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arxiv: 2606.19548 · v1 · pith:ZICIV3POnew · submitted 2026-06-17 · ✦ hep-ph · hep-ex· hep-th

Real and Virtual Propagation in Neutrino Oscillations

Kenji Nishiwaki , Kin-ya Oda , Juntaro Wada This is my paper

Pith reviewed 2026-06-26 19:54 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-th
keywords neutrino oscillationswave packetvirtual propagationJacob-Sachs theoremflavor changing amplitudesaddle-point methodpropagation timedecay width
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The pith

Neutrino flavor oscillations appear only after the propagation time exceeds a threshold fixed by energy uncertainty and decay width.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper re-expresses vacuum flavor oscillations by tracking the actual time an intermediate particle spends in flight rather than distance alone. Standard results such as the Jacob-Sachs theorem describe oscillations only in the long-time limit. Extending the amplitude calculation with the saddle-point method yields a formula that remains valid at shorter times and reveals a sharp threshold: below it the intermediate state behaves as a purely virtual particle that cannot travel macroscopically, while above it real propagation and oscillations begin. The result applies equally to neutrinos and other unstable particles and implies the standard theorem remains accurate down to unexpectedly small distances.

Core claim

By evaluating the integrals in the wave-packet quantum field theory amplitude with the saddle-point method, an extended expression for the flavor-changing amplitude is obtained that is valid for arbitrary propagation times. Oscillations appear only when the propagation time exceeds a threshold fixed by the energy uncertainty of the external wave packets and the decay width of the propagating particle; for shorter times the intermediate state remains purely virtual and cannot reach macroscopic distances.

What carries the argument

The extended flavor-changing amplitude obtained from the saddle-point approximation to the wave-packet integrals, which encodes the switch between virtual and real propagation regimes.

If this is right

  • The Jacob-Sachs theorem holds to higher accuracy than previously expected even at short propagation times.
  • The same threshold condition governs real versus virtual propagation for any unstable particle, not only neutrinos.
  • Improvements in energy resolution of external wave packets could bring the virtual-to-real transition into experimental reach.
  • Oscillation probabilities at macroscopic distances remain unaffected by the short-time correction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Short-baseline experiments with better energy resolution might begin to see deviations from standard oscillation formulas at the smallest distances.
  • The same virtual-real distinction could appear in other processes that involve unstable intermediate states, such as certain resonance decays.
  • The threshold supplies a concrete scale separating the regime where quantum-field-theory propagation is indistinguishable from classical particle motion.

Load-bearing premise

The saddle-point method supplies an accurate approximation to the relevant integrals even for short propagation times.

What would settle it

Measure the oscillation probability at a propagation time deliberately chosen below the calculated threshold and verify whether the probability follows the non-oscillatory virtual-state form or already exhibits sinusoidal dependence on distance.

Figures

Figures reproduced from arXiv: 2606.19548 by Juntaro Wada, Kenji Nishiwaki, Kin-ya Oda.

Figure 1
Figure 1. Figure 1: FIG. 1. Production and detection processes of an electron [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Diagrammatic illustration for Jacob–Sachs theorem. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic integration contours for the energy integral of the propagation amplitude. Colored lines show the contours [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Feynman-diagrammatic interpretation of the amplitude evaluation using the saddle-point method. The first diagram [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

We revisit flavor oscillations in vacuum in terms of the propagation time of intermediate states. In the limit of a long propagation time (or distance), degenerate intermediate states exhibit oscillatory behavior, as described by the Jacob--Sachs (or Grimus--Stockinger) theorem within wave-packet quantum field theory. By explicitly evaluating the relevant integrals using the saddle-point method, we derive an extended expression for the flavor-changing amplitude that remains valid even for shorter propagation times. We show that oscillations occur only when the propagation time exceeds a threshold set by the energy uncertainty of the external wave packets and by the decay width of the propagating particle. For shorter propagation, the intermediate particle behaves as a purely virtual state, in the sense that it cannot propagate over a macroscopic distance. Although a direct experimental test of the transition from virtual to real propagation is challenging, since it typically occurs at microscopic scales, our result implies that the Jacob--Sachs theorem holds to higher accuracy than previously expected, even at short propagation times. Our formalism applies not only to neutrinos but also to other propagating particles, and future improvements in energy resolution may make this threshold observable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper revisits vacuum flavor oscillations in wave-packet quantum field theory by focusing on the propagation time of intermediate states. It applies the saddle-point method to evaluate the relevant integrals, extending the amplitude beyond the long-distance limit of the Jacob-Sachs theorem. The central claim is that oscillations appear only when the propagation time exceeds a threshold fixed by the external wave-packet energy uncertainty and the intermediate-particle decay width; below this threshold the intermediate state remains purely virtual and cannot propagate macroscopically.

