SU(n)对称性破裂：三阶与二阶反对称张量场的作用

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本文探讨了在物理学领域中的一个重要课题——SU(n)对称性的破裂，特别关注的是由三阶和二阶反对称张量场所引发的这一过程。SU(n)对称性是粒子物理学中的基本对称性之一，它在标准模型中扮演着核心角色。当这种对称性被特定类型的张量场破坏时，理论预测出新的物理现象和粒子态。首先，作者 Stephen L. Adler 在论文中运用了张量分析方法，这是在高维数学中处理协变和反对称张量的重要工具。他研究了 SU(n)群的两个关键代表：adjoint（对角）张量表示和反对称张量表示。在SU(n)中，反对称张量可以分为不同的类型，其行为受到群的特殊性质影响。通过分支规则，他揭示了这些张量在对称性破缺时如何分裂为不同组分，这有助于理解理论中的粒子谱和相互作用。值得注意的是，Adler观察到，在一般的SU(n)群下，与之前特殊情况下类似的现象出现：存在一个U(1)子群的生成器不匹配。这种不匹配通常与自发对称性破缺有关，意味着在某些特定条件下，系统的对称性可能不是完全对称的，而是部分保留了一些子群。接着，论文的核心内容转向了对反对称张量场的可重整化势参数的计算。可重整化势是量子场论中用来描述粒子相互作用强度的函数，其参数决定了这些场在对称性破缺后的能量分布和粒子质量。通过这些参数，Adler计算了分支扩张中不同标量场的具体质量值，这对于理解新物理粒子的性质和实验可探测性至关重要。最后，这篇开放获取的文章强调了版权和许可问题。它是由 Elsevier B.V. 发表，并遵循 CC BY 许可协议，这意味着读者可以在合法范围内自由分享和引用该研究。此外，文章是在 SCOAP3 计划的支持下发布的，这是一种旨在降低科研文献访问成本的全球合作项目。总结来说，本文是一篇深入研究SU(n)对称性破坏机制的学术论文，特别是在涉及反对称张量场的情况下。它提供了关于张量表示、分支规则、U(1)生成器的匹配以及反对称张量场标量质量计算的关键见解，对于粒子物理学家理解和探索高能物理的新边界具有重要意义。

Physics Letters B 744 (2015) 380–384

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

SU(n) symmetry breaking by rank three and rank two

antisymmetric

tensor scalars

Stephen L. Adler

Institute for Adv anced Study, Einstein Drive, Princeton, NJ 08540, USA

a r t i c l e i n f o a b s t r a c t

Article history:

Received

25 March 2015

Received

in revised form 7 April 2015

Accepted

9 April 2015

Available

online 14 April 2015

Editor:

M. Cveti

We study SU(n) symmetry breaking by rank three and rank two antisymmetric tensor ﬁelds. Using tensor

analysis, we derive branching rules for the adjoint and antisymmetric tensor representations, and explain

why for general SU(n) one ﬁnds the same U (1) generator mismatch that we noted earlier in special

cases. We then compute the masses of the various scalar ﬁelds in the branching expansion, in terms of

parameters of the general renormalizable potential for the antisymmetric tensor ﬁelds.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

The most familiar case of symmetry breaking for grand uniﬁed

theories, such as minimal SU(5) ⊃ SU(2) × SU(3) × U (1), utilizes

a scalar ﬁeld in the adjoint representation, with a gauge singlet

component with U (1) generator zero that receives a vacuum ex-

pectation.

The symmetry breaking mechanism is then straightfor-

ward:

since the gauge ﬁelds and the symmetry breaking scalar are

both in the adjoint representation, the same representations ap-

pear

in their branching expansions. As a consequence, the massless

gauge ﬁelds that pick up masses, and the scalars that supply their

longitudinal components, have the same group theoretic quantum

numbers.

recently noted [1,2] that when the symmetry breaking

scalar is in a totally antisymmetric representation, the situation is

more complicated. Using as explicit examples SU(8) broken by a

rank three antisymmetric tensor scalar, and SU(5) broken by a rank

two antisymmetric tensor scalar, we showed that there is a mis-

match

between the U (1) generator values of the massless gauge

ﬁelds that obtain masses, and the scalars that supply their longi-

tudinal

components. We noted that this mismatch is related to the

fact that the gauge singlet component of the antisymmetric ten-

sor

ﬁeld that receives a vacuum expectation has a nonzero U (1)

generator N, requiring a modular ground state that is periodic in

integer divisors p of N.

The

purpose of this paper is twofold. First, we show that the

mismatch found in [1,2] appears in the case of general SU(n), and

can be traced to the fact that invariant tensors lying in SU(3) or

SU(2) subgroups are available to lower subgroup indices. This anal-

E-mail address: adler@ias.edu.

ysis is given in Section 2, where we use tensor methods to com-

pute

the relevant branching expansions and U (1) generator values.

The second aim of this paper is to calculate the masses of the

various scalar ﬁeld components in the branching expansions, ob-

tained

by expanding the general renormalizable scalar ﬁeld poten-

tial

around the generic symmetry breaking minimum. This analysis

is given in Section 3, and a brief summary of our results follows in

Section 4.

Our notation is to deﬁne the upper index totally antisymmetric

tensor with R components to be a basis for the representation R,

and the corresponding lower index tensor to be a basis for the con-

jugate

representation R. Thus in SU(n) the tensor φ

, α = 1, ..., n,

is a basis for the fundamental representation n, and φ

is a ba-

sis

for the conjugate representation n. In SU(3), the tensor φ

a basis for the 3, and since the totally antisymmetric tensor 

αβγ

is invariant and can be used to lower indices, both the tensors φ

and φ

[αβ]

give a basis for the 3, and the tensor φ

[αβγ ]

∝ 

αβγ

a singlet. Similarly, in SU(2), since the invariant tensor 

αβ

can

be used to lower indices the representations 2 and 2are equiva-

lent,

and can be represented by either φ

or φ

, and the tensor

[αβ]

∝ 

αβ

is a singlet [3].

2. Branching rules for the SU(n) antisymmetric tensor and

adjoint representations

2.1. Branching under SU(n) ⊃ SU(3) × SU(n − 3) × U(1) for the rank

three antisymmetric tensor and adjoint representations

We assume that SU(n) is broken by the ground state expecta-

tion

of a single component φ

[123]

= a = 0, corresponding to the

simplest case considered by Cummins and King [4], which applies

http://dx.doi.org/10.1016/j.physletb.2015.04.016

0370-2693/

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