Algorithmic boundedness-from-below conditions for generic scalar potentials
Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar sectors. Exact analytic BFB conditions are known only in simple cases. In this work, we present a novel approach to algorithmically establish the BFB condi...
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| Published in | The European physical journal. C, Particles and fields Vol. 78; no. 5; pp. 1 - 15 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.05.2018
Springer Springer Nature B.V SpringerOpen |
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| Online Access | Get full text |
| ISSN | 1434-6044 1434-6052 1434-6052 |
| DOI | 10.1140/epjc/s10052-018-5893-y |
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| Abstract | Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar sectors. Exact analytic BFB conditions are known only in simple cases. In this work, we present a novel approach to algorithmically establish the BFB conditions for any polynomial scalar potential. The method relies on elements of multivariate algebra, in particular, on resultants and on the spectral theory of tensors, which is being developed by the mathematical community. We give first a pedagogical introduction to this approach, illustrate it with elementary examples, and then present the working Mathematica implementation publicly available at GitHub. Due to the rapidly increasing complexity of the problem, we have not yet produced ready-to-use analytical BFB conditions for new multi-scalar cases. But we are confident that the present implementation can be dramatically improved and may eventually lead to such results. |
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| AbstractList | Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar sectors. Exact analytic BFB conditions are known only in simple cases. In this work, we present a novel approach to algorithmically establish the BFB conditions for any polynomial scalar potential. The method relies on elements of multivariate algebra, in particular, on resultants and on the spectral theory of tensors, which is being developed by the mathematical community. We give first a pedagogical introduction to this approach, illustrate it with elementary examples, and then present the working Mathematica implementation publicly available at GitHub. Due to the rapidly increasing complexity of the problem, we have not yet produced ready-to-use analytical BFB conditions for new multi-scalar cases. But we are confident that the present implementation can be dramatically improved and may eventually lead to such results. Abstract Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar sectors. Exact analytic BFB conditions are known only in simple cases. In this work, we present a novel approach to algorithmically establish the BFB conditions for any polynomial scalar potential. The method relies on elements of multivariate algebra, in particular, on resultants and on the spectral theory of tensors, which is being developed by the mathematical community. We give first a pedagogical introduction to this approach, illustrate it with elementary examples, and then present the working Mathematica implementation publicly available at GitHub. Due to the rapidly increasing complexity of the problem, we have not yet produced ready-to-use analytical BFB conditions for new multi-scalar cases. But we are confident that the present implementation can be dramatically improved and may eventually lead to such results. |
| ArticleNumber | 413 |
| Audience | Academic |
| Author | Ivanov, Igor P. Köpke, Marcel Mühlleitner, Margarete |
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| Cites_doi | 10.1103/PhysRevD.76.039902 10.1103/PhysRevD.76.119901 10.1007/BF01034825 10.1016/j.jsc.2005.05.007 10.1016/j.physletb.2012.08.020 10.1007/JHEP07(2010)020 10.1016/0550-3213(82)90040-2 10.1007/JHEP02(2013)125 10.1007/JHEP10(2015)149 10.1007/JHEP03(2018)061 10.1007/JHEP02(2017)121 10.1016/0370-2693(84)92024-0 10.1006/jsco.1993.1009 10.1016/j.cpc.2018.11.006 10.1103/PhysRevD.18.2574 10.1112/plms/s1-35.1.3 10.1016/j.physletb.2005.10.015 10.1140/epjc/s10052-012-2093-z 10.1093/comjnl/36.5.485 10.1016/0370-2693(81)90578-5 10.1112/plms/s2-21.1.14 10.1103/PhysRevD.76.055013 10.1007/978-1-4757-6911-1 10.1016/0550-3213(82)90160-2 10.1007/978-3-642-15155-2_42 10.1103/PhysRevD.89.095008 10.1103/PhysRevD.92.075017 10.1145/780506.780516 10.1016/0003-4916(83)90017-9 10.1007/JHEP01(2018)065 10.1137/1.9781611974751 10.1007/JHEP10(2017)014 10.1007/JHEP03(2011)035 10.1007/JHEP03(2018)014 10.1016/S0022-4049(99)00005-5 10.1103/PhysRevD.92.035017 10.1140/epjc/s10052-016-4160-3 10.1140/epjc/s10052-006-0016-6 10.1016/j.ppnp.2017.03.001 10.1016/j.physletb.2012.08.021 |
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| Snippet | Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar sectors. Exact... Abstract Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar... |
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| SubjectTerms | Algebra Algorithms Analysis Astronomy Astrophysics and Cosmology Elementary Particles Hadrons Heavy Ions Mathematical analysis Mathematical models Measurement Science and Instrumentation Nuclear Energy Nuclear Physics Physics Physics and Astronomy Quantum Field Theories Quantum Field Theory Regular Article - Theoretical Physics Resultants Spectral theory String Theory Tensors |
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| Title | Algorithmic boundedness-from-below conditions for generic scalar potentials |
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