Abstract

1. Introduction

Quantum tunneling in strongly driven backgrounds is a paradigmatic non-perturbative effect, with the Schwinger process providing the textbook QED example. Superfluid films furnish controlled analog platforms for related physics, while the Berezinskii–Kosterlitz–Thouless (BKT) framework fixes the logarithmic vortex energetics.

Here we develop a worldline-instanton treatment of vortex tunneling in a thin ⁴He film under uniform superflow, obtain the closed-form exponent, and then embed the result into the Universal Tuple–Matrix Framework (UTMF). The UTMF expansion yields sector-indexed mass corrections and a falsifiable prediction: curvature in ln Γ versus 1/|v_s| originating from a velocity-dependent effective mass.

2. Vortex Loop Action in a Superfluid Film

2.1 Microscopic Origin of the Effective Mass

We define the effective vortex mass in a thin ⁴He film as:

where M_core is the core contribution, χ_M encodes geometry/boundary effects, a is the core radius, and L an infrared cutoff set by the device. The dynamic dressing δM arises from kelvon/phonon coupling and depends on frequency and velocity. Thus M_eff is a bona fide mass (SI units kg), not per length, for the 2D vortex.

Magnus Work Density

For a loop of radius R expanding at rate Ṙ in uniform flow v_s, the Magnus force on the loop is F_M = ρ_s κ ẑ × v_s. The power is P = ∫₀²π R dφ F_M · Ṙ = ρ_s κ v_s (2π R) Ṙ. Integrating from 0 to R yields the work W = ∫₀ᴿ ρ_s κ v_s (2π r) dr = π ρ_s κ v_s R², which identifies F ≡ ρ_s κ |v_s| as the area density of work.

3. Stationary Action and Tunneling Exponent

3.1 Stationary Action and Exponent (Corrected)

Extremizing S_E(R) = 2π R M_eff - π R² F gives R_★ = M_eff/F and:

so that:

At the saddle, N is constant and Ṅ = 0; thus β₂ Ṅ² contributes at fluctuation level (prefactor).

This is formally identical to the Schwinger exponent in QED, under the replacement:

4. UTMF Expansion of the Effective Mass

4.1 UTMF Primer and Parameter Provenance

As developed in our prior UTMF work, the mass operator on tuple sectors decomposes as:

where N ∈ ℤ indexes tuple sectors, w is a linking number from the boundary algebra, and T² encodes the Kosterlitz–Thouless tension scale, T² ~ (ρ_s κ²/4π) ln(L/a). Here Λ_c, λ_c arise from the sector spectral gap and fusion rules (instantonic growth), α_c from the linking form coupling, and κ_c from the KT logarithmic stiffness.

Within the UTMF, the effective mass becomes:

where:

• N is the tuple/phase index (mapping to vortex winding q_v)
• w is the linking number of vortex worldlines
• T² ~ (ρ_s κ²/4π) ln(L/a) encodes thickness/core via KT scaling
• β₁,₂ capture dynamical renormalizations from motion (velocity–dependent mass)

Substituting, the tunneling probability becomes:

(Note: In large-N regimes, M_0 may be subleading.)

4.2 Fluctuation Prefactor (Roadmap)

The one–loop prefactor A follows from the fluctuation determinant around the circular saddle. A Gel'fand–Yaglom evaluation in the radial mode with zero modes factored by collective coordinates transparently yields A ∝ F up to renormalization by dynamic dressing in M_eff. We defer its explicit closed form to a companion technical note; the present work focuses on the exponent.

5. Regime, Limitations, and Observables

We assume T ≪ T_λ, t ≪ ξ (2D limit), uniform v_s, and device size L ≫ a. The circular loop minimizes area at fixed perimeter; boundary anisotropies modify the prefactor.

A key observable is the curvature of ln Γ versus 1/|v_s|, directly probing the velocity dependence of M_eff. A critical drive F_c is defined by S_E★/ℏ ~ 1, yielding the scale estimate:

which connects the instanton threshold to critical-velocity scales in thin films. (For orientation: κ = h/m_4; ρ_s is the areal superfluid density in the 2D limit.)

6. Back–Map to QED

6.1 Back–Map to QED (Semantics)

Replacing (ρ_s κ v_s, M_eff) → (qE, m_eff) yields the Schwinger exponent with:

Here N labels vacuum sectors in an effective worldline sum; we do not claim a modified prefactor at leading order. The analogy is formal yet conservative, in the spirit of superfluid analogs and Schwinger's original non-perturbative framework.

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Key Results Summary

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