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Automorphisms in Gauge Theories and the Definition of CP and P



UWThPh-1995-7 June 1995

arXiv:hep-ph/9506272v1 8 Jun 1995

Automorphisms in Gauge Theories and the De?nition of CP and P

W. Grimus and M.N. Rebelo* Institut f¨ ur Theoretische Physik Universit¨ at Wien Boltzmanngasse 5, A–1090 Wien, Austria

Abstract We study the possibilities to de?ne CP and parity in general gauge theories by utilizing the intimate connection of these discrete symmetries with the group of automorphisms of the gauge Lie algebra. Special emphasis is put on the scalar gauge interactions and the CP invariance of the Yukawa couplings.

*) Presently supported by Fonds zur F¨ orderung der wissenschaftlichen Forschung, Project No. P08955–PHY. Permanent address: Dept. de F? ?sica and CFIF-IST, Instituto Superior T? ecnico, Lisbon, Portugal (on leave until September 1995).

1

Contents
1 Introduction 2 Examples 3 Conditions for CP–type transformations: gauge bosons and fermions 3.1 The pure gauge sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fermions and gauge interactions . . . . . . . . . . . . . . . . . . . . . . . 4 Automorphisms of Lc 4.1 Types of automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Some examples of non–simple groups . . . . . . . . . . . . . . . . . . . . 4.3 Irreps and automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The existence of CP in gauge theories and Condition B 5.1 The automorphism associated with CP . . . . . . . . . . . . . . . . . . . 5.2 Canonical and generalized CP transformations . . . . . . . . . . . . . . . 6 On the general solution of Condition B 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 On the de?nition of parity . . . . . . . . . . . . . . . . . . . . . . . . . . 7 CP–type transformations in the Higgs sector 7.1 Pseudoreal scalars and CP transformations . . . . . . . . . . . . . . . . . 7.2 Real scalars in complex disguise and CP transformations . . . . . . . . . 7.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Yukawa couplings and CP–type symmetries 8.1 The condition on Yukawa couplings . . . . . . . . . . . . . . . . . . . . . 8.2 Real Clebsch–Gordan coe?cients and the generalized CP condition . . . 8.3 Solutions of the generalized CP condition . . . . . . . . . . . . . . . . . . 9 C–type transformations 9.1 Charge conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Compatibility of CP and C . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 An example with G = SU (2)L × SU (2)R × U (1)B?L . . . . . . . . . . . . 10 Comments and conclusions Appendices A Notation and conventions B Facts about semisimple Lie groups 3 6 12 12 14 16 16 18 19 20 20 21 23 23 26 28 28 30 33 35 35 36 37 41 41 43 45 47 51 51 52

2 C so(N ) and the spinor representations D On the isomorphisms so(4) = su(2) ⊕ su(2) and so(6) = su(4) E Irreps of G? F On the existence of a CP basis G Basis transformations for pseudoreal scalars H How to solve the generalized CP conditions I On the case DΛ ? DΛ ? ψd in so(2?)
? ?

58 59 60 62 63 66 69 73 74 77

Tables References Figure

3

1

Introduction

The discrete transformations charge conjugation (C), parity (P) and CP played an important r? ole in the development of particle physics. In particular, the hypothesis of P (C) non–conservation [1] and its subsequent experimental con?rmation [2] and the discovery of CP violation [3] constituted both substantial progress in our understanding of weak interactions and gave incentives to important theoretical developments culminating in the advent of the Standard Model (SM) [4]. It is obvious, nowadays, that the suitable framework for discussing CP and P is given by spontaneously broken gauge theories with chiral ?elds as the basic fermionic degrees of freedom. In the SM, CP and P non–invariance appear in two totally di?erent ways: parity is broken because left and right–handed fermionic multiplets belong to di?erent representations of the gauge group whereas complex Yukawa couplings entail CP non–conservation through the Kobayashi– Maskawa mechanism [5] for three quark families. Thus the spontaneous breakdown of the gauge group SU (2) × U (1) only a?ects the manifestation of CP and P violation but has nothing to do with the non–invariance itself. On the other hand, on a more fundamental level, in an appropriate extension of the SM or in a Grand Uni?ed Theory (GUT) [6] one might prefer to break CP or both CP and P spontaneously to provide the same footing for the breaking of continuous and discrete symmetries. This o?ers the possibility to relate the breaking of CP (and P) to the breaking of the gauge group and thus to have a more intimate relationship between CP (and P) non–conservation and the gauge structure in addition to the aesthetic point of view addressed to before. Adopting the premise of spontaneous breakdown of CP and P we are immediately led to the question when and how CP and P transformations can be de?ned in a gauge theory before spontaneous symmetry breaking. This problem is particularly acute for P when left and right–handed irreducible fermionic representations do not match. It is the purpose of this paper to provide a thorough discussion of these questions, i.e. to consider the symmetry aspects leading to CP, P and C transformations in general gauge theories. Thereby we are motivated by the striking structural similarity between CP and P when formulated in a framework where all fermion ?elds have the same chirality. Such a formulation is always possible since it is equivalent, e.g., to the use of the right–handed ?eld T ? (χL )c ≡ Cγ0 χL (1.1) instead of χL (for conventions and the de?nition of the matrix C , see app. A). In this work we will use right–handed ?elds for de?niteness. It is well known that the key to an understanding of the relation between CP and P is given by the automorphisms of the gauge group G [7] and that a gauge theory containing only the gauge bosons and fermions is always CP invariant [7, 8]. These points will be worked out and commented upon in detail in the ?rst chapters of this paper. Furthermore, gauge interactions of scalar ?elds, the second focus of this work, are dealt with in the same way and this part of the interaction is CP invariant as well. However, complications in the de?nition of CP and P transformations arise for Higgs ?elds in real and pseudoreal irreducible representations (irreps) of G. Finally, the third focus is on Yukawa interactions LY . In this connection not

4 only the group structure of the couplings represented by Clebsch–Gordan coe?cients is important but also the couplings pertaining to each group singlet contained in LY . In the case of replication of irreps of G like the well–known families of quarks and leptons there is freedom of de?ning CP and P with respect to unitary rotations among the “families” (see refs. [9, 10] for the context of left–right symmetric models). Choosing one of these “horizontal rotations”, e.g. in the de?nition of CP, and requiring CP invariance of LY we impose conditions on the Yukawa coupling constants. In the simplest case (when the horizontal rotations are proportional to unit matrices) and with appropriate phase conventions they have to be real . Thus the Yukawa interactions are not automatically CP invariant in contrast to the gauge interactions. In the discussions of the three main subjects of this work special attention will be paid to the existence of certain bases with respect to the group structure and to the horizontal structure to provide “canonical” forms of CP where its properties are especially transparent. General remarks on the scope of the paper: To clarify the range of validity of this paper some remarks are in order. We will always deal with compact Lie groups G as gauge groups which entails unitary representations (reps) and the real Lie algebra Lc of G is compact (see app. B). On the other hand, requiring unitary reps on quantum theoretical grounds, it can be shown that one can con?ne oneself to compact groups [11]. We will not only cover simple groups but also groups of the type G′ × G′′ and G′ × U (1) with G′ and G′′ simple. In physical terms these cases require two independent gauge coupling constants. More complicated gauge structures like the SM gauge group can easily be discussed by an extension of these considerations. Special emphasis will be put on G = G′ × G′ (G′ simple) extended by a discrete element such that only one gauge coupling constant is present. A typical case would be left–right symmetric models [12, 13]. In general there will also be symmetries of the Lagrangian which are not gauged, whether discrete or continuous. Therefore the total symmetry group will have the structure G × H where H is the group of these additional symmetries. Multiplets identical with respect to the gauge group may be distinguished by di?erent transformation properties under H . In certain cases this can forbid operations to be discussed in what follows which involve non–trivial multiplicities of irreps of G. For instance, if in a model there is a complex scalar multiplet transforming according to a real irrep of G but to a complex irrep of H then this multiplet cannot be split into two real multiplets. We leave it with this caveat and concentrate on the gauge group from now on. Although we will be mainly concerned with reps of Lc it will be tacitly assumed that these reps can be extended to reps of G. Furthermore, in our discussion of the various terms of the Lagrangian we will not cover the scalar potential because we think that in concrete cases it can be treated with the methods of this paper though its general discussion is involved. Finally, since for gauge theories the CPT theorem holds [14] (see also ref. [15]) and since CP is discussed extensively time reversal is only mentioned shortly in this work.

