Field Theoretic Approach to Interacting Two Body Tunneling

This paper develops an analytic field-theoretic approach to interacting two-body tunneling by deriving the Bethe-Salpeter equation for a Yukawa-coupled tunneling field, demonstrating a closed-form solution in the 1+1 dimensional instantaneous positive-energy regime, and confirming physical consistency through the recovery of the Lippmann-Schwinger equation.

The Gist

The Big Picture: The "Ghost" Problem

Imagine you have two tiny, invisible ghosts (particles) trying to walk through a solid brick wall (a potential barrier). In the world of quantum physics, this is called tunneling.

Usually, if you have just one ghost, physicists can calculate exactly how likely it is to pass through. But what happens if you have two ghosts that are holding hands (interacting) and trying to walk through the wall together?

This is incredibly hard to solve.

• The Problem: Standard math tools (perturbation theory) work like a ladder: you take small steps to get an answer. But tunneling is a "non-perturbative" event—it's like trying to jump over a canyon. You can't take small steps; you have to leap.
• The Complication: When the two ghosts hold hands, they change each other's behavior. If one gets stuck, the other might get pulled back. If they push each other, they might both get through. This interaction makes the math explode into complexity.

The Authors' Solution: A New Map

The authors, led by Ye Guo, decided to stop trying to solve the problem with standard "particle" math. Instead, they used Field Theory.

The Analogy: The Ocean vs. The Swimmers

• Old Way: Imagine the ghosts are swimmers trying to cross a river. You try to track every splash and wave they make.
• New Way: Imagine the river itself is a living, breathing ocean. The "ghosts" are just ripples in this ocean. The wall isn't a brick wall; it's a sudden change in the ocean's depth.

By treating the particles as ripples in a field, the authors could use a powerful mathematical tool called the Bethe-Salpeter equation. Think of this equation as a recipe book for building complex interactions. It allows them to stack "ladder diagrams" (visual representations of how the particles exchange energy) to see the whole picture, rather than just looking at one step at a time.

The "Instantaneous" Shortcut

The math they derived is still very heavy. To make it solvable, they made a clever simplification called the Instantaneous Positive-Energy Regime.

The Analogy: The Fast-Forward Button
Imagine the two ghosts are running a relay race.

1. Relativistic Reality: In the real world, the baton (interaction) takes time to pass, and the runners might run backward or forward in time (virtual particles).
2. The Shortcut: The authors said, "Let's assume the baton passes instantly, and let's only count the runners who are moving forward."

This is like hitting the "Fast Forward" button on a movie. You skip the confusing parts where the characters are confused or moving backward, and you focus only on the main action: the two particles moving forward together. This turned a 4-dimensional nightmare into a manageable 1-dimensional puzzle that they could actually solve on paper.

What Did They Find? (The "Heat Map")

Once they solved the math, they looked at the results, specifically how the two particles behave when they interact. They found some surprising patterns:

1. The "High Five" Effect (Same Direction): When the two particles are moving in the same direction, the interaction actually helps them tunnel through the wall. It's like two people pushing a heavy car; if they push together, it's easier to get over the hill.
2. The "Cancel Out" Effect (Opposite Directions): When the particles are moving in opposite directions (one left, one right), the interaction stops them from tunneling. It's like two people trying to push a car from opposite sides; they cancel each other out, and the car doesn't move.
3. Low Energy Wins: Surprisingly, the interaction helps the most when the particles are moving slowly. Usually, in physics, you need high speed to break through barriers. Here, the "holding hands" effect makes the slow, gentle approach the most effective way to tunnel.

Why Does This Matter?

This paper is a bridge between two worlds:

1. The Micro World: It explains how nuclear particles (like protons) might escape a nucleus together (a process called two-proton decay).
2. The Tech World: It helps us understand devices like tunnel diodes and scanning tunneling microscopes, where electrons move through barriers.

The Takeaway:
The authors built a new "field theoretic" engine to drive two interacting particles through a wall. They proved that when these particles interact, they don't just act like two separate individuals; they act as a single, coordinated unit. Sometimes this teamwork helps them pass through barriers they couldn't cross alone, and sometimes it holds them back.

This work is a crucial step toward understanding the complex dance of particles in our universe without having to rely on guesswork or oversimplified models. It shows that even in the chaotic quantum world, there is a hidden order that can be mapped out with the right mathematical tools.

Technical Summary

1. Problem Statement

The paper addresses the theoretical challenge of interacting two-body quantum tunneling. While single-body tunneling and many-body tunneling in discrete models or large-$N$ limits are well-understood, the continuous two-particle tunneling problem remains analytically intractable due to:

• Non-perturbative nature: Tunneling cannot be treated via standard perturbation theory.
• Lack of classical solutions: The Euclidean Lagrangian for continuous systems with barriers lacks classical solutions, thwarting semiclassical expansions (e.g., instanton methods).
• Symmetry breaking: The presence of a static barrier breaks translational symmetry, while particle interactions ($U(\hat{x}_1 - \hat{x}_2)$) introduce inseparability, preventing exact solutions for Hamiltonians of the form:
  $$\hat{H} = \hat{h}(1) + \hat{h}(2) + \alpha(W(\hat{x}_1) + W(\hat{x}_2)) + \beta U(\hat{x}_1 - \hat{x}_2)$$
  where $W$ is the barrier potential and $U$ is the interparticle interaction.

