrow vs. Wide Packets):
   • They identify that for realistic electron packets in linacs (coherence length $\sigma \sim 1\text{--}10$ nm), the system operates in the narrow-packet regime.
   • In this regime, the emission probability is practically independent of the electric field strength because the packet is effectively point-like compared to the formation length.
   • In the wide-packet regime, the probability becomes sensitive to the field strength, but this is less relevant for current high-energy facilities.
4. Hierarchy of Transitions: The study reveals a strict hierarchy in emission channels:
   • Transitions with small OAM loss ($\Delta l = l - l' \sim 1$) dominate.
   • Transitions with large OAM loss ($\Delta l \gg 1$) or changes in the radial quantum number ($\Delta n \neq 0$) are strongly suppressed.

4. Key Results

• Extreme Stability: For fields typical of accelerator facilities (e.g., $E \sim 10\text{--}100$ MV/m, $H \sim 10$ T), the effective lifetime ($\tau$) of the vortex state is orders of magnitude larger than the acceleration time ($t_{out}$).
  • Even for kilometer-long linacs, the probability of losing OAM during a single pass is negligible.
  • The lifetime increases as the magnetic field decreases (due to the scaling of the Landau radius $\rho_H \propto 1/\sqrt{H}$).
• Dependence on Momentum: Contrary to naive classical time-dilation expectations, the lifetime $\tau$ decreases as the initial longitudinal momentum $p_z$ increases in the non-relativistic regime. This highlights the genuinely quantum nature of the process, where higher momentum leads to a higher probability of OAM-changing radiation.
• Soft Photon Dominance: The emission is dominated by soft photons (low energy, low transverse momentum). The differential probability peaks sharply when the final longitudinal momentum $p'_z$ is close to the initial $p_z$.
• Robustness: The vortex quantum state is highly robust against photon emission. The "twisted" nature of the particle is preserved throughout the acceleration process to relativistic energies.

5. Significance and Implications

• Feasibility of Vortex Acceleration: The results prove that it is physically possible to accelerate vortex electrons, ions, and muons to relativistic energies in conventional linacs without destroying their OAM. This validates the experimental pursuit of high-energy vortex beams.
• New Physics Opportunities: Once accelerated, these stable vortex beams can be used to study quantum phenomena inaccessible to standard plane-wave (Gaussian) beams, such as:
  • High-energy collisions with twisted particles.
  • Novel scattering cross-sections dependent on OAM.
  • "Superkick" effects and other topological interactions.
• Theoretical Foundation: The paper provides a necessary theoretical framework for non-stationary radiation from structured quantum states, bridging the gap between standard QED and the specific dynamics of accelerated wave packets. It sets the stage for extending these studies to Dirac particles (spin-1/2) and more complex accelerating structures like wakefields.

In conclusion, the paper demonstrates that radiative losses do not pose a barrier to the acceleration of vortex particles, confirming the stability of their orbital angular momentum in realistic accelerator environments.