5 Plan of the paper: In sect. 2 the main features of CP and P are worked out in detail by means of three examples: QED, QCD and the SO (10)–GUT. CP–type transformations comprising CP and P are introduced in sect. 3 where Conditions A and B for invariance of the Lagrangian containing gauge ?elds and fermions are derived. Since Condition A requires that a CP–type transformation on the gauge ?elds corresponds to an automorphism of Lc sect. 4 is devoted to a detailed discussion of Aut (Lc ). Sects. 5 and 6 treat Condition B. In sect. 5 CP transformations are introduced as special solutions of Condition B and the existence of the “CP basis” is discussed where all the generators of Lc are either symmetric or antisymmetric matrices. In sect. 6 the general solution of Condition B is expounded and parity transformations are de?ned by characteristic properties of the associated automorphisms. Several notions of CP–type transformations are introduced in sects. 5 and 6 to clarify the subject. They are related to each other in the following way:
? ? ? ? ? ? ?

generalized CP { canonical CP parity internal parity external parity

The brace { signi?es that the more general notion is found to the left of it. In the examples of sect. 2 on the one hand, the structural similarity between CP and P will become apparent, on the other hand, their distinctive features will be elucidated as well. QED and QCD provide examples of external parity whereas the SO (10)–GUT, where all fermionic degrees of freedom are in a 16–dimensional irrep, allows to de?ne an internal parity. In sect. 7 the whole discussion is carried over to scalars with emphasis on the special cases of real and pseudoreal irreps. Sect. 8 is devoted to the invariance of the Yukawa couplings under CP–type transformations. Furthermore, the general solutions of the condition imposed on these couplings by a generalized CP transformation are derived. Transformations of the charge conjugation type are considered in sect. 9. Finally, in sect. 10 a summary is presented. In app. A our conventions concerning the γ matrices and the charge conjugation matrix C can be found. All properties of Lie algebras necessary for our purposes are collected in app. B. In app. C spinor reps of so(N ) are de?ned via the Cli?ord algebra and app. D describes the Lie algebra isomorphisms so(4) ? = su(2) ⊕ su(2) and so(6) ? = su(4). The remaining appendices E–I give proofs which are not carried out in the main body of the paper. Notational remarks: For convenience we will often leave out the arguments x and x = (x0 , ?x) in CP–type transformations. Equivalence of reps will be denoted by ?. The letter D will be used for reps of Lc and also of G. Its use should be clear from the context. Yet ?D T refers only to the complex conjugate of the Lie algebra rep D . Finally let us collect the abbreviations used in this work: CSA, GUT, ON, SM, rep and irrep denote Cartan subalgebra, Grand Uni?ed Theory, orthonormal, Standard Model, representation and irreducible representation, respectively.

CP–type transformation ? ?

6

2

Examples

Before coming to the general discussion it is quite instructive to consider some examples which enable us to get a feeling for the main features. In this light we will discuss now QED, QCD and the GUT based on the spinor rep of SO (10) (strictly speaking, an irrep of Spin(10), its universal covering group, see e.g. ref. [16]). QED: Denoting the electron ?eld by e(x) we know that the Lagrangian is invariant under CP : P : e(x) → ?Ce(x)? e(x) → γ 0 e(x)

(2.1)

where x = (x0 , ?x). In terms of chiral ?elds eq. (2.1) reads CP : P : eL,R (x) → ?CeL,R (x)? eL,R (x) → γ 0 eR,L (x) (2.2)

expressing the fact that, having started with Dirac ?elds, CP does not mix chiralities whereas parity connects ?elds with di?erent chirality. De?ning χR1 ≡ eR ,
T ? χR2 ≡ Cγ0 eL

(2.3)

we can get yet another view of eq. (2.1) [17], namely CP : P : With UCP ≡ χR1 χR2 χR1 χR2 → → ?1 0 0 1 0 ?1 1 0 C C χR1 χR2 χR1 χR2 0 1 ?1 0
?

?

.

(2.4)

we observe that the form of CP and P is only distinguished by the matrix U in the CP–type transformation ? χR1 χR1 → ?UC . (2.6) χR2 χR2 How are UCP and UP characterized with respect to the gauge group U (1)em ? The chiral ?elds χR1,2 transform with complex conjugate phases, i.e. χR1 → eiα χR1 , χR2 → e?iα χR2 , e = χR1 + (χR2 )c → eiα e (2.7)

1 0 0 ?1

,

UP ≡

(2.5)

and the chiral ?eld vector consists of two irreps which are exchanged under UP whereas UCP acts within the respective irreps. Thus, in the formulation where the previously

7 mentioned structural similarity becomes obvious, CP and P can be distinguished by group theoretical properties of the matrix U . Eq. (2.4) has to be supplemented by the transformation properties of the photon ?eld CP : P : A? (x) → ?ε(?)A? (x) A? (x) → ε(?)A? (x)

(2.8)

where the ε(?) are the signs +1 for ? = 0 and ?1 for ? = 1, 2, 3. QCD: We con?ne ourselves to a single quark ?avour q . It is well known that QCD is invariant under the transformations analogous to eq. (2.2) and with the same reasoning as before we get for the “CP–type” transformation χR → ?UCχ? R and UCP = Eq. (2.7) now reads χR1 → DχR1 , χR2 → D ? χR2 , q = χR1 + (χR2 )c → Dq (2.11) with χR = χR1 χR2 = qR (qL )c (2.9)

1 0 0 ?1

,

UP =

0 1 ?1 0

.

(2.10)

with D ∈ SU (3)c and eq. (2.8) is adapted to QCD by
a a CP : W? (x) → ηa ε(?)W? (x) a a P : W? (x) → ε(?)W? (x)

(2.12)

with a = 1, . . . , 8. The signs ηa depend on the transposition properties of the generators λa /2 where λa are the Gell–Mann matrices [18] and are obtained by ? λT a = ηa λa . (2.13)

As is well–known this choice of signs in eq.(2.12) also leaves invariant the pure gauge part of QCD. In sect. 3 (see also app. B) this fact will be connected with the mapping λa → ηa λa being an automorphism of the Lie algebra su(3). SO(10) – GUT: A more involved example of CP and P is provided by a Grand Uni?ed Theory based on SO (10) [19, 20, 21, 22] or actually on its universal covering group Spin(10). Here, all 15 fundamental fermions of one SM family together with an additional right–handed neutrino carrying lepton number +1 ?t nicely into a 16–dimensional spinor irrep. In order to implement CP and P a more elaborate analysis of the spinor irrep has to be performed. For our purpose, it is appropriate to take advantage of so(10) ? so(6) ⊕ so(4) ? = su(4) ⊕ su(2) ⊕ su(2) (2.14)

8 since the classi?cation of the states according to SU (4)c × SU (2)L × SU (2)R is well known. Here, SU (4)c denotes the colour or Pati–Salam–SU (4) [12] where lepton number is treated as “fourth colour”. Furthermore, eq. (2.14) also represents a step towards a realistic pattern of spontaneous symmetry breaking. All necessary ingredients we are using in the following are explained in apps. C and D. Following refs. [20, 21], we denote the basis vectors of the 32–dimensional space C2 ? C2 ? C2 ? C2 ? C2 by | s1 s2 s3 s4 s5 , where sj = ±1 or, abbreviated, ±, and |s1 s2 s3 s4 s5 = es1 ? es2 ? es3 ? es4 ? es5 with e+ = 1 0 , e? = 0 1 . (2.15)

The 16–dimensional subspaces for the spinor irreps {16} and {16} are given by the two projectors P± on the 32–dimensional space where P± = 1 ± Γ11 , 2
5 j =1

Γ11 = σ3 ? σ3 ? σ3 ? σ3 ? σ3 .