2. Methodology

The author employs a relativistic field-theoretic approach based on the formalism developed by Zielinski et al. [9, 10].

• Field Lagrangian: The system is modeled using a scalar field $\psi$ (tunneling particles) coupled to a meson field $\phi$ (mediating interaction) and an external barrier potential $u(x) = aV_0\delta(x^3)$. The Lagrangian includes Yukawa coupling ($g\phi\psi^2$).
• Resummed Propagator: Instead of treating the barrier perturbatively, the author utilizes a resummed tunneling propagator ($D_\psi$) derived in previous work. This propagator accounts for the infinite series of interactions with the barrier (delta function potential) exactly.
• Bethe-Salpeter (BS) Equation: The four-point correlation function (representing the two-particle scattering amplitude) is approximated by resumming all ladder diagrams (meson exchanges). This leads to a BS equation where the kernel includes the resummed tunneling propagator.
• Dimensional Reduction & Instantaneous Approximation:
  • The problem is reduced to 1+1 dimensions (energy and the tunneling direction $x^3$) to find a closed-form solution.
  • An instantaneous approximation (Salpeter regime) is applied to the interaction kernel, assuming the meson mass is small or the system is in the non-relativistic (NR) limit. This allows the conversion of the integral equation into a solvable form using Laplace transforms and the Wiener-Hopf technique.
• Perturbative Consistency Check: To validate the formalism, the author perturbs the interparticle interaction ($g$) to derive the scattering amplitude and recovers the Lippmann-Schwinger equation in the NR limit.

3. Key Contributions

1. Derivation of the BS Equation for Tunneling: The paper successfully derives the Bethe-Salpeter equation for two interacting particles tunneling through a barrier, incorporating the exact barrier effects via the resummed propagator.
2. Closed-Form Solution in 1+1D: A non-perturbative, closed-form approximate solution is obtained in the instantaneous positive-energy (Salpeter) regime. This is achieved by:
   • Reducing the 4D BS equation to 1+1D.
   • Applying the instantaneous approximation to the kernel.
   • Solving the resulting integral equation via Laplace transforms and matrix inversion in the frequency domain.
3. Recovery of Standard Limits: The formalism is shown to be consistent with known physics by:
   • Recovering the Lippmann-Schwinger equation in the non-relativistic limit.
   • Demonstrating that the model reduces to the free resolvent when the interaction vanishes or the barrier is removed.
4. Analysis of Interaction Effects: The paper provides an explicit calculation of the T-matrix (scattering amplitude) for interacting tunneling particles, analyzing how interparticle forces modify tunneling probabilities.

4. Key Results

• Closed-Form Structure: The solution for the scattering amplitude $L(p)$ in Laplace space is expressed as $L = (1-A)^{-1}(G + BK)$, where $A$ and $B$ are operators derived from the tunneling propagator and interaction kernel. This provides a pathway to calculate non-perturbative amplitudes without relying on phenomenological potentials.
• Tunneling Amplitude Features:
  • Momentum Dependence: The scattering amplitude $|t|$ exhibits a "cross" structure.
  • Suppression at Opposite Momenta: There is a significant suppression of tunneling when the incoming and outgoing particles have opposite momenta ($p'_a \approx -p'_b$). This is attributed to destructive interference between forward- and backward-propagating virtual states, causing the correlated contribution to vanish when total momentum is zero.
  • Enhancement at Similar Momenta: Tunneling is enhanced when particles have similar momenta ($p'_a \approx p'_b$), showing a ridge along the diagonal.
  • Interaction Strength: For weak interactions ($g/e a V_0 \ll 1$), the system behaves like non-interacting tunneling. As interaction strength increases, low-momentum tunneling becomes dominant.
• Physical Consistency: The derived T-matrix contains pole-like structures that align with the expected behavior of bound states and resonances in the non-relativistic limit.

5. Significance

• Bridging QM and QFT: The work bridges the gap between few-body quantum mechanics (Hamiltonian approaches) and relativistic field theory. It demonstrates that correlated tunneling effects can be reproduced without phenomenological potentials, arising naturally from the underlying propagator structure.
• Handling Non-Perturbative Regimes: By using resummed propagators, the method bypasses the failure of standard perturbation theory in tunneling problems, offering a rigorous framework for interacting tunneling.
• Experimental Relevance: The findings are relevant to cold atom physics, nuclear decay (e.g., two-proton decay), and solid-state devices (tunnel diodes, Josephson junctions). The predicted suppression of tunneling for opposite momenta could explain frequency splitting observed in numerical studies of symmetric boson systems.
• Foundation for Future Work: The closed-form solution in the instantaneous regime serves as a crucial stepping stone for tackling the full non-perturbative problem in higher dimensions and for exploring the interplay between strong interactions and tunneling dynamics.

In summary, this paper provides a rigorous field-theoretic framework for interacting two-body tunneling, deriving a solvable Bethe-Salpeter equation and demonstrating its consistency with established non-relativistic limits while revealing novel interference effects in the tunneling amplitude.