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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 883 (2014) 267–305

www.elsevier.com/locate/nuclphysb

CP violation from ﬁnite groups

Mu-Chun Chen

, Maximilian Fallbacher

, K.T. Mahanthappa

Michael Ratz

b,∗

, Andreas Trautner

b,d

Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA

Physik Department T30, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany

Department of Physics, University of Colorado, Boulder, CO 80309, USA

Excellence Cluster Universe, Boltzmannstraße 2, 85748 Garching, Germany

Received 6 February 2014; received in revised form 17 March 2014; accepted 25 March 2014

Available online 31 March 2014

Editor: Tommy Ohlsson

Abstract

We discuss the origin of CP violation in settings with a discrete (ﬂavor) symmetry G.

We show that phys-

ical CP transformations always have to be class-inverting automorphisms of G. This allows us to categorize

ﬁnite groups into three types: (i) Groups that do not exhibit such an automorphism and, therefore, in generic

settings, explicitly violate CP. In settings based on such groups, CP violation can have pure group-theoretic

origin and can be related to the complexity of some Clebsch–Gordan coefﬁcients. (ii) Groups for which one

can ﬁnd a CP basis in which all the Clebsch–Gordan coefﬁcients are real. For such groups, imposing CP

invariance restricts the phases of coupling coefﬁcients. (iii) Groups that do not admit real Clebsch–Gordan

coefﬁcients but possess a class-inverting automorphism that can be used to deﬁne a proper (generalized) CP

transformation. For such groups, imposing CP invariance can lead to an additional symmetry that forbids

certain couplings. We make use of the so-called twisted Frobenius–Schur indicator to distinguish between

the three types of discrete groups. With (27),T



,andΣ(72) we present one explicit example for each

type of group, thereby illustrating the CP properties of models based on them. We also show that certain

operations that have been dubbed generalized CP transformations in the recent literature do not lead to

physical CP conservation.

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

Corresponding author.

http://dx.doi.org/10.1016/j.nuclphysb.2014.03.023

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

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

1. Introduction and outline

It is well known that the simultaneous action of parity and charge conjugation (CP) is not a

symmetry of Nature. This fact has been established experimentally in oscillations and decays

of K, B, and D mesons. Furthermore, CP violation is a necessary condition to generate the

observed matter–antimatter asymmetry of the universe [1]. The origin of CP violation is, thus,

one of the most fundamental questions in particle physics. Currently, all direct evidence for CP

violation in Nature can be related to the ﬂavor structure of the standard model (SM) of particle

physics [2].

Since it is conceivable that the ﬂavor structure may be explained by an (explicitly or sponta-

neously

broken) horizontal or ﬂavor symmetry, it appears natural to seek a connection between

the fundamental origins of CP violation and ﬂavor. In the past, it has been argued that the ap-

pearance of complex Clebsch–Gordan (CG) coefﬁcients in some of these groups gives rise to

(explicit) CP violation [3], thus relating CP violation to some intrinsic properties of the ﬂavor

symmetry.

There are many ways to check whether or not CP is (explicitly) violated in a given setting.

the low-energy effective theory, an unambiguous check of the existence of CP violation is

the computation of so-called weak basis invariants, which, if vanishing, guarantee the absence

of (ﬂavor related) CP violation [4–6]. However, in order to decide whether CP violation is ex-

plicit or spontaneous in the high energy theory, one has to identify the corresponding symmetry

transformation that, if unbroken, guarantees the absence of CP violation — which is typically

less straightforward. Especially in settings with a discrete (ﬂavor) symmetry, the true physical

CP transformation may be obscured by the fact that, say, complex Clebsch–Gordan coefﬁcients

are present.

To decide whether a particular transformation really conserves CP, one has to check if it can be

“undone”

by a symmetry or basis transformation. If this is the case, CP is conserved, otherwise

CP is violated (cf. e.g. [7]). This then leads to the notion of a so-called generalized CP transfor-

mation [8–10], where one amends the canonical quantum ﬁeld theory (QFT) transformation laws

by this operation.

The main purpose of this study is to explore the relation between discrete (ﬂavor) sym-

metries G and

physical CP invariance guaranteed by generalized CP transformations in more

detail.

The outline of this paper is as follows. In Section 2 we

discuss the general properties

of generalized CP transformations. In particular, we will show that physical CP transfor-

mations are always connected to class-inverting automorphisms of G. We will classify dis-

crete groups G based on the existence and the speciﬁc properties of such transformations.