(2.16)

This means we are taking as basis vectors those with sj = +1 or ? 1 (2.17)

for the {16} or the {16}, respectively. The subalgebra so(4) of so(10) is assumed to be generated by Mij , 1 ≤ i < j ≤ 4, c and so(6) by Mij , 5 ≤ i < j ≤ 10. The CSA {F3 , Yc , B ? L} of su(4)c is easily carried over to the spinor representation of so(10) by using the Cli?ord algebra (see app. C) and the mapping of the Gell–Mann basis into so(6) as given in app. D.1 Thus we obtain 1 1 c F3 = λ3 = diag (1, ?1, 0, 0) 2 2 1 c → (F3 )s = (?1 ? 1 ? σ3 + 1 ? σ3 ? 1) ? 1(2) 4 1 1 Yc = √ λ8 = diag (1, 1, ?2, 0) 3 3 1 → (Yc )s = (?1 ? 1 ? σ3 ? 1 ? σ3 ? 1 + 2σ3 ? 1 ? 1) ? 1(2) 6 1 B ? L = 2/3λ15 = diag (1, 1, 1, ?3) 3 1 → (B ? L)s = (1 ? 1 ? σ3 + 1 ? σ3 ? 1 + σ3 ? 1 ? 1) ? 1(2) . 3
1

(2.18)

The Mij in eq. (D.4) have to be replaced by M4+i 4+j to comply with the embedding of so(6) into so(10) chosen above.

9
c Here, B and L are baryon and lepton number, respectively, and {F3 , Yc } generate the (p) CSA of su(3)c. The symbol 1 denotes the p-fold tensor product of the 2 × 2 unit matrix 1 and the subscript s refers to the spinor rep. By a completely analogous procedure the two su(2) subalgebras of so(4) can be obtained:

a1 = a2 = a3 = b1 = b2 = b3 =

i (3) 1 ? (σ2 ? σ2 + σ1 ? σ1 ) 4 i (3) 1 ? (σ2 ? σ1 ? σ1 ? σ2 ) 4 i (3) 1 ? (1 ? σ3 ? σ3 ? 1) 4 i ? 1(3) ? (σ2 ? σ2 ? σ1 ? σ1 ) 4 i (3) 1 ? (σ2 ? σ1 + σ1 ? σ2 ) 4 i ? 1(3) ? (1 ? σ3 + σ3 ? 1). 4

(2.19)

T Comparing eq. (2.19) with eq. (D.1), we see the correspondences Aj ? aj and ?Bj ? bj . T The advantage of the choice {?Bj } compared with the equivalent {Bj } will become clear at the end of this section. The algebra {aj } generates SU (2)L whereas {bj } generates SU (2)R . The remaining two elements of the Cartan subalgebra in the spinor rep of so(10) are given by

(I3L )s = ia3 = (I3R )s = ib3

1 (3) 1 ? (σ3 ? 1 ? 1 ? σ3 ) 4 1 (3) 1 ? (σ3 ? 1 + 1 ? σ3 ). = 4

(2.20)

This completes the de?nition of the CSA of so(10) in the 32–dimensional spinor rep. We can now easily construct SU (2)L × SU (2)R multiplets by means of the lowering operators a? = i(a1 ? ia2 ) = ?1(3) ? σ? ? σ+ b? = i(b1 ? ib2 ) = ?1(3) ? σ? ? σ?
1 with σ± = 2 (σ1 ± iσ2 ) and ?nd the basis vectors, omitting the so(6) part,

(2.21)

(2, 1) : (1, 2) :

| + ? , ?| ? + | + + , ?| ? ? .

(2.22)

c Calling the quarks with (F3 , Yc ) quantum numbers (1/2, 1/3), (?1/2, 1/3), (0, ?2/3) red, yellow and blue, respectively, we can start with a right–handed red up–quark state ur = | + + ? ++ . Then, applying b? we obtain dr = ?| + + ? ?? . The su(4) multiplet

10 is completed by the raising and lowering operators 1 (λ1 ± iλ2 ) 2 1 (λ4 ± iλ5 ) 2 1 (λ6 ± iλ7 ) 2 1 (λ9 ± iλ10 ) 2 1 (λ11 ± iλ12 ) 2 1 (λ13 ± iλ14 ) 2 → ?1 ? σ± ? σ? ? 1(2) → σ± ? σ3 ? σ? ? 1(2) → ?σ± ? σ? ? 1 ? 1(2) → σ± ? σ± ? 1 ? 1(2) → σ± ? σ3 ? σ± ? 1(2) → 1 ? σ± ? σ± ? 1(2) . (2.23)

Thus, having embedded SU (4)c × SU (2)L × SU (2)R in Spin(10) we ?nd for its (4, 1, 2) multiplet of right–handed quarks ur uy ub N dr dy db e | + + ? ++ ?| + + ? ?? = ?| + ? + ++ | + ? + ?? | ? + + ++ ?| ? + + ?? | ? ? ? ++ ?| ? ? ? ?? . (2.24)

The electric charges of the states can easily be checked by means of the charge operator 1 Qem = I3L + I3R + (B ? L) → 2 1 (3) 1 (Qem )s = 1 ? σ3 ? 1 + (1 ? 1 ? σ3 + 1 ? σ3 ? 1 + σ3 ? 1 ? 1) ? 1(2) . 2 6 (2.25)
c From this and using F3 , Yc (see eq. (2.18)) we ?nd that N has the quantum numbers of a right–handed SM singlet neutrino, whereas e is the right–handed electron state. The state | ? ? + ?+ has the quantum numbers of an anti–red quark belonging to the (2,1) multiplet of SU (2)L × SU (2)R with eigenvalue ?1/2 of (I3L )s . This leads to the (? 4, 2, 1) multiplets c c c uc r uy ub ν c c c dc r dy db e

= ?| ? + ? ?+ ?| ? + ? +? ?| + ? ? ?+ ?| + ? ? +? | + + + ?+ | + + + +? . (2.26)

?| ? ? + ?+ ?| ? ? + +?

The choice of signs in eq. (2.26) relative to eq. (2.24) will become clear from the discussion of parity. We see that with our states (2.24) and (2.26) we have projected out the space corresponding to j sj = ?1. All the quantum numbers correspond to the choice of right– handed ?elds in the {16} irrep. The construction of the gauge theory based on so(10) and

11 the spinor representation {16} for the multiplet χR , the physical content of which has been described above, is now straightforward using the hermitian generators Tij = iσij /2, (1 ≤ i < j ≤ 10) with σij = (Mij )s (see eqs. (C.3) and (C.4)). Anticipating sect. 3 and trying CP : χR (x) → ?CχR (x)? (2.27)

we merely have to check that invariance of the Lagrangian is achieved by an appropriate transformation of the gauge ?elds. In our case it is convenient to number the 45 gauge ij ?elds by W? (x), corresponding to the generators Tij . Analogously to the case of QCD it is easy to see that it is su?cient for CP invariance to transform the gauge ?elds as
ij ij CP : W? (x) → ε(?)ηij W? (x)

(2.28)

where the signs ηij are obtained through
T ? σij = ηij σij .

(2.29)

It is easy to calculate the ηij recalling the de?nition of σij in terms of the elements of the Cli?ord algebra (see app. C). We have ΓT i = ξ i Γi , therefore and ηij = (?1)i+j = ξi ξj . (2.32) In sect. 5 it will be shown that one can always choose a basis in representation space such T that Ta = ±Ta is valid for the generators. Eq. (2.29) tells us that Mij → ηij Mij is an automorphism of so(10) (see app. B and sect. 3). We will see later that this guarantees a consistent choice of signs in eq. (2.28) such that the pure gauge part without fermions is also invariant under our CP transformation. Turning to parity one is led to presume that this gauge theory is also parity invariant because the states (2.26) emerge from eq. (2.24) by a kind of “antiparticle formation”. Whereas in QCD the states of {3} and {? 3} correspond to each other, we now have such a correspondence within one so(10) multiplet. In sect. 6 we will call the ?rst kind of parity “external” and the second one “internal”. One can indeed formulate a parity transformation by P : χR (x) → ?UP CχR (x)? (2.33) with UP = σ1 ? σ2 ? σ1 ? σ2 ? 1 (2.34) ξi = (?1)i+1 , (2.30)

1 T ? σij = ? [Γi , Γj ]T = ξi ξj σij 2

(2.31)

12 and where the signs ρij are now obtained by (to be derived in sect. 3)
T ?1 UP (?σij )UP = ρij σij ij ij P : W? (x) → ε(?)ρij W? (x)

(2.35)

(2.36)

or else, using eqs. (2.31) and (2.32), by
?1 UP σij UP = ηij ρij σij .

