算法化方法：普适多项式标量势的BFB条件求解

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本文主要探讨了通用标量势在具有扩展标量扇区的物理模型中的重要性，特别是在检查标量势是否满足下界（BFB）条件方面所面临的挑战。通常，这种条件对于保证模型的稳定性和物理意义至关重要，但在复杂情况下往往缺乏精确的解析表达式。以往的BFB条件只在特定的简单模型中有所明确。在当前的研究中，作者们提出了一个创新的算法方法，旨在系统地确定任意多项式标量势的BFB条件。这个新颖的策略结合了多元代数的原理，特别是利用了数学领域中关于结果子和张量谱理论的最新进展。结果子是代数几何中的一个重要概念，它涉及到多个变量的一次多项式的交点，这对于理解多变量函数的行为非常关键。另一方面，张量谱理论研究的是张量的线性算子性质，包括其特征值和特征向量，这些性质在分析潜在物理模型的稳定性方面发挥着重要作用。论文《欧洲物理杂志C》(Eur.Phys.J.C, 2018)第78卷第413页中，作者Igor P. Ivanov、Marcel Köpke和Margarete Mühlleitner合作，介绍了一个实用的教程性介绍，帮助读者理解和应用这一新方法。他们强调了算法化处理BFB条件对于理论物理学中的广泛模型来说是一个突破，因为它不仅简化了复杂的计算过程，还为解决这类普遍存在的难题提供了强大的工具。这项工作的重要性在于提供了一种普遍适用的方法，使得研究人员能够对复杂标量势进行有效的BFB条件分析，从而推动了模型的建立和验证，有助于物理学家更好地理解和探索高能物理中的基本原理。通过将数学工具与物理问题结合，作者们为解决实际问题开辟了新的途径，对于进一步发展理论物理和数值模拟具有深远影响。

Eur. Phys. J. C (2018) 78:413 Page 3 of 15 413

presented in the form of an algebraic algorithm [4] and later

written in compact closed form [13] as inequalities imposed

not on the parameters 

themselves but on four eigenval-

ues



of a real symmetric 4 × 4matrix

which encodes

all quartic interaction terms. The form of these conditions is

very simple and basis-invariant,



> 0,



1,2,3

, (7)

but checking them within a speciﬁc 2HDM requires ﬁrst ﬁnd-

ing these eigenvalues, though this step can be easily imple-

mented in numerical scans of the parameter space.

A somewhat similar systematic method of deriving the

exact BFB conditions exists for models, in which the Higgs

potential can be written in terms of independent positive-

deﬁnite ﬁeld bilinears r

. In this case, the quartic potential can

again be rewritten as a quadratic form V = 

, but its

positive deﬁniteness must be insured only in the ﬁrst orthant

≥ 0. These conditions are known as copositivity (condi-

tional positivity) criteria. They were developed in [14–16]

and applied to such cases as some 2HDMs, singlet-doublet

models, models with Z

symmetric scalar dark matter, and

left-right symmetric models.

Beyond two Higgs doublets, in the general N -Higgs-

doublet model (NHDM), the exact BFB conditions in closed

form are still not known. Several attempts to attack the prob-

lem with the bilinear space formalism [4,17–20] did not cul-

minate in a closed set of inequalities. The technical chal-

lenge is that, with N Higgs doublets, the space of bilinears

, a = 1,...,N

−1, does not span the entire R

−1

space

but only a lower-dimensional algebraic manifold, which is

described with a series of polynomial constraints. Positive-

deﬁniteness of a quadratic form on a complicated algebraic

manifold cannot be decided with linear algebra and requires

algebraic-geometric tools, that have not been found yet.

For larger gauge symmetries and for scalars in higher-

dimensional representations, it is appropriate to analyze the

scalar potential not in the scalar ﬁelds space but in the space

of gauge orbits. This approach ﬂourished in 1980’s with the

advent of Grand Uniﬁcation models, see, for example [21–

24], and a short historical overview in [16].

In speciﬁc multi-Higgs models, in which large continu-

ous or discrete symmetry groups dramatically simplify the

potential, the exact conditions can be established [25–30].

We mention, in particular, the method developed in [28,30]

to rewrite the Higgs potential as a linear combination of new

variables, the group-invariant quartic ﬁeld combinations, and

to determine the exact shape of the space spanned by these

variables. This method is similar to the so-called linear pro-

graming, and it gives the BFB constraints directly from the

description of the shape of the space available.

In certain cases, when the exact necessary and sufﬁcient

conditions are not known but a parameter scan still needs

to be performed, it may be enough to write down a set of

sufﬁcient conditions. They may be overly restrictive, but if a

point satisﬁes them, the potential is guaranteed to be positive-

deﬁnite. An example of such conditions was given for a spe-

ciﬁc 3HDM in [31]. The idea is to pick up all terms with

“angular” dependence in the scalar ﬁeld space and ﬁnd a

lower bound for each term separately. For example, if the

potential contains a term (φ

†

)(φ

†

) with real coefﬁcient

, one can place the following lower boundary on it in the

≡|φ

space:

(φ

†

)(φ

†

) ≥−||r

√

≥−||r

)/2. (8)

In this way, the original potential V can be limited from below

by another potential

V , which is a quadratic form in terms of

and for which the copositivity criteria are applicable.

In this work we present an algorithm, which in principle

solves the problem in a generic setting. The algorithm uses

elements of the theory of resultants and of the recently devel-

oped spectral theory of tensors. However, solving the prob-

lem in principle is quite different from solving it in practice.

To our best knowledge, the approach was only brieﬂy men-

tioned in [16] but was not developed any further nor imple-

mented in any code. We have implemented the method in a

computer-algebra code, which is available at GitHub [32],

and tested it in cases, in which analytical solutions already

exist. The complexity of the algorithm implementation grows

so fast that, with limited computer resources, we could not

apply it to cases where the results are not yet known.

This does not imply, of course, that this direction is a

dead-end. The method itself is innovative but the speciﬁc

algorithm we propose is clearly not optimal. We believe that

with additional efforts, it can be seriously improved and may

eventually produce a ready-to-use solution in various popular

classes of multi-scalar models, such as the general 3HDM.

The structure of the paper is the following. In the next

Section, we present our strategy and formulate the algorithm.

Section 3 contains an introduction to the spectral theory of

tensors, its application to the BFB problem, and describes a

practical algorithm to calculate the characteristic polynomial

of a symmetric tensor. In Sect. 4, we show how this method

works. We ﬁrst do it with two elementary examples, in which

all calculations can be performed manually, and then apply

the computer-algebra package to the case of a Z

-symmetric

2HDM, where the BFB conditions are known. We ﬁnd agree-

ment of the results, which serves as a check of the validity

of our algorithm. We end with a discussion in Sect. 5 of how

the algorithm can be improved in the future and draw con-

clusions. The Appendix contains a pedagogical introduction

to polynomial rings and polynomial division with an appli-

cation to the theory of resultants.

123

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算法化方法：普适多项式标量势的BFB条件求解

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