Existence of a Mass Gap in Four-Dimensional SU(N) Yang-Mills Theory

Existence of a Mass Gap in Four-Dimensional SU(N) Yang-Mills Theory | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Existence of a Mass Gap in Four-Dimensional SU(N) Yang-Mills Theory Moustafa Amin Radwan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9601622/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract A rigorous framework for establishing the mass gap in pure Yang–Mills theory on R⁴ for every compact simple gauge group is presented. The four-dimensional gauge theory is embedded as the low-energy sector of eleven-dimensional supergravity compactified on a seven-dimensional manifold X₇ of G₂ holonomy possessing conical singularities of ADE type. The existence of the mass gap depends only on the existence of X 7 supporting the volume cancellation property and not on the specific Betti numbers of X 7; numerical predictions are derived under the additional choice X 7 = T 7 /(Z 3 ⋊ I ) with (b 2 , b 3) = (27, 451) and stabilized volume modulus K 0 = 90.0085. Gauge fields are identified with the reduction of the M-theory three-form C₃ on exceptional two-cycles localized at the resolved singularity, following the constructions of Acharya and of Atiyah and Witten. The central new result is a volume cancellation theorem: the product of the Kaluza–Klein spectral density and the harmonic two-form norm at the singularity is independent of the compactification volume V₇, using the hyper-Kähler ALE metric of Kronheimer. Three analytically proven properties of the one-loop scalar vertex function—positivity, continuity, and power-law decoupling—combined with the infrared divergence at the vanishing mass parameter yield, via the intermediate value theorem, the existence of a strictly positive self-consistent gluon mass Σ * > 0 in the rainbow-resummed Schwinger–Dyson equation. Through Källén–Lehmann analysis and Osterwalder–Schrader reconstruction, the unconditional mass gap bound ∆ ≥ Λ QCD > 0 is established from exponential clustering of the continuum two-point function; in the rainbow truncation, the gap saturates to √ Σ . Reflection positivity is proven from the positive definiteness of the transfer matrix, exploiting the non-negativity of both the Yang–Mills Hamiltonian and the massive Kaluza–Klein Hamiltonian guaranteed by the G 2 holonomy structure. Tightness of the regulated family is derived from the volume-cancellation-induced uniformity of the correlation length, establishing the continuum limit via Prohorov's compactness. The decompactification limit K → ∞ recovers a pure Yang–Mills quantum field theory on R⁴ satisfying the full Osterwalder–Schrader axioms, with mass gap preserved at exponential rate e⁻²K/7 via Appelquist–Carazzone decoupling. The extension to all compact simple Lie groups follows from the McKay correspondence (simply-laced cases) combined with folding under outer automorphisms (non-simply-laced cases). The resulting continuum theory possesses a mass gap ∆G ≥ Λ QCD (G) > 0 and Wilson area law (confinement) for every compact simple gauge group G. All assumptions are made explicit in Section 11. Full Text Additional Declarations No competing interests reported. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9601622","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":634015898,"identity":"e4600318-2005-44b0-9220-c069a439757c","order_by":0,"name":"Moustafa Amin Radwan","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBUlEQVRIiWNgGAWjYPACCQZ+BgYDJC4xWiQbSNQCtOIAsVp029sfMPzMscgzPt68TYLhz2E5+Qbmg7d5GGzsGnBoMTtzxoCxd5tEsdmZY2USjG2HjRkb2JKteRjSknFquZHDwMC7TSJx240cMwnGhsOJzQw8ZtI8DIeTcTnM7Eb6A8a/QC2b578xAzmsvo2B/xtQy388WhIMmEG2bJDgAWphO5zAw8DDBtRywA6nFqBfmGWBWmacSSu2SGxLN5zBzGZsOccgOQGnluPtDxjfbqtL7G8/vPHGhz/W8vLtzQ9vvKmws8elBQjYf8CZYJOZQYQBQ2IDHj3YAT5bRsEoGAWjYGQBAFZlT64eaE9pAAAAAElFTkSuQmCC","orcid":"","institution":"Suez Canal University","correspondingAuthor":true,"prefix":"","firstName":"Moustafa","middleName":"Amin","lastName":"Radwan","suffix":""}],"badges":[],"createdAt":"2026-05-03 17:09:51","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9601622/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9601622/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109087216,"identity":"c3066527-a9cd-4a78-8975-66bee432228a","added_by":"auto","created_at":"2026-05-12 13:13:59","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":560897,"visible":true,"origin":"","legend":"","description":"","filename":"ExistenceofaMassGap.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9601622/v1_covered_c8f587dd-90aa-4a4c-91c4-b3d0690e9871.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eExistence of a Mass Gap in Four-Dimensional SU(N) Yang-Mills Theory\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-9601622/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9601622/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"A rigorous framework for establishing the mass gap in pure Yang–Mills theory on R⁴ for every compact simple gauge group is presented. The four-dimensional gauge theory is embedded as the low-energy sector of eleven-dimensional supergravity compactified on a seven-dimensional manifold X₇ of G₂ holonomy possessing conical singularities of ADE type. The existence of the mass gap depends only on the existence of X 7 supporting the volume cancellation property and not on the specific Betti numbers of X 7; numerical predictions are derived under the additional choice X 7 = T 7 /(Z 3 ⋊ I *) with (b 2 , b 3) = (27, 451) and stabilized volume modulus K 0 = 90.0085. Gauge fields are identified with the reduction of the M-theory three-form C₃ on exceptional two-cycles localized at the resolved singularity, following the constructions of Acharya and of Atiyah and Witten. The central new result is a volume cancellation theorem: the product of the Kaluza–Klein spectral density and the harmonic two-form norm at the singularity is independent of the compactification volume V₇, using the hyper-Kähler ALE metric of Kronheimer. Three analytically proven properties of the one-loop scalar vertex function—positivity, continuity, and power-law decoupling—combined with the infrared divergence at the vanishing mass parameter yield, via the intermediate value theorem, the existence of a strictly positive self-consistent gluon mass Σ * \u003e 0 in the rainbow-resummed Schwinger–Dyson equation. Through Källén–Lehmann analysis and Osterwalder–Schrader reconstruction, the unconditional mass gap bound ∆ ≥ Λ QCD \u003e 0 is established from exponential clustering of the continuum two-point function; in the rainbow truncation, the gap saturates to √ Σ*. Reflection positivity is proven from the positive definiteness of the transfer matrix, exploiting the non-negativity of both the Yang–Mills Hamiltonian and the massive Kaluza–Klein Hamiltonian guaranteed by the G 2 holonomy structure. Tightness of the regulated family is derived from the volume-cancellation-induced uniformity of the correlation length, establishing the continuum limit via Prohorov's compactness. The decompactification limit K → ∞ recovers a pure Yang–Mills quantum field theory on R⁴ satisfying the full Osterwalder–Schrader axioms, with mass gap preserved at exponential rate e⁻²K/7 via Appelquist–Carazzone decoupling. The extension to all compact simple Lie groups follows from the McKay correspondence (simply-laced cases) combined with folding under outer automorphisms (non-simply-laced cases). The resulting continuum theory possesses a mass gap ∆G ≥ Λ QCD (G) \u003e 0 and Wilson area law (confinement) for every compact simple gauge group G. 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