离散群CP违反的三种类型与克莱布斯-哥丹系数

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本文主要探讨了在具有离散（风味）对称群G的物理系统中的规范对称性CP违反（Charge-Parity violation, CPV）现象。作者首先强调了物理上的CP变换必须是G的类反转自同构，这是理解CP违反来源的关键。根据这个性质，有限群被划分为三个基本类别： 1. **不具有类反转自同构的群**：这类群在常规物理设置中会导致显式CP违反，因为它们不具备物理上实现CP不变所需的内在结构。CP违反在这种情况下可能是由群论特性驱动的，例如Clebsch-Gordan系数的复杂性，这些系数在量子力学的耦合过程中起到重要作用。 2. **存在实数克莱布斯-哥丹系数的群**：对于这类群，CP不变性会限制耦合常数的相位，这意味着如果选择一个CP不变的基底，所有的Clebsch-Gordan系数将是实数，从而避免了明显的CP违反。这表明在这样的群框架下，对CP不变性的实施可以对模型参数产生约束。 3. **具有类反转自同构但非实数克莱布斯-哥丹系数的群**：虽然这些群不能直接支持真实的Clebsch-Gordan系数，但通过引入广义的CP转换，它们仍然可以用于构建物理模型。在这种情况下，对CP不变性的应用可能会导致某些耦合的禁止，从而间接地影响模型的对称性。文章利用Frobenius-Schur扭曲指标来进一步区分这些离散群，这是一种数学工具，有助于理解群论在物理中的具体作用。论文引用了《Nuclear Physics B》的一篇文章，发表在2014年883期，267-305页，可在线访问ScienceDirect和Elsevier的网站获取。作者Mu-Chun Chen等人来自美国加州大学欧文分校、德国慕尼黑工业大学和科罗拉多大学波尔德分校，他们的研究工作展示了离散对称群在解释CP违反现象中的潜在丰富性和多样性。编辑为Tommy Ohlsson，该成果于2014年3月接收并修订，最终在同年3月31日在线发布。

M.-C. Chen et al. / Nuclear Physics B 883 (2014) 267–305 273

First we start with the consistency condition equation (2.16) for a class-inverting u.Byre-

placing g by u(g) (and u(g) by u

(g))inEq.(2.16) and bringing the U

’s to the other side, we

obtain



u(g)





(g)



∗

(g)

∗

∀g ∈G and ∀i. (2.21)

This shows that with U

also the transpose U

fulﬁlls equation (2.16). Since ρ

is an irreducible

representation, Schur’s Lemma implies that

iα

∀i, (2.22)

which is only possible if each U

is either symmetric or anti-symmetric, i.e. α is either 0 or π .

Thus, V

∗

=±1. Hence, V consists of blocks identical to ±1.

Now assume that all V

are ±1. Then, by inserting Eq. (2.16) into itself,



(g)





∗



(g)



∗



†

=ρ

(g) ∀g ∈G and ∀i. (2.23)

Since this equation is true for all irreducible representations, it follows that u

(g) = g for all g

in G and the order of u is thus either one or two (i.e. u is involuntary). This completes the proof

that V is different from a diagonal matrix with only ±1 on the diagonal if and only if u is of

order n>2.

We therefore conclude that, if an involutory u is

imposed as a symmetry, G may be amended

by an additional Z

symmetry. This is possible if and only if V

= +1 for some r

. We will

discuss this case in more detail in Section 2.7. In what follows, we refer to such an enlargement

as “trivial” extension of G to G × Z

. Note that the assignment of Z

charges to the ﬁelds of

a model is not arbitrary but is given by the signs of the V

for their respective representations

under G. We will also discuss this Z

factor in an example in Section 3.3.

The second logical possibility, case (ii), is that v is

an inner automorphism.

In this case,

the order of u is larger than two but one can still show that the ﬂavor group only gets enlarged

by some Abelian factor. However, CP transformations connected to automorphisms that square

to an inner automorphism do not seem to yield any CP transformations which are physically

different from those that are connected to involutory automorphisms. The reason is that if two

automorphisms u and u



are related by an inner automorphism,

u(g) = bu



(g)b

−1

∀g ∈G and some b ∈ G, (2.24)

the resulting CP transformations only differ by a transformation with the group element ρ(b).

