Real and Virtual Propagation in Neutrino Oscillations

[2606.19548] Real and Virtual Propagation in Neutrino Oscillations Skip to main content Search Submit Donate Log in High Energy Physics - Phenomenology arXiv:2606.19548 (hep-ph) [Submitted on 17 Jun 2026 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: Real and Virtual Propagation in Neutrino Oscillations Authors: Kenji Nishiwaki , Kin-ya Oda , Juntaro Wada View PDF HTML (experimental) 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. Comments: 30 pages, 5 figures. v2: Discussion clarified and expanded; several equations and typos corrected; references added; v3: Discussion of the crossover behavior added, including a new figure; minor corrections Subjects: High Energy Physics - Phenomenology (hep-ph) ; High Energy Physics - Experiment (hep-ex); High Energy Physics - Theory (hep-th) Report number: TU-1309 Cite as: arXiv:2606.19548 [hep-ph] (or arXiv:2606.19548v3 [hep-ph] for this version) https://doi.org/10.48550/arXiv.2606.19548 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Juntaro Wada [ view email ] [v1] Wed, 17 Jun 2026 19:39:44 UTC (135 KB) [v2] Tue, 28 Jul 2026 08:16:09 UTC (154 KB) [v3] Thu, 27 Aug 2026 07:22:49 UTC (162 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: hep-ph < prev | next > new | recent | 2026-06 Change to browse by: hep-ex hep-th References & Citations INSPIRE HEP SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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