This will allow us to conclude that in theories based on a certain type of symmetry CP

is generically violated since one cannot deﬁne a proper CP transformation. Section 3 con-

tains three examples illustrating our results. In particular, we demonstrate that both explicit

CP violation and spontaneous CP violation with a phase predicted by group theory can arise

based on a decay example in an explicit toy model. As we shall see, some of the transfor-

mations that were dubbed “generalized CP transformations” in the recent literature do not

correspond to physical CP transformations. Finally, in Section 4 we summarize our results.

In various appendices we collect some of the more technical details relevant to our discus-

sion.

M.-C. Chen et al. / Nuclear Physics B 883 (2014) 267–305 269

2. Generalized CP transformations

2.1. The canonical CP transformation

We start out by reviewing the standard transformation laws of quantum ﬁeld theory. By def-

inition, charge conjugation reverses the sign of conserved currents, j

−→ −j

. For a scalar

ﬁeld operator

φ(x) =



p



a( p)e

−ip·x

†

( p)e

ip·x



, (2.1)

this implies that the creation and annihilation operators a and b for particles and anti-particles get

exchanged. Combining this with a spatial inversion, i.e. a parity transformation, the combined

transformation is given by (e.g. [11])

(CP )

−1

a( p)CP =η

b(−p), (CP )

−1

†

( p)CP =η

∗

†

(−p), (2.2a)

(CP )

−1

b( p)CP = η

∗

a(−p), (CP )

−1

†

( p)CP =η

†

(−p). (2.2b)

Here η

is a phase factor. As a consequence, scalar ﬁeld operators transform as

(CP )

−1

φ(x)CP =η

†

(Px) (2.3)

with Px = (t, −x). At the level of the Lagrangean, this corresponds to a transformation

φ(x)

−→η

∗

(Px) (2.4)

for the ﬁelds, and we see that η

represents the freedom of rephasing the ﬁelds. Analogous

considerations for Dirac spinor ﬁelds result in the transformation

Ψ(x)

−→η

∗

(Px), (2.5)

where C is the charge conjugation matrix.

A Lagrangean, which is invariant under CPT, is schematically given by

L = cO(x) +c

∗

†

(x), (2.6)

where c is a coupling constant and O is an operator. Under a physical CP transformation,

O(x)

−→η

†

(Px) and c

−→c. (2.7)

Demanding the Lagrangean to be invariant under the CP transformation then restricts the phase

of the coupling constant c. In this case the physical CP asymmetry of scattering amplitudes

i→f

|Γ(i→f)|

−|Γ(ı →f)|

|Γ(i→f)|

+|Γ(ı →f)|

(2.8)

will vanish to all orders in perturbation theory. Here ı and f denote the CP conjugate states of

i and f , and are composed out of the corresponding anti-particles. As discussed above, anti-

particles are, per deﬁnition, related to the particles via (2.2).

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

2.2. Generalizing CP transformations

If the setting enjoys a discrete symmetry G, such that φ furnishes a non-trivial representation

of G, the phase factor η

in (2.2) may (and, as we shall see shortly, in general has to) be

promoted to a unitary matrix U

representing an automorphism transformation of G [12]

(CP )

−1

a( p)CP =U

b(−p), (CP )

−1

†

( p)CP =b

†

(−p)U

†

, (2.9a)

(CP )

−1

b( p)CP = a(−p)U

†

,(CP )

−1

†

( p)CP =U

†

(−p), (2.9b)

thus leading to a generalized CP transformation [8–10]

φ(x)



−→U

∗

(Px). (2.10)

Let us brieﬂy explain, following Holthausen, Lindner and Schmidt (HLS) [12], why it is

necessary to generalize CP. Consider a model based on the symmetry group T



with two triplets

x and y as well as a ﬁeld φ transforming as non-trivial one-dimensional representation 1

. Then

the coupling (see Appendix A.1.3 for our conventions for T



)



⊗(x

⊗y

)