(2.37)

Again, one can check that the pure gauge terms of the Lagrangian are invariant under the transformation (2.35). Note that the element σ12 of the CSA commutes with UP , whereas σ34 , σ56 , σ78 , σ9 10 anticommute. In other words, UP transforms eigenvectors of (I3L ? I3R )s into states with the same eigenvalue, whereas the eigenvalues of (I3L + I3R )s , c (B ? L)s , (F3 )s and (Yc )s are reversed. Our sign conventions are such that UP veri?es UP ur uy ub N dr dy db e =
c c c uc r uy ub ν c c c dc r dy db e

(2.38)

where this notation means that UP is applied to all the states in the parentheses. Taking into account eq. (2.19) and ρij = 1 for 5 ≤ i < j ≤ 10 we obtain
?1 T UP σij UP = ?σij ?1 UP aj UP ?1 UP bj UP

=

=

?bT j , ?aT j .

for 5 ≤ i < j ≤ 10,

(2.39)

From this it follows that the ?elds χR1 and χR2 corresponding to (4,1,2) and (4, 2, 1), respectively, in the irrep {16} transform with SU (4) matrices which are exactly complex conjugate to each other. The same is true for the “diagonal SU (2)” generated by {aj + bj }. Thus χR1 and (χR2 )c transform alike under SU (4)c × SU (2)diag and correspond to eR and eL in QED, or qR and qL in QCD. Thus our conventions eqs. (2.19) and (2.38) were suggested to exhibit the common features of the three examples.

3
3.1

Conditions for CP–type transformations: gauge bosons and fermions
The pure gauge sector

To have a starting point for the discussion of CP–type transformations it is appropriate to repeat shortly the construction of gauge theories. Let {Ta } be the hermitian generators in the fermionic representation such that [Ta , Tb ] = ifabc Tc (3.1)

13 and Tr (Ta Tb ) = kδab with totally antisymmetric coe?cients fabc (see app. B, ?rst subsection). De?ning
a W? ≡ Ta W?

(3.2)

(3.3)

the pure gauge Lagrangian is given by (see, e.g., ref. [23]) LG = ? with Under a gauge transformation, the fermionic multiplet ωR and the ?elds W? transform as i W? (x) → U (x)W? (x)U (x)? + (?? U (x))U (x)? g ωR (x) → U (x)ωR (x), (3.6) (3.7) G?ν = ?? Wν ? ?ν W? + ig [W? , Wν ] ≡ Ta Ga ?ν . (3.5) 1 Tr (G?ν G?ν ) 4k (3.4)

with U (x) = exp{?iTa αa (x)}. As a result the ?eld strength tensor G?ν transforms according to the adjoint representation written as G?ν (x) → U (x)G?ν (x)U (x)? (3.8)

therefore leaving LG invariant. Having ?xed our notation we will now examine in detail the e?ect of a CP–type transformation in the gauge sector. The general form of such a transformation acting on the gauge boson multiplet is given by
a b W? (x) → ε(?)Rab W? (x) with R ∈ O (nG )

(3.9)

where nG is the number of gauge bosons and thus equal to the number of group generators. a The ?elds W? are real and therefore R is a real matrix. It will shortly become clear that the orthogonality condition (3.2) requires R to be orthogonal. Let us now consider the e?ect of the transformation (3.9) on the ?eld strength tensor. For the terms with derivatives we have a d d ?? Wν (x) → ε(ν )Rad ?? (Wν (x)) = ε(?)ε(ν )Rad ?? Wν (x) (3.10) where ?? is the derivative with respect to x. The commutator transforms according to
b c b c (x)Wν (x) ? gfabc W? (x)Wν (x) → ?gfab′ c′ ε(?)ε(ν )Rb′ b Rc′ c W? b c (x)Wν (x). = ?gRad fa′ b′ c′ Ra′ d Rb′ b Rc′ c ε(?)ε(ν )W?

(3.11)

Consequently, under a CP–type transformation Ga ?ν behaves as
b c d d Ga ?ν (x) → ε(?)ε(ν )Rad (?? Wν ? ?ν W? ? g fdbc W? Wν )(x)

(3.12)

14 with fdbc = fa′ b′ c′ Ra′ d Rb′ b Rc′ c . (3.13) This leads us to the ?rst condition for invariance of L under a CP–type transformation: Condition A: fabc = fa′ b′ c′ Ra′ a Rb′ b Rc′ c . (3.14) In what follows eq. (3.14) will be referred to as Condition A. If it is ful?lled we get
d Ga ?ν (x) → ε(?)ε(ν )Rad G?ν (x)

(3.15)

and d4 x LG is clearly invariant under such a transformation. Note that eq. (3.10) together with eq. (3.2) already leads to R orthogonal in order to get invariance of the quadratic part of LG under the transformation (3.9).

3.2

Fermions and gauge interactions

As mentioned before we choose to represent all the fermionic degrees of freedom by a single right–handed Weyl ?eld vector ωR transforming according to the rep {Ta } (see app. A). Hence the fermionic Lagrangian is given by LF = ωR iγ ? (?? + igTa W? a )ωR . (3.16)

The general form of a CP transformation acting on the fermionic multiplet ωR is given by (3.17) ωR (x) → Uγ 0 CωR (x)T = ?UCωR (x)? where U , here, is a constant unitary matrix in representation space, i.e., in the same space on which the rep {Ta } operates. It can be easily checked that the kinetic part of eq. (3.16) transforms as LF kin(x) → LF kin (x) under eq. (3.17) whilst the interaction term leads to the invariance condition ?(U ? Tb Rba U )T = Ta which can readily be cast into the form Condition B: U (?Tb T Rab )U ? = Ta . (3.18)

In what follows we will refer to this equation as Condition B. It is easily veri?ed that {?Tb T Rab } ful?lls the commutation relations (3.1) for R satisfying Condition A. This fact will be further exploited. Note that every CP–type transformation acting in a gauge theory with fermions can be described by a pair of matrices (R, U ) de?ned above. As we have seen R is an orthogonal nG × nG matrix and U a unitary matrix with the dimension of {Ta }. In general the rep {Ta } will not be irreducible and we can therefore perform a decomposition into irreps Ta = i
r

(1mr ? Dr (Xa )),

dim Dr = dr

(3.19)

15 where {Xa } is an ON basis of the real compact Lie algebra Lc (see app. B); mr is the multiplicity of the irrep Dr in {Ta } of dimension dr ; the direct sum runs over all irreps included in {Ta } and its total dimension is r mr dr . We will call the degeneracy spaces “horizontal spaces” and the indices associated with them “horizontal indices”. The decomposition (3.19) leads immediately to the following statement: Theorem I: Let (R, U0 ) be a solution of Conditions A and B and let (R, U1 U0 ) be another solution. Then U1 = (3.20) (ur ? 1dr )
r

where the ur are unitary mr × mr matrices. Proof: Inserting U1 U0 into Condition B we obtain
? ? ? = U1 Ta U1 . ]U1 Ta = U1 [U0 (?Tb T Rab )U0

Then Schur’s lemma forces U1 to be of the form (3.20). Since U0 and U1 U0 are both unitary the matrices ur are unitary as well. 2 This simple theorem together with Theorem II of subsect. 6.1 will prove very useful in the discussion of solutions of Conditions A and B. In fact Theorem I shows that in order to solve Condition B one can concentrate on the determination of the group theoretical aspects of U with the freedom in the horizontal component simply given by eq. (3.20). De?ning a linear operator on Lc by (see app. B) ψR : Xa → Rba Xb we infer from Condition A that [ψR (Xa ), ψR (Xb )] = fabc ψR (Xc ). (3.22) (3.21)

Therefore ψR is an automorphism of Lc as well as ψR?1 since the set of automorphisms of Lc forms a group (this can also be veri?ed by examining Condition A). As a result Condition A can be formulated as ψR ∈ Aut (Lc ). (3.23)

T For every irrep Dr the complex conjugate irrep is given by ?Dr . Clearly, Dr ? ψR de?ned by (Dr ? ψR )(X ) ≡ Dr (ψR (X )) is also an irrep. Thus Condition B can be read as T ? ψR?1 ) ? 1mr ? (?Dr

r

r

(1mr ? Dr ) .

(3.24)

Eqs. (3.23) and (3.24) are purely Lie algebra theoretical conditions and will be discussed in sect. 4 and sects. 5 and 6, respectively.

16 Finally, let us write down the e?ect of changing the basis of the fermionic multiplet ′ on a CP–type transformation. Let ωR be the ?eld vector in the new basis and
′ ωR = ZωR .

(3.25)

As a result the new matrix U ′ in eq. (3.17) is given by U ′ = Z ? UZ ? . (3.26)

It is important to note the complex conjugation on the right–hand side of eq. (3.26). This prevents the use of the well–known theorems for normal matrices (see subsect. 8.3 for a further discussion on basis transformations).

