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Teoreticheskaya i Matematicheskaya Fizika, 2024, Volume 220, Number 3, Pages 591–604
DOI: https://doi.org/10.4213/tmf10645 (Mi tmf10645) 		 
Boundedness-below conditions for a general scalar potential of two real scalar fields and the Higgs boson

Yisheng Songa, Liqun Qibc

a School of Mathematical Sciences, Chongqing Normal University, Chongqing, China
b Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, China
c Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
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DOI: https://doi.org/10.4213/tmf10645
Abstract: The most general scalar potential of two real scalar fields and a Higgs boson is a quartic homogeneous polynomial in three variables, which defines a $4$th-order three-dimensional symmetric tensor. Hence, the boundedness of such a scalar potential from below involves the positive (semi-)definiteness of the corresponding tensor. In this paper, we therefore mainly discuss analytic expressions of positive (semi-)definiteness for such a special tensor. First, an analytically necessary and sufficient condition is given to test the positive (semi-)definiteness of a $4$th-order two-dimensional symmetric tensor. Furthermore, by means of such a result, the analytic necessary and sufficient conditions of the boundedness from below are obtained for a general scalar potential of two real scalar fields and the Higgs boson.
Keywords: scalar potentials, boundedness from below, 4th-order tensors, positive definiteness, homogeneous polynomial, analytic expression.
Funding agency Grant number
National Natural Science Foundation of China 12171064
Team Project of Innovation Leading Talent in Chongqing CQYC20210309536
Foundation of Chongqing Normal University 20XLB009
This work was supported by the National Natural Science Foundation of China (grant No. 12171064), Team Project of Innovation Leading Talent in Chongqing (grant No. CQYC20210309536), and Foundation of Chongqing Normal University (grant No. 20XLB009).
Received: 16.11.2023
Revised: 18.12.2023
English version:
Theoretical and Mathematical Physics, 2024, Volume 220, Issue 3, Pages 1567–1579
DOI: https://doi.org/10.1134/S0040577924090101
Document Type: Article
Language: Russian
Citation: Yisheng Song, Liqun Qi, “Boundedness-below conditions for a general scalar potential of two real scalar fields and the Higgs boson”, TMF, 220:3 (2024), 591–604; Theoret. and Math. Phys., 220:3 (2024), 1567–1579
Citation in format AMSBIB
\Bibitem{SonQi24}
\by Yisheng~Song, Liqun~Qi
\paper Boundedness-below conditions for a~general scalar potential of two real scalar fields and the~Higgs boson
\jour TMF
\yr 2024
\vol 220
\issue 3
\pages 591--604
\mathnet{http://mi.mathnet.ru/tmf10645}
\crossref{https://doi.org/10.4213/tmf10645}
\transl
\jour Theoret. and Math. Phys.
\yr 2024
\vol 220
\issue 3
\pages 1567--1579
\crossref{https://doi.org/10.1134/S0040577924090101}
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    		Теоретическая и математическая физика Theoretical and Mathematical Physics
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