⚛️ high-energy theory

Composite worldline instantons and the nonperturbative particle decay in constant external electric and magnetic fields

This paper validates the composite worldline instanton approach for calculating the nonperturbative decay of charged particles in constant electric and magnetic fields by demonstrating its agreement with self-energy and wave function overlap methods, while also exploring its potential application to proton decay and the effects of final-state entanglement.

Original authors: Alexander Gorsky, Ivan Poluboyarinov

Published 2026-07-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexander Gorsky, Ivan Poluboyarinov

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 a proton, the tiny, heavy workhorse inside an atom's nucleus, sitting in a room filled with a super-strong, invisible electric wind. Normally, this proton is stuck. It's too heavy to break apart on its own. But if the electric wind blows hard enough, it can act like a cosmic tunneling machine, giving the proton a little push to sneak through a wall it couldn't climb.

This paper is about figuring out exactly how likely that sneaky escape is. The authors, Alexander Gorsky and Ivan Poluboyarinov, are testing a specific mathematical tool called the "composite worldline instanton" to predict this escape rate. Think of this tool as a way to draw the "ghost path" a particle takes when it tunnels through a barrier. Usually, these ghost paths are simple loops. But when a proton breaks into pieces (like a neutron and a charged pion), the path gets complicated. It's no longer one loop; it's a "composite" shape made of several different tracks meeting at junction points, like a complex knot of string.

The Big Discovery: Three Ways to Draw the Same Map
The authors' main finding is that this "composite knot" method works perfectly. They checked their results against two other very different ways of calculating the same thing:

The Self-Energy Check: Looking at the "imaginary part" of the proton's energy in the electric field (a bit like listening for a faint hum that signals instability).
The Wave-Function Overlap: Checking how much the "ghost waves" of the starting proton and the ending pieces overlap in space.

In the "leading exponential approximation" (which means looking at the biggest, most dominant factor that makes the event rare or likely), all three methods gave the exact same answer. It's as if three different cartographers drew maps of a hidden cave using different tools, and when they compared notes, the cave entrances and exits matched perfectly. This suggests that the "composite worldline instanton" is a valid and reliable way to predict how charged particles decay in strong electric fields.

The Magnetic Twist: Tunneling Between Levels
The paper also looks at what happens in a magnetic field instead of an electric one. Here, the rules change slightly. In an electric field, the particle tunnels through a barrier in space. In a magnetic field, the particle is stuck in "energy rungs" (called Landau levels), like a ladder. Decay happens when the particle jumps down a rung, but it has to tunnel through a gap in momentum, not just space.

The authors found that if you use a special mathematical trick called the "Routhian action" (a way of rewriting the energy equations to focus on momentum), the "ghost path" calculation matches the results from overlapping wave functions. They also compared this to a process called "synchrotron emission" (where particles emit light while spiraling in a magnetic field). They showed that their math is a generalized version of that known process. If the particle emits a "hard" photon (a high-energy burst of light), the decay is exponentially suppressed (very rare), just like their instanton math predicts.

The "What Ifs" and the "What Nots"
The authors are careful to point out what their math doesn't do.

No Magic for Light Particles: They found that if the charged piece flying off is too light (lighter than the mass difference between the proton and neutron), the "ghost path" becomes impossible to draw in the real world. The math says the path turns "complex" (involving imaginary numbers), meaning the simple tunneling picture breaks down, and the suppression is determined by the mass of the flying piece alone, regardless of the other particles.
No Instant Fixes: They explicitly state that their results are for the "leading exponential approximation." This means they calculated the main factor that makes the event rare, but they haven't calculated the tiny, messy details (the "prefactors") that would give the exact probability down to the last decimal. They suggest that future work needs to look at "higher winding modes" and fluctuations to get those details.
No Real-World Proton Decay (Yet): While they mention that this could theoretically apply to protons near black holes or in particle accelerators, they don't claim to have measured a real proton decaying. They are testing the math of the decay, not the event itself.

The Entanglement Puzzle
One of the cooler side notes is about "entanglement." When the proton splits, the two new particles fly off with opposite momenta. The authors suggest that if you don't know exactly where they are going, they are "entangled"—their fates are linked. They calculated an "entropy" (a measure of this linked uncertainty) and found it depends on how precisely you can measure the particles' momentum. If you can't measure the momentum very well (a large "coherence scale"), the uncertainty (and entropy) goes up. It's like two dancers spinning away from each other; if you can't see their feet, you can't be sure who is leading, and the "confusion" (entropy) is high.

The Bottom Line
The paper suggests that the "composite worldline instanton" is a robust tool for understanding how particles tunnel and decay in strong fields. It confirms that this method agrees with other established theories in the most important limits. However, it also highlights that for very light particles or for getting the exact probability (not just the main trend), more complex math is needed. The authors are essentially saying, "We've drawn the map, and it matches the other maps we have. Now we need to add the street names and traffic lights to get the full picture."