4
4.1

Automorphisms of Lc
Types of automorphisms

The reformulation of Conditions A and B into eqs. (3.23) and (3.24), respectively, makes plain that the group of automorphisms of Lc , Aut (Lc ), plays an important r? ole in our discussion. Therefore we want to set forth in this section all the details we will need in the following. The basic notions of Lie algebras necessary for this section can be found in app. B. Inner and outer automorphisms [16, 24]: For each element Y of Lc we can de?ne an automorphism ψY of the form ψY (X ) ≡ (exp ad Y )(X ) = eY Xe?Y with (ad Y )X ≡ [Y, X ]. (4.1)

Automorphisms of this kind are called inner. The representation of exp ad by exponentials, second equality of eq. (4.1), comes about due to the fact that all Lie algebras can be seen as matrix Lie algebras (Theorem of Ado [25]). It is clear from eq. (4.1) that the inner automorphisms of Lc from a group denoted by Int (Lc ). For a connected Lie group G it is isomorphic to G/Z where Z is the centre of G [26]. Moreover, Int (Lc ) is a normal subgroup of Aut (Lc ) since if ψ ∈ Aut (Lc ) then ψ (ad Y )ψ ?1 = ad ψ (Y ) and therefore ψ (exp ad Y )ψ ?1 = exp ad ψ (Y ). Outer automorphisms are automorphisms which are not inner. Clearly, it is su?cient to consider one representative of each coset in Aut (Lc )/Int (Lc ) in order to make a complete study of outer automorphisms. Root rotations [16]: The group of root rotations Aut (?) is de?ned as a mapping of ?, the set of (non–zero) roots of L (complexi?cation of Lc ), onto itself which ful?lls a) τ (α + β ) = τ (α) + τ (β ) ? α, β ∈ ? such that α + β ∈ ? and b) τ (?α) = ?τ (α).

17 Aut (?) has an important normal subgroup, the Weyl group W . It is the group generated by the elements Sα (α ∈ ?) which act on ? as Sα β = β ? 2 β, α α. α, α (4.2)

One can show that Sα ∈ Aut (?) ? α ∈ ?. Associated with each root rotation τ ∈ Aut (?) there is an automorphism ψτ of the corresponding algebra L, which is also an automorphism of Lc . The connection between Aut (?) and Aut (Lc ) is given by the following theorem. Theorem: For every τ ∈ Aut (?) there is a mapping ψτ of L onto itself de?ned by ψτ (hα ) = hτ (α) where χα = ±1 ? α ∈ ? such that χα = 1 for all simple roots, χα+β = Nτ (α)τ (β ) χα χβ Nαβ for α, β, α + β ∈ ?+ (4.4) and ψτ (eα ) = χα eτ (α) (4.3)

and χ?α = χα . Then ψτ is an automorphism of L and, moreover, restricted to Lc we also have ψτ ∈ Aut (Lc ). See app. B for the notation in eqs. (4.3) and (4.4). In what follows we will need the automorphism induced by root re?exion ψ △ ≡ ψτr with τr (α) = ?α. (4.5)

It is clear that τr ∈ Aut (?) and χα = 1 ? α ∈ ?. The index △ stands for contragredient. Another important class of root rotations in the case of simple Lie algebras is de?ned by those which are symmetries of the Dynkin diagram or, equivalently, of the Cartan matrix Ajk (see app. B). The group of these rotations is thus given by Aut (DD ) = {τ ∈ Aut (?)| Aτ (j )τ (k) = Ajk ? j, k = 1, . . . , ?}. From the set of Dynkin diagrams depicted in ?g. 1 it is clear that Aut (DD ) can only be isomorphic to {e} or Z2 or S3 . The latter case is only possible for D4 , the complexi?cation of so(8), where the three outer simple roots α1 , α3 , α4 can be permutated leading to the permutation group S3 . For A1 , B? , C? and all exceptional algebras except E6 there is no trivial diagram symmetry and therefore Aut (DD ) is reduced to {e}. For A? (? ≥ 2) there is symmetry under the inversion of the order of the simple roots, i.e. τ (αj ) = α?+1?j , j = 1, 2, . . . , ?. For D? (? ≥ 5) exchange of α??1 and α? is the unique Dynkin diagram symmetry, i.e., τ (αj ) = αj (j = 1, 2, . . . , ? ? 2), τ (α??1 ) = α? , τ (α? ) = α??1 . In E6 we can reverse the order of the roots in the line containing the ?ve roots α1 , . . . , α5 , i.e., there the Dynkin diagram is symmetric under τ (αj ) = α6?j (j = 1, 2, . . . , 5), τ (α6 ) = α6 . The automorphisms associated with non–trivial elements of Aut (DD ) will be called diagram automorphisms and denoted by ψd .

18 The essence of the above discussion is related to the facts that for a simple Lie algebra Lc , ψτ is an inner automorphism if and only if the root rotation τ is an element of the Weyl groyp W of L and diagram automorphisms are always outer automorphisms so that we have [16, 24] Aut (Lc )/Int (Lc ) ? (4.6) = Aut (?)/W ? = Aut (DD ). For our purposes it will be su?cient to use as representations of the cosets in Aut (Lc )/Int (Lc ) the automorphisms ψd , ψ △ or id (see table 1) depending on the algebra Lc under consideration. This covers the automorphism structure of simple Lie algebras.

4.2

Some examples of non–simple groups

We want to go a little beyond simple Lie algebras to include the most frequent cases occurring in model building. To determine Aut (Lc ) in more complicated cases we need two trivial observations: i) Let I be an ideal of L and ψ ∈ Aut (L). Then ψ (I ) is again an ideal. ii) Let I1 , I2 be ideals of L then also I1 ∩ I2 is an ideal. Let us now consider a few cases:
′ ′′ i) Lc = L′c ⊕ L′′ c with Lc , Lc simple and non–isomorphic:

Here we have

′′ ′ ? Aut (L′c ⊕ L′′ c ) = Aut (Lc ) × Aut (Lc ).

(4.7)

Proof: L′c ⊕ 0 is an ideal of Lc . Then (L′c ⊕ 0) ∩ ψ (L′c ⊕ 0) is an ideal of Lc and also of ′ ′ ? ′ L′c . Since L′c is simple and L′′ c = Lc we have ψ (Lc ⊕ 0) = Lc ⊕ 0. The same reasoning is valid for the other summand L′′ 2 c. In the following we will adopt the physical point of view that even when mathematically L′c ? = L′′ c , if the associated gauge couplings are di?erent we will consider both Lie algebras as being di?erent from each other. ii) Lc = L′c ⊕ L′c , L′c simple and associated to a group G: In analogy with the previous case we have now Aut (L′c ⊕ L′c )/(Aut (L′c ) × Aut (L′c )) ? = Z2 with the Z2 generated by ψE ((X, Y )) = (Y, X ). (4.9) Now we can imagine that the Lie algebra is actually associated to a group G? de?ned by enlarging G × G by an element E such that G? = (G × G) ∪ {E }, E 2 = (e, e), E (g, g ′)E = (g ′ , g ). (4.10) (4.8)

19 The physical background for this construction is that here the reps we are interested in are actually reps of G? with the representation of E giving a symmetry reason for equal coupling constants for both group factors in G? . In app. E the irreps of G? are derived. Here we give the full list. Let Dr denote the irreps of G. Then all irreps of G? are given by one of the following constructions:
± Dr : (g1 , g2 ) ∈ G × G is represented by Dr (g1 ) ? Dr (g2 ) and D (E )v ? w = ±w ? v .

Drr′ (r = r ′ ) : (g1 , g2 ) ∈ G × G is represented by (Dr (g1 ) ? Dr′ (g2 )) ⊕ (Dr′ (g1 ) ? Dr (g2 )) and D (E )(v ? w, x ? y ) = (y ? x, w ? v ).