√





+ω

+ωx



, (2.11)

is T



invariant. A canonical CP transformation

−→x

∗

−→y

∗

, and φ

−→φ

∗

, (2.12)

would map the 1

representation φ to a 1

such that the contraction (2.11) gets mapped to a term

which is not T



invariant. This can be repaired by imposing a generalized CP transformation



CP ,

which we discuss in more detail later in Section 3.2, and under which

⎛

⎝

⎞

⎠



−→

⎛

⎜

⎝

∗

⎞

⎟

⎠

⎛

⎝

⎞

⎠



−→

⎛

⎜

⎝

∗

⎞

⎟

⎠

, and φ



−→φ

∗

. (2.13)

Under this transformation, the contraction (2.11) gets mapped to its Hermitean conjugate. The

Lagrangean then respects the generalized CP symmetry if the coupling coefﬁcient is real. The

crucial property of the transformation (2.13) is that it is not composed of a canonical CP and a



symmetry transformation. Rather, it involves an outer automorphism of this group [12].

The heart of the above problem seems to be related to the complexity of the Clebsch–Gordan

(CG)

coefﬁcients appearing in equation (2.11). One may then speculate that one might have to

switch to a basis in which all CGs are real, and impose the canonical CP transformation there.

The purpose of our discussion is to show that the true picture is somewhat more subtle. First

of all, we will see that there are groups which do not admit real CGs but nevertheless allow for

a consistent CP transformation, which, if it is a symmetry of the Lagrangean, ensures physical

CP conservation. Second, we shall show that there are symmetry groups that do not allow for

a transformation which ensures physical CP conservation. The CGs in such groups are always

complex, and models based on such symmetries will, at least generically, violate CP. In other

words, for such groups CP violation originates from group theory [3], thus providing us with

very interesting explanation for why CP is violated in Nature relating CP violation to ﬂavor.

M.-C. Chen et al. / Nuclear Physics B 883 (2014) 267–305 271

2.3. What are the proper constraints on generalized CP transformations?

Let us now discuss the general properties of generalized CP transformations. As discussed

in HLS [12] (see also [13]) and above, generalized CP transformations are given by automor-

phisms of the group G, since otherwise the transformation would map G-invariant terms in the

Lagrangean to non-invariant terms.

However, the only way to generalize CP in a model-independent way is to demand that the

operators a and b in (2.9) get

interchanged. Imposing a “generalized CP transformation” that

does not have this property will, in general, not warrant physical CP conservation. This is because

it does not map ﬁeld operators to their own Hermitean conjugates. In fact, as we shall discuss

in an explicit example (see Section 3.1.3), such a “generalized CP symmetry” does not lead to a

vanishing decay asymmetry. That is, in models with very speciﬁc ﬁeld content one may re-deﬁne

CP such that it contains a non-trivial interchange of ﬁelds in representations which are not related

by complex conjugation. The violation of the thus “generalized CP” is then, however, no longer

a prerequisite for, say, baryogenesis. We therefore prefer to refer to such transformations as

“CP-like” transformations. As we are interested in the origin of physical CP violation, we will

from now on impose that the operators a and b in (2.9) get interchanged. This implies that a

true (generalized) CP transformation has to map all complex (irreducible) representations of G

to their conjugates.

Let us now discuss CP transformations that generalize the canonical CP transformation (2.4)

and

act on scalar ﬁelds as

Φ(x)



−→U

∗

(Px), (2.14)

where U

is a unitary matrix and Φ contains, in principle, all ﬁelds of the model. Here and in

what follows, we will only discuss the transformation of scalar ﬁelds; the extension to higher–

spin ﬁelds is straightforward. HLS [12] showed that this generalized CP transformation is only

consistent with the ﬂavor symmetry group G if U

is non-trivially related to an automorphism

u : G →G. In fact, U

has to be a solution to the consistency equation (cf. Eq. (2.8) in HLS [12]

and see also [13])



u(g)



ρ(g)

∗

†

∀g ∈G, (2.15)

where ρ(g) is the (in general reducible) matrix representation in which Φ transforms under G.

However, if Eq. (2.14) is to be a physical CP transformation, u has to have, in generic settings,

some further properties:

u has

to be class-inverting

As discussed above, we demand that u maps

every irreducible representation r

to its own

conjugate. Therefore, the matrix realizations ρ

fulﬁll



u(g)



(g)

∗

†

∀g ∈G and ∀i, (2.16)

with some unitary matrices U

. This implies, of course, that (pseudo-)real representations get

mapped to themselves. The matrix U

of Eq. (2.14) is given by the direct sum of the U

corresponding to the particle content of the model at hand, or, more explicitly, U

is composed

of blocks consisting of the U

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