Technical Summary: Composite Worldline Instantons and Nonperturbative Particle Decay

Problem Statement
This paper investigates the validity of the composite worldline instanton approach for describing the nonperturbative decay of charged particles in constant external electric and magnetic fields. While the Schwinger mechanism for pair creation is well-understood via worldline instantons, the decay of a single charged particle (energetically forbidden without the field) into multiple daughter particles involves more complex trajectories. Specifically, the authors address the decay of a proton (modeled as a scalar particle for simplicity) into a neutron and a charged meson (or positron and neutrino) in external fields. The central problem is to verify whether the leading exponential suppression factors derived from composite worldline instantons agree with established methods: the imaginary part of the self-energy in the external field and the overlap of exact wave functions.

Methodology
The authors employ three complementary approaches to calculate the decay rate in the leading exponential approximation:

Composite Worldline Instantons:
- Electric Field: The decay is modeled as a "bounce" in Euclidean space-time. The instanton configuration is composite, consisting of the worldlines of the initial particle (proton) and the daughter particles (neutron and charged meson) meeting at two junction points. The action is evaluated in two frames:
  - Rindler Coordinates: The initial proton is static, and daughters evolve along equations of motion.
  - Euclidean Coordinates: The initial proton is accelerated. The trajectory is a composite of circular arcs (cyclotron motion) and straight lines (neutral particle).
- Magnetic Field: The decay is treated as a transition between Landau levels. The authors utilize a Routhian action (Legendre transform of the proper-time action with respect to energy and conserved momentum) to handle the tunneling between states with prescribed energy and momentum, rather than coordinate shifts.
Imaginary Part of Self-Energy:
- The decay rate is extracted from the imaginary part of the proton self-energy ( $\text{Im } \Sigma$ ) in the external field. The calculation involves Schwinger proper-time propagators and evaluating residues at Schwinger poles to isolate the leading exponential suppression in the weak-field limit.
Wave Function Overlap:
- For the magnetic field case, the decay rate is derived from the spatial overlap of the initial proton wave function (Landau level state) and the final state wave functions (neutron plane wave and charged particle Landau level). This involves calculating matrix elements of ladder operators between harmonic oscillator states.

Key Contributions and Results

Validation of Composite Instantons in Electric Fields:
The authors derive an analytical expression for the instanton action $S_{inst}$ for the decay $p \to n + \pi^+$ (or similar scalar analogs) in a constant electric field (Eq. 17). They demonstrate that this result exactly matches the asymptotic behavior of the imaginary part of the self-energy (Eq. 53) in the weak-field limit. The agreement holds across different mass regimes, including equal masses, slightly different masses, and the limit of a light charged daughter particle.
- Complex Instantons: In the regime where the charged daughter mass $m_c$ is less than the mass difference $\delta = m_n - m_p$ , the real instanton solution ceases to exist. The authors show that the analytic continuation of the action yields a complex saddle point. The real part of the action, which determines the tunneling suppression, becomes solely dependent on the mass of the charged daughter particle ( $S_{inst} \approx \pi m_c^2 / |eE|$ ), independent of the initial and neutral particle masses.
Magnetic Field Decay and Generalized Synchrotron Emission:
The paper establishes the equivalence between the wave-function overlap method and the semiclassical Routhian action approach for magnetic field decay. The resulting decay exponent is given by a system of equations relating the momenta and rapidities of the particles (Eq. 92).
- The process is identified as a generalization of synchrotron emission. While standard synchrotron emission (soft photons) is not exponentially suppressed, the emission of hard photons (or massive particles) is exponentially suppressed, matching the derived instanton action.
- The authors calculate the critical photon energy for induced decay ( $\gamma + p \to c + n$ ), showing that above this threshold, the process becomes kinematically allowed with a power-law rate, while below it, it remains exponentially suppressed.
Entanglement in the Final State:
The authors briefly analyze the entanglement entropy of the daughter particles. Due to momentum conservation, the transverse momenta of the daughter particles are correlated. In the leading exponential approximation, the decay produces a momentum-entangled state. The calculated von Neumann entropy depends logarithmically on the momentum resolution scale and the field strength (Eq. 63).
Three-Body Decay:
The framework is extended to three-body decays (e.g., $p \to n + e^+ + \nu_e$ ). The authors argue that at the level of the leading exponential action, the two neutral particles (neutron and neutrino) can be treated as a single composite neutral object with mass $m_n + m_\nu$ , reducing the problem effectively to the two-body case.

Significance and Claims
The paper claims that the composite worldline instanton approach is a valid and robust method for calculating nonperturbative decay rates in external fields, as confirmed by its agreement with self-energy and wave-function overlap methods in the leading exponential approximation.

Consistency: The exact match between the instanton action and the imaginary part of the self-energy validates the use of composite trajectories involving multiple worldlines for particle decay.
Universality: The approach is shown to be applicable to various mass hierarchies and final state configurations (two or three particles).
Physical Insight: The analysis provides physical interpretations for the suppression factors, such as the geometric meaning of the "half-circle" trajectory in specific mass limits and the transition to complex instantons when kinematic thresholds are crossed.
Future Directions: The authors suggest that while the leading exponential term is established, future work is needed to calculate prefactors (higher winding modes, fluctuations) and to extend the formalism to inhomogeneous fields. They also propose investigating the thermodynamic interpretation of entanglement entropy in decay and the application of these instantons to holographic QCD and Skyrmion decay.

The authors explicitly state that their primary goal was to test the validity of the composite worldline instanton approach, and the results confirm its applicability for arbitrary numbers of particles in the final state within the leading exponential accuracy.

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