These irreps serve as a guideline for the construction of theories with gauge group G × G and equal coupling constants. In speci?c models ψE can be used as the automorphism associated with P (see sect. 6) or C (see sect. 9) and de?ne in that way a transformation forcing equal gauge coupling constants. iii) Lc = L′c ⊕ u(1), L′c simple: Again one can show that Aut (L′c ⊕ u(1)) ? = Aut (L′c ) × Z2 with Z2 generated by Actually, since u(1) is abelian any transformation Xu → aXu with a ∈ R\{0} would be an automorphism but the gauge Lagrangian LG can only be invariant under a = ±1 since Ru = diag (1, . . . , 1, ±a) associated with ψu must be orthogonal. ψu ((X, Xu )) = (X, ?Xu ). (4.12) (4.11)

4.3

Irreps and automorphisms

As discussed in subsect. 3.2 composition of irreps with automorphisms plays a crucial r? ole in solving Condition B. Con?ning ourselves to simple Lie groups we know that any ψ ∈ Aut (Lc ) can be written as ψ = ψY or ψY ? ψd . If D is an arbitrary rep then D ? ψY = eD(Y ) De?D(Y ) , D ? ψY ? ψd = eD(Y ) D ? ψd e?D(Y ) . (4.13)

Therefore only outer automorphisms can give non–equivalent reps through composition. To explore the e?ects of diagram automorphisms we consider ?rst the action of τ ∈ Aut (DD ) on the fundamental weights (see app. B):
? ?

τ (Λj ) = =

(A?1 )jk τ (αk ) =
k =1 ? k =1

(A?1 )jk ατ (k)
k =1 ?

(A?1 )τ (j )τ (k) ατ (k) =

(A?1 )τ (j )k αk = Λτ (j ) .
k =1

(4.14)

20 For convenience we did not distinguish between the diagram symmetry and its ensuing permutation of the indices 1, . . . , ?, i.e. τ (αk ) ≡ ατ (k) . Next we ?nd the weights of the irrep D ? ψτ by comparing it with D . Let e(λ, q ) be the eigenvector with the weight λ of D and q = 1, . . . , mλ where mλ is the multiplicity of λ. Then D (hα )e(λ, q ) = λ(hα )e(λ, q ) = λ, α e(λ, q ) (D ? ψτ )(hα )e(λ, q ) = D (hτ (α) )e(λ, q ) = λ, τ (α) e(λ, q ). Since τ is a root rotation one can show that [16] τ (β ), τ (γ ) = β, γ ? β, γ ∈ ?. (4.16) (4.15)

Therefore, if λ is a weight of D then τ ?1 (λ) is a weight of D ? ψτ . This is valid for any root rotation τ . If in addition τ ∈ Aut (DD ) we obtain the highest weight of D ? ψτ as (see eq. (4.14)) τ ?1 (Λ) = n1 Λτ ?1 (1) + . . . + n? Λτ ?1 (?) = nτ (1) Λ1 + . . . + nτ (?) Λ? (4.17)

if Λ = n1 Λ1 + . . . + n? Λ? is the highest weight of D . Now we can list the simple complex Lie algebras with non–trivial diagram symmetries and give the conditions for DΛ ? DΛ ? ψd where we indicate the highest weight as a subscript. In all cases except D4 the automorphism is unique. Thus we have DΛ ? DΛ ? ψd if and only if nj = n?+1?j j = 1, 2, . . . , ? for A? (? ≥ 2) n??1 = n? n1 = n5 , n2 = n4 for D? (? ≥ 5) for E6 . (4.18)

For D4 there are ?ve non–trivial diagram automorphisms ψτ and DΛ ? DΛ ? ψτ only if n1 = nτ (1) , n3 = nτ (3) , n4 = nτ (4) . (4.19) In subsect. 4.2 we have also de?ned the automorphism ψ △ associated with root re?exT ion. Clearly, DΛ ? ψ △ has the weight ?λ if λ is a weight of DΛ . Therefore DΛ ? ψ △ ? ?DΛ △ and ψ relates a rep to its contragredient rep. It is outer for A? (? ≥ 2), D? (? = 5, 7, 9, . . .) T is given and E6 . In these cases DΛ ? ψ △ ? DΛ ? ψd and thus the condition for DΛ ? ?DΛ △ T by eq. (4.18). For all other simple Lie algebras ψ is inner and DΛ ? ?DΛ for all their irreps (see table 1).

5
5.1

The existence of CP in gauge theories and Condition B
The automorphism associated with CP

In this section we will show that the fermionic and vector boson sectors of gauge theories (the terms LF and LG of the Lagrangian) are always CP symmetric. The same is true for

21 the couplings LH of scalars with the gauge ?elds (see sect. 7). Imposing CP symmetry to the full Lagrangian L containing the terms LG , LF , LH , LY , the Yukawa interactions, and LS , the scalar potential, will induce conditions on the only terms that may violate this symmetry — LY and LS . In sect. 8 we will discuss the conditions induced on LY — the simplest will be reality of Yukawa couplings but in general the situation is more complicated (see full classi?cation in subsect. 8.3). We will not discuss invariance of LS in this paper. In our formulation a CP transformation of the fermionic and gauge boson multiplets is completely speci?ed by the two matrices R and U de?ned in sect. 3. Requiring that CP reverses all quantum numbers of each ?eld we see that RCP corresponds to an automorphism which induces a re?exion of all the roots or, in other words, acts as h → ?h on the CSA (as required by its physical interpretation). Hence we identify RCP with the matrix representation of the contragredient automorphism de?ned by eq. (4.5) [7, 8]: ψCP ≡ ψ △ . (5.1)

In what concerns the pure gauge term LG of the Lagrangian we can conclude from the derivation of Condition A that it is always invariant under any automorphism of the Lie algebra irrespective of whether this is an inner or outer automorphism, contragredient or not. One might think that identifying ψCP with ψ △ is too restrictive. This is, however, not the case because any two automorphisms that act in the same way on the CSA will only di?er by an inner automorphism generated by an element of the CSA itself [24]. As we shall see in subsect. 6.1 using ψCP instead of ψ △ leads to equivalent conditions for CP invariance.

5.2

Canonical and generalized CP transformations

We consider now how to de?ne a CP transformation on the fermion ?elds. In sect. 3, eq. (3.19) shows how the representation {Ta } can be decomposed into irreps Dr . We know from the discussion of sect. 4 that
T ? Dr ? ψ △ ? Dr

(5.2)

because the weights of both irreps are the same. This follows immediately from ψ △ (hα ) = h?α = ?hα . Therefore there exist unitary matrices Vr such that
T Vr (?Dr ? ψ △ )Vr? = Dr

(5.3)

? r.

(5.4)

Thus we de?ne a “canonical CP transformation” by [7, 8] (R△ , U0 ) with ψR△ ≡ ψ △ , U0 =
r

1mr ? Vr .

(5.5)

22 Since (R△ , U0 ) solves Conditions A and B we have a symmetry of LG + LF . This means that a gauge theory without Yukawa interactions and a scalar potential is automatically CP invariant (the scalar couplings to the gauge ?elds, LH , are treated like the fermionic couplings in sect. 7). As already mentioned the physical reason to call the above transformation CP is that it turns around the signs of all members of the CSA in any rep. Thus it transforms ?elds into those with opposite quantum numbers. In order to see that this is indeed the case let U and Dr (g ) be the operator implementations of CP and Dr (g ), respectively, on the Hilbert space of states and χR be a second quantized irreducible ?eld multiplet. If CP is conserved we have U χR (x)U ?1 = ?Vr CχR (x)? and Dr (g )χR (x)Dr (g )?1 = Dr (g )χR (x) (5.6)

and, together with eq. (5.4), we obtain Dr (g )U χR (x)U ?1 Dr (g )?1 = ?Vr Dr (g )? CχR (x)? = (Dr ? ψ △ )(g )(?Vr CχR (x)? ). (5.7)

As we know, in the irrep Dr ? ψ △ the CSA is represented with opposite signs compared to Dr thus verifying the above statement. Theorem I (sect. 3) tells us that given R = R△ the most general solution of Condition B is obtained by U = U1 U0 where U1 is a horizontal transformation: U1 =
r

ur ? 1dr .

