Extremal fixed points and Diophantine equations
Abstract The coupling constants of fixed points in the ϵ expansion at one loop are known to satisfy a quadratic bound due to Rychkov and Stergiou. We refer to fixed points that saturate this bound as extremal fixed points. The theories which contain such fixed points are those which undergo a saddle...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2024-09-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP09(2024)165 |
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| author | Christopher P. Herzog Christian B. Jepsen Hugh Osborn Yaron Oz |
| author_facet | Christopher P. Herzog Christian B. Jepsen Hugh Osborn Yaron Oz |
| author_sort | Christopher P. Herzog |
| collection | DOAJ |
| description | Abstract The coupling constants of fixed points in the ϵ expansion at one loop are known to satisfy a quadratic bound due to Rychkov and Stergiou. We refer to fixed points that saturate this bound as extremal fixed points. The theories which contain such fixed points are those which undergo a saddle-node bifurcation, entailing the presence of a marginal operator. Among bifundamental theories, a few examples of infinite families of such theories are known. A necessary condition for extremality is that the sizes of the factors of the symmetry group of a given theory satisfy a specific Diophantine equation, given in terms of what we call the extremality polynomial. In this work we study such Diophantine equations and employ a combination of rigorous and probabilistic estimates to argue that these infinite families constitute rare exceptions. The Pell equation, Falting’s theorem, Siegel’s theorem, and elliptic curves figure prominently in our analysis. In the cases we study here, more generic classes of multi-fundamental theories saturate the Rychkov-Stergiou bound only in sporadic cases or in limits where they degenerate into simpler known examples. |
| format | Article |
| id | doaj-art-2c680f2f70674293b792f5f760b7c813 |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-2c680f2f70674293b792f5f760b7c8132024-12-08T12:14:27ZengSpringerOpenJournal of High Energy Physics1029-84792024-09-012024915310.1007/JHEP09(2024)165Extremal fixed points and Diophantine equationsChristopher P. Herzog0Christian B. Jepsen1Hugh Osborn2Yaron Oz3Department of Mathematics, King’s College LondonSchool of Physics, Korea Institute for Advanced StudyDepartment of Applied Mathematics and Theoretical Physics, University of CambridgeSchool of Physics and Astronomy, Tel Aviv UniversityAbstract The coupling constants of fixed points in the ϵ expansion at one loop are known to satisfy a quadratic bound due to Rychkov and Stergiou. We refer to fixed points that saturate this bound as extremal fixed points. The theories which contain such fixed points are those which undergo a saddle-node bifurcation, entailing the presence of a marginal operator. Among bifundamental theories, a few examples of infinite families of such theories are known. A necessary condition for extremality is that the sizes of the factors of the symmetry group of a given theory satisfy a specific Diophantine equation, given in terms of what we call the extremality polynomial. In this work we study such Diophantine equations and employ a combination of rigorous and probabilistic estimates to argue that these infinite families constitute rare exceptions. The Pell equation, Falting’s theorem, Siegel’s theorem, and elliptic curves figure prominently in our analysis. In the cases we study here, more generic classes of multi-fundamental theories saturate the Rychkov-Stergiou bound only in sporadic cases or in limits where they degenerate into simpler known examples.https://doi.org/10.1007/JHEP09(2024)165Global SymmetriesRenormalization GroupScale and Conformal SymmetriesDifferential and Algebraic Geometry |
| spellingShingle | Christopher P. Herzog Christian B. Jepsen Hugh Osborn Yaron Oz Extremal fixed points and Diophantine equations Journal of High Energy Physics Global Symmetries Renormalization Group Scale and Conformal Symmetries Differential and Algebraic Geometry |
| title | Extremal fixed points and Diophantine equations |
| title_full | Extremal fixed points and Diophantine equations |
| title_fullStr | Extremal fixed points and Diophantine equations |
| title_full_unstemmed | Extremal fixed points and Diophantine equations |
| title_short | Extremal fixed points and Diophantine equations |
| title_sort | extremal fixed points and diophantine equations |
| topic | Global Symmetries Renormalization Group Scale and Conformal Symmetries Differential and Algebraic Geometry |
| url | https://doi.org/10.1007/JHEP09(2024)165 |
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