Since the latter transformation certainly is a symmetry of the Lagrangean, the two CP transfor-

mations are indistinguishable. In fact, it turns out that we were not able to ﬁnd an example where

there is a class-inverting automorphism of higher order that is not related to an involutory class-

inverting automorphism in the prescribed way. We were able to prove that such an automorphism

cannot exist for some cases, see Appendix C, and have explicitly checked this for all non-Abelian

groups of order less than 150 (with the exception of some groups of order 128) with the group

theory program GAP [15].

The last logical possibility, case (iii), is that u

=v is an outer automorphism. Then the addi-

tional generator h with ρ

(h) =V

does not commute with all group elements of G, and, hence,

The property that u should square to the identity or an inner automorphism has also been stressed in [14].However,

the discussion there misses the point that this does not imply U

∗

= ρ

(g) for some g in G but that the group still

might be extended by a Z

factor.

274 M.-C. Chen et al. / Nuclear Physics B 883 (2014) 267–305

genuinely enlarges the original ﬂavor symmetry group non-trivially to the larger semi-direct

product group H = G 

, where Z

is the cyclic group generated by h. As a consequence,

terms which are allowed by G but prohibited by H are absent from a Lagrangean if CP conserva-

tion is imposed. The representation content, however, still coincides with the one of G. Although

the structure of the extended group H is more complicated than the direct product in case (i), the

physical implications are similar to the case of the trivial Z

extension.

Even though we have no general argument for their absence, we were not able to ﬁnd an

xample in which case (iii) is realized. In more detail, a GAP scan for class-preserving outer

automorphisms that are the square of a class-inverting inner automorphism did not yield any

result for groups up to order 150 (some groups of order 128 were not checked). Case (iii), hence,

seems to be very rare.

In summary, we ﬁnd that u should

be a class-inverting automorphism of G in order for the re-

lated CP transformation to be physical. Moreover, for practical purposes, one can usually restrict

the discussion to involutory automorphisms.

2.4. The Bickerstaff–Damhus automorphism (BDA)

As shown by Bickerstaff and Damhus [16], the existence of a basis in which all CG coefﬁ-

cients are real can be related to the existence of an automorphism u which fulﬁlls



u(g)



(g)

∗

†

unitary and symmetric, ∀g ∈G and ∀i (2.25)

for some U

with the given properties. From our discussion in Section 2.3 we know that such a u

is involutory and class-inverting. In what follows, we will refer to an automorphism u which satis-

ﬁes Eq. (2.25) as Bickerstaff–Damhus automorphism (BDA). In short, a BDA is a class-inverting

involutory automorphism that fulﬁlls the consistency condition (2.16) with some symmetric uni-

tary matrices U

The important property of the BDA is that its existence is equivalent to the existence of a basis

of G in

which all CG’s are real,

∃ BDA u fulﬁlling (2.25) ⇐⇒

⎧

⎪

⎨

⎪

⎩

existence of a

basis in which

all CG coefﬁcients

are real

⎫

⎪

⎬

⎪

⎭

. (2.26)

The basis in which the CGs can be chosen real is exactly the basis for which all U

in Eq. (2.25)

are unit matrices, i.e. for which



u(g)



=ρ

(g)

∗

∀g ∈G and ∀i. (2.27)

More precisely, this deﬁnes a whole set of bases which are related by real orthogonal basis

transformations.

An automorphism u that

fulﬁlls this equation in a certain basis is unique. However, there

can be several different BDAs which fulﬁll Eq. (2.27) for different bases. The different BDAs

also do not have to be related by inner automorphisms, see for example the group SG(32, 43) of

the SmallGroups library which is part of GAP. Note that, as shown in Appendix C.2, odd order

non-Abelian groups do not admit a BDA and, hence, do not have a basis with completely real

Clebsch–Gordan coefﬁcients.

How can one tell whether or not a given automorphism u is

a BDA? In what follows, we will

discuss a tool which allows us to answer this question.

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离散群CP违反的三种类型与克莱布斯-哥丹系数

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