(5.8)

In the following we will call such transformations associated with (R△ , U1 U0 ) “generalized CP transformations” (see refs. [10, 27] for the context of left–right symmetric models and refs. [28, 29] for general considerations). Considering the example of QCD in sect. 2 it is obvious that U0 can be regarded as the identity matrix. Clearly this result can be generalized to the de?ning rep of SU (N ) for arbitrary N by generalizing the Gell–Mann matrices. The reason for U0 = 1 is that these matrices are either symmetric or antisymmetric, therefore ?λT a = ηa λa and the signs ηa a a are compensated by the transformation of the gauge bosons W? → ηa W? . Thus we have △ R = diag (η1 , . . . , ηnG ). It turns out that this is a general feature for any irrep of any semisimple compact Lie group if we choose the standard basis of Lc (see eq. (B.23)) and a suitable basis in representation space. To be speci?c, in app. F the following theorem is proved. Theorem: For every irrep D of LC there is an ON basis of Cd (d = dim D ) such that D (Xa )T = ?ηa D (Xa ),
2 =1 ηa

(a = 1, . . . , nG )

(5.9)

for the antihermitian generators of LC in D . Furthermore the generators {Xa } are those of the compact normal form of LC and therefore the root re?exion ψ △ is given by ψ △ (Xa ) = ηa Xa . (5.10)

23 In other words, the matrices D (Xa ) are either imaginary and symmetric (ηa = ?1) or real and antisymmetric (ηa = 1) and therefore a generalization of ?iσa /2 (Pauli matrices) for SU (2) and ?iλa /2 (Gell–Mann matrices) for SU (3) to arbitrary irreps of semisimple compact Lie algebras. The above basis will be called “CP basis” in the following. In this basis a canonical CP transformation, (CP)0 , can be simply represented by To be more explicit (see app. F) we have the following situation in a CP basis: D (?iHj ) (j = 1, . . . , ?), (imaginary and symmetric) D (real and antisymmetric). Therefore we get 1 e +e e ?e √ D α √ ?α + iD α √ ?α = D (eα ) real ? α ∈ ?. (5.13) 2 2 i 2 These results will be of importance when we consider the Yukawa couplings in sect. 8. As a conclusion we have seen that a fermionic gauge theory (without scalars) is always CP invariant. The pure gauge term by itself is always invariant under any automorphism of the Lie algebra. In what concerns LF , the reason for invariance is due to the fact that the fermionic multiplet transforms in such a way that the CP transformed fermion ?elds are associated to the complex conjugate rep whilst the same transformation in the gauge boson sector will reverse this e?ect through R△ (this is obvious from the derivation of Condition B in sect. 3). This is the reason why invariance under a canonical CP transformation is veri?ed irrespective of whether the automorphism is inner or outer. In all the examples of sect. 2, QED, QCD and the SO (10)–GUT, ψ △ is outer (see table 1). eα + e?α √ 2 (α ∈ ?) : ηa = 1 (5.12) D eα ? e?α √ i 2 (α ∈ ?) : ηa = ?1 (CP )0 → (R△ , 1) with R△ = diag (η1 , . . . , ηnG ). (5.11)

6
6.1

On the general solution of Condition B
Introduction

In sect. 5 we discussed the solution of Condition B together with the requirement that R be such that it represents a CP transformation. Yet as we have seen in the examples of sect. 2 the form of CP and P transformations is the same in a formalism with fermion ?elds of a single chirality so that the di?erence between CP and P lies in the properties of R and U . This justi?es that we study now Condition B with a general R. Therefore in this section we will discuss invariance of LF , the fermionic part of the Lagrangian of a gauge theory, under a CP–type transformation without constraining R to be the contragredient automorphism. The following theorem will simplify our discussion.

24 Theorem II: Let (R, U ) be a pair of matrices that verify Conditions A and B and let RI represent an inner automorphism of Lc . Then the pairs (RI R, eiya Ta U ) where ψRI (X ) = e?ya Xa Xeya Xa are also solutions of Conditions A and B. Proof: RI R and RRI are solutions of Condition A because the set of automorphisms form a group. Since ψRI is inner there are real numbers ya (a = 1, . . . , nG ) such that eq. (6.2) is ful?lled. Translating eq. (6.2) into the fermion representation we get eiyc Tc Ta e?iyc Tc = RIba Tb . Writing Condition B in terms of the second pair of eq. (6.1) we get U (eiyc Tc )? (?TbT (RRI )ab )(e?iyc Tc )? U ? = = ?U (eiyc Tc Tb e?iyc Tc (RRI )ab )? U ? = = U (?TcT RIcb (RRI )ab )U ? = U (?TbT Rab )U ? = Ta . The other pair can be dealt with in an analogous way. 2 Theorem II means for the discussion of invariance of LF under CP–type transformations that if for a given automorphism there is a solution of Condition B (i.e., there is a matrix U that veri?es the equality) then for any other automorphism di?ering from this one by an inner automorphism there will also be a solution of Condition B. The theorem also tells us how to relate the new matrix U to the initial one. In mathematical terms this means that it is only the quotient group Aut (Lc )/Int (Lc ) that needs to be considered so that we can con?ne ourselves to a particular representative of each coset. Let us consider now the simple Lie algebras Lc in the light of Theorem II and distinguish three classes of Lie algebras (see table 1) [7]: a) Lc = su(2), so(2? + 1) (? ≥ 2), sp(2?) (? ≥ 3), cE7 , cE8 , cF4 , cG2 (6.3) (6.2) and (RRI , U (eiya Ta )? ) (6.1)

Here no outer automorphisms exist and we take ψR = id as representative. Since ψ △ is inner ?D T ? D is valid for all irreps D and therefore there is a matrix W such that W (?D T )W ? = D. (6.4)

Consequently Condition B is solvable without restrictions on the irrep content of {Ta }. b) Lc = su(? + 1) (? ≥ 2), so(2?) (? = 5, 7, 9, . . .), cE6 Here we have both inner and outer automorphisms. For the inner automorphisms we can consider again ψR = id as representative, for the outer automorphisms we

25 can choose the contragredient one, ψR = ψ △ , since in this case ψ △ is outer. In these Lie algebras any outer automorphism will be a composition of ψ △ with an inner automorphism ψR = id: In general an irrep is not equivalent to its complex conjugate. As a result Condition B requires that for any irrep D also its complex conjugate ?D T must be contained in {Ta }.

ψR = ψ △ : Choosing a basis where the CSA in the irrep D is diagonal it is clear that the weights of ?D T ? ψ △ are identical with those of D . Therefore ?D T ? ψ △ ? D and Condition B is solvable without restriction on {Ta }. c) Lc = so(2?) (? = 4, 6, 8, . . .)

In this case ψ △ is an inner automorphism, therefore all irreps D are equivalent to ?D T . Hence it is appropriate to represent Aut (Lc ) by the two cases ψR = id and ψR = ψd , the unique diagram automorphism for ? = 6, 8, 10, . . .. As we know for so(8) (? = 4) there are ?ve non–trivial diagram automorphisms. ψR = id: Analogously to case a) Condition B is solvable without restriction on {Ta }. ψR = ψd : Condition B is solvable only if for any irrep D also D ? ψd is contained in {Ta }. The relationship between D and D ? ψd was discussed in subsect. 4.3.

We may also consider non–simple Lie groups in the light of Theorem II. For that purpose we take into account the results of sect. 4 and split the discussion into three generic classes. i) Lc = L′c ⊕ u(1), L′c simple: We know that there is the automorphism ψu : Xu → ?Xu eq. (4.12) on u(1) in addition to Aut (L′c ). Thus we can represent Aut (Lc )/Int (Lc ) u by ψ = (ψ ′ , idu ) or (ψ ′ , ψu ) where ψ ′ ∈ {id, ψ △ , ψd } ? Aut (L′c ). Each irrep Dk of ′ u(1) is characterized by the generator ik , k ∈ Z. Denoting by D an irrep of L′c u u then the irreps of Lc are given by D ′ ? Dk . For ψ = (ψ ′ , idu ) also (?D ′ T ? ψ ′ ) ? D? k must be contained in {Ta } whereas for ψ = (ψ ′ , ψu ) it is su?cient to consider L′c .
′ ′′ ii) Lc = L′c ⊕ L′′ c , Lc , Lc simple: This case leads to the discussion of each simple summand. Note that the irreps of Lc are given by the tensor products of the irreps of each summand [30] as for i). One should remember that here we have in mind ′ ? ′′ di?erent gauge coupling constants for L′c and L′′ c even for Lc = Lc and therefore no ′′ ′ automorphisms other than Aut (Lc ) × Aut (Lc ) exist.

iii) Lc = L(G? ) with G? de?ned in subsect. 4.2: The only additional automorphism is ψE given by eq. (4.9). It is easy to check that D ? ψE = D (E )DD (E ) (6.5)

± for all irreps D = Dr , Dr,r′ . Consequently ψE is as good as inner and one only has to consider a single summand.