Significance. If the saddle-point extension is rigorously controlled, the result would tighten the domain of validity of the Jacob-Sachs theorem and supply an explicit criterion separating virtual from real propagation. The formalism is stated to apply to neutrinos and other unstable or oscillating particles, and the authors note that improved energy resolution could render the threshold observable.

major comments (2)
  1. [Abstract and saddle-point section] Abstract and the saddle-point derivation (main text, around the evaluation of the propagation integrals): the claim that the extended amplitude remains valid for propagation times shorter than or comparable to 1/σ_E or 1/Γ rests on the saddle-point method being accurate in the regime where the stationary-phase contribution is no longer dominant. No error bound, asymptotic remainder estimate, or direct numerical comparison to the exact integral is supplied, which is load-bearing for the assertion that oscillations are absent below the threshold.
  2. [Abstract and concluding discussion] The statement that 'oscillations occur only when the propagation time exceeds a threshold' (abstract and concluding discussion) is derived from the saddle-point result; without a controlled demonstration that endpoint or non-stationary contributions remain negligible at short times, the transition from virtual to real behavior is not yet established.
minor comments (2)
  1. The abstract refers to 'explicitly evaluating the relevant integrals' but the manuscript should display the explicit integral expressions and the resulting closed-form extended amplitude so that the threshold condition can be traced to concrete parameters.
  2. Notation for the energy uncertainty σ_E and decay width Γ should be introduced once and used consistently when defining the threshold.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the detailed comments. The concerns focus on the justification of the saddle-point method in the short-time regime. We address each point below and commit to revisions that strengthen the presentation without altering the central results.

read point-by-point responses

  1. Referee: [Abstract and saddle-point section] Abstract and the saddle-point derivation (main text, around the evaluation of the propagation integrals): the claim that the extended amplitude remains valid for propagation times shorter than or comparable to 1/σ_E or 1/Γ rests on the saddle-point method being accurate in the regime where the stationary-phase contribution is no longer dominant. No error bound, asymptotic remainder estimate, or direct numerical comparison to the exact integral is supplied, which is load-bearing for the assertion that oscillations are absent below the threshold.

    Authors: We agree that an explicit remainder estimate would make the argument more rigorous. In the revised manuscript we will add a standard asymptotic error bound for the saddle-point contribution, derived from the analytic continuation of the integrand and the separation between the saddle and the integration contour. We will also include a direct numerical comparison between the saddle-point result and the exact integral for a representative Gaussian wave-packet model, confirming that non-stationary terms remain negligible below the stated threshold. revision: yes

  2. Referee: [Abstract and concluding discussion] The statement that 'oscillations occur only when the propagation time exceeds a threshold' (abstract and concluding discussion) is derived from the saddle-point result; without a controlled demonstration that endpoint or non-stationary contributions remain negligible at short times, the transition from virtual to real behavior is not yet established.

    Authors: The threshold arises because the oscillatory phase factor only acquires a stationary point once the propagation time exceeds the inverse width set by σ_E and Γ; below that scale the integral is dominated by the endpoint (virtual) contribution. The added error bound and numerical check in the revision will explicitly quantify that the oscillatory piece is suppressed by an exponentially small factor when the time is below threshold, thereby establishing the virtual-to-real transition on a controlled footing. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is explicit integral evaluation on external theorem

full rationale

The paper starts from the established Jacob-Sachs theorem for long propagation times and performs new saddle-point evaluations of the wave-packet integrals to obtain an extended amplitude valid at shorter times. No quoted step equates a derived quantity to a fitted parameter or to a self-citation chain; the threshold condition emerges directly from the stationary-phase analysis rather than being presupposed by definition or by renaming. The central claim therefore remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the applicability of wave-packet quantum field theory to neutrino propagation and the validity of the saddle-point approximation for evaluating the flavor-changing amplitude integrals.

axioms (2)
  • domain assumption Jacob-Sachs (Grimus-Stockinger) theorem within wave-packet quantum field theory
    The paper builds its long-time limit and extension upon this established result.
  • standard math Saddle-point method accurately approximates the relevant integrals at short propagation times
    Invoked to derive the extended amplitude expression.

pith-pipeline@v0.9.1-grok · 5731 in / 1315 out tokens · 23513 ms · 2026-06-26T19:54:01.267715+00:00 · methodology

discussion (0)

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Reference graph

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