Positive definiteness for 4th order symmetric tensors and applications
- PDF / 352,136 Bytes
- 17 Pages / 439.37 x 666.142 pts Page_size
- 83 Downloads / 192 Views
Positive definiteness for 4th order symmetric tensors and applications Yisheng Song1 Received: 30 October 2019 / Revised: 2 November 2020 / Accepted: 23 November 2020 © The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature 2020
Abstract In particle physics, the vacuum stability of scalar potentials is to check positive definiteness (or copositivity) of its coupling tensors, and such a coupling tensor is a 4th order and symmetric tensor. In this paper, we mainly discuss precise expressions of positive definiteness of 4th order tensors. More specifically, two analytically sufficient conditions of positive definiteness for 4th order 2 dimensional symmetric tensors are given by reducing orders of tensors, and applying these conclusions, some sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors are derived. We also present several other sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors. Finally, we test and verify the vacuum stability of general scalar potentials of two real singlet scalar fields and the Higgs boson by using these results. Keywords Tensors · Positive definiteness · Homogeneous polynomial · Analytical expression Mathematics Subject Classification 70S20 · 15A72 · 15A63 · 90C20 · 90C30 · 81T60
1 Introduction Recently, Kannike [19–21] studied the vacuum stability of general scalar potentials of a few fields. The most general scalar potential of n real singlet scalar fields φi
This author’s work was supported by the National Natural Science Foundation of P.R. China (Grant Nos. 11571095, 11601134) and by the Foundation of Chongqing Normal university (20XLB009).
B 1
Yisheng Song [email protected] School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, People’s Republic of China 0123456789().: V,-vol
10
Page 2 of 17
Y. Song
(i = 1, 2, · · · , n) can be expressed as V (φ) =
n
λi jkl φi φ j φk φl = φ 4 ,
(1.1)
i, j,k,l=1
where = (λi jkl ) is coupling tensor and φ = (φ1 , φ2 , · · · , φn ) . So, the vacuum stability of such a system is equivalent to the positive definiteness of the tensor = (λi jkl ), that is, the positivity of the polynomial (1.1). The most general scalar quartic potential of the SM Higgs H1 , an inert doublet H2 and a complex singlet S is V = Ax 4 =
3
ai jkl xi x j xk xl ,
(1.2)
i, j,k,l=1
where x = (x1 , x2 , x3 ) with x1 = |H1 |, x2 = |H2 |, H2† H1 = x1 x2 ρeiφ , S = x3 eiφ S , A = (ai jkl ) is a 4th order 3 dimensional symmetric tensor. Clearly, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0. So, the vacuum stability of Z3 scalar dark matter V is equivalent to the (strict) copositivity of A. Also see Faro–Ivanov [12], Belanger– Kannike–Pukhov–Raidal [1,2], Ivanov–Köpke–Mühlleitner [18] for more details. So, the positive definiteness and copositivity are different properties of tensors. Recently, Song–Qi [45] and Liu–Song [26] have respectively gave the differently sufficient condition of copositivity for 4th order 3 dimensi
Data Loading...