26

6.2

On the de?nition of parity

In the de?nition of a CP transformation we had the automorphism ψ △ (changing the sign of all elements of the CSA H) with (ψ △ )2 = id. This suggests a mathematical de?nition of 2 a parity transformation via an involutive automorphism ψP (i.e., verifying ψP = id) which maps the CSA into itself and does not change the sign of the whole algebra. Therefore we can ?nd an ON basis of H where [7] ψP (Hj ) = ?j Hj , ?j = 1 , j = 1, . . . , p ?1, j = p + 1, . . . , ?. (6.6)

From the discussion in subsect. 6.1 it is obvious that if parity is a symmetry of the theory the matrix UP associated with this transformation either maps a given irrep into itself or connects the irrep D to (?D T ) ? ψP ? D ? ψ △ ? ψP . Both irreps have to be present in {Ta } with the same multiplicity to ful?ll Condition B. In the ?rst case we call such a parity transformation internal with respect to D and in the second external. In sect. 2 we had the external cases of QED and QCD and the {16} of so(10) as an internal case. Let us for the time being consider the internal case with a single irrep D with dimension d. Then we can de?ne a mapping: f : x → UP x? Cd → Cd .

(6.7)

Given an ON basis e(λ, q ) of weight vectors (see app. B) in the Hilbert space Cd associated with the irrep D , where q = 1, . . . , m(λ) and m(λ) denotes the multiplicity of λ, we have D (Hj )[f (e(λ, q ))] = ?UP D (ψP (Hj ))? e(λ, q )? = ??j λ(Hj )f (e(λ, q )). (6.8)

To derive this relation we have used eq. (6.6) together with the fact that D (Hj ) is hermitian and that Condition B can be rewritten as
? ? UP [D (ψP (iX ))]? UP = D (iX )

? X ∈ Lc .

(6.9)

As a result eq. (6.8) shows that f relates states characterized in general by di?erent weights λ and λP given by λ ? (λ(H1 ), . . . , λ(Hp ), λ(Hp+1), . . . , λ(H? )), λP ? (?λ(H1 ), . . . , ?λ(Hp ), λ(Hp+1), . . . , λ(H? )). (6.10)

Thus weight vectors associated with λ are transformed into weight vectors associated with λP . Only vectors with λ(H1 ) = . . . = λ(Hp ) = 0 have no partner. Since D (Hj ) is hermitian e(λ, q ) is orthogonal to e(λP , q ) for λ = λP . Obviously the contragredient automorphism ψ △ does not establish such a relationship between weight vectors because for ψP = ψ △ we would have λ = λP for all weights. In the external case restricting oneself to single copies of D and ?D T ? ψP , respectively, the above discussion goes through without change, but now e(λ, q ) and e(λP , q ) lie in

27 di?erent spaces, namely in the spaces associated with D and ?D T ? ψP , respectively. Taking into account non–trivial multiplicities in both cases, internal and external, does not alter the essence of our discussion but for simplicity we stick to single copies of irreps for the rest of this section. We can make an analysis which has some resemblance to the one for CP in eqs. (5.6) and (5.7). As just before it will be valid for internal and external parity. De?ning
λ,q ωR (x) ≡ e(λ, q )? ωR (x)

(6.11)

we get and

λ,q λ,q D (e?isHj )ωR D (eisHj ) = e?isλ(Hj ) ωR λP ,q λP ,q D (e?isHj )ωR D (eisHj ) = e?isλP (Hj ) ωR λ,q λP ,q T ωL (x) ≡ Cγ0 (ω R (x))?

This allows to de?ne

? s ∈ R.

(6.12) (6.13)

λ,q for λ = λP which has the same quantum numbers λ(Hj ) (j = 1, . . . , p) as ωR (x). Then P transforms right into left and vice versa: λ,q ?1 UP ωR (x)UP = ?e(λ, q )? UP CωR (x)? λP ,q ? ? = ?(UP e(λ, q ))? UP UP CωR (x)? = ?eiδ C (ωR (x))? λ,q (x ) = eiδ γ0 ωL

and, similarly,

To derive this equation we have taken into account eq. (6.8) and
? UP UP = e?iδ 1d

λ,q λ,q ?1 UP ωL (x)UP = ?γ0 ωR (x).

(6.14)

or

eiδ 1d 0 ?iδ 0 e 1d

(6.15)

for internal and external parity, respectively2 . Eq. (6.15) can be cast into a more general form: If (R, U ) is a CP–type transformation with R2 = 1 then [UU ? , Ta ] = 0 Proof:
? ? ? ? U ) U = ?UTb? Rba U ? = U (?TbT Rab )U ? = Ta . (UU ? )Ta (UU ? )? = U (UTa

? a = 1, . . . , nG .

2 In contrast to CP there is no canonical way to de?ne P in general. With eq. (6.6) a mathematical de?nition was given but what is considered as parity in a model also
2 ? δ is the phase of UP UP in D or ?DT ? ψP for the internal or external case, respectively.

28 depends on the physical situation. A plausible requirement would be that the signs of the electromagnetic charge and the two colour charges are not changed under ψP . For a deeper connection of P and C with group theory the reader is referred to the discussion of charge conjugation in ref. [7] (see also sect. 9). The external case of parity with ψP = id, D ? D ? is the type of parity we are used to (QED, QCD). In a basis where the matrices of D ? are just the complex conjugate matrices of D we simply form the Dirac ?eld χD ≡ χR +(χ′R )c where χR and χ′R transform according to D and D ? , respectively. In the so(10) example (sect. 2) ψP is an inner automorphism.

7
7.1

CP–type transformations in the Higgs sector
Pseudoreal scalars and CP transformations

Considering a pseudoreal irrep D [16] there are two unitary matrices associated with it which are important in our context. The ?rst one is given by the equivalence of the irrep and its complex conjugate. In Lie algebra form it is written as W (?D T )W ? = D, W T = ?W (7.1)

where antisymmetry of W follows from pseudoreality [16, 31]. The other unitary matrix is associated with the (CP)0 symmetry: U (?D T ? ψ △ )U ? = D. The matrix U is symmetric: U T = U. (7.3) (7.2)

Proof: The symmetry of U is actually contained in the proof of app. F. There it is ?rst demonstrated that U T = ±U . Then antisymmetry is excluded for irreps of semisimple compact Lie algebras. 2 Eq. (7.3) is, of course, valid for arbitrary irreps. Clearly, in a CP basis U ? 1 and part of the discussion here would be super?uous. However, we think that it is very instructive not to restrict our discussion to a CP basis but instead to have the general setting. The matrix W allows to de?ne φ ≡ W φ? with φ = ?φ (7.4)

which transforms in exactly the same way as the scalar multiplet φ under the gauge group. It is more di?cult to see that both ?elds also transform alike under CP: CP : φ → Uφ? , φ → U φ? . (7.5)

This follows from the fact that W and U are related by W U ? W = ?U with an appropriate choice of phase, e.g., for W . (7.6)

29 Proof of eq. (7.6): Putting ψ △ on the right–hand side of eq. (7.2) and using eq. (7.1) one obtains U (?D T )U ? = (UW ? )D (UW ? )? = D ? ψ △ . [(UW ? )2 , D ] = 0 and thus (UW ? )2 = ?1.

Taking advantage of (ψ △ )2 = id one further derives

Choosing the phase of W such that ? = ?1 and making some algebraic manipulations one arrives at eq. (7.6). 2 Proof of eq. (7.5): Under CP φ transforms as where in the last step eq. (7.6) has been taken into account. 2 Let us assume now that there are m copies φ1 , . . . , φm all in the same irrep D . Putting the m multiplets into a vector χ then LH , the scalar kinetic and gauge Lagrangian, can be rewritten as [9, 10] 1 1 LH (χ) = LH (χ) + LH (χ). (7.7) 2 2 This allows for a wider class of generalized CP transformations for pseudoreal scalars leaving LH invariant de?ned by CP : χ χ → HU χ χ
?

φ → W U ? φ = W U ? W T φ? = ?W U ? W φ? = U φ?

(7.8)

where H is a unitary “horizontal” 2m × 2m matrix. One has to take into account, however, that χ is related to χ via de?nition (7.4) which gives a restriction on H . With H= A B C D (7.9)

this restriction is derived by the requirement W (AUχ? + BU χ? )? = CUχ? + DU χ? . (7.10) Exploiting this condition by using eq. (7.6) in the form W U ? = UW ? we ?nally obtain D = A? , C = ?B ? and therefore H= A B ? ?B A? ∈ Sp(2m) (7.11)

where Sp(2m) = {H ∈ U (2m)| H T Jm H = Jm } is the unitary symplectic group [16]. Jm is de?ned by 0 1m Jm = . ?1m 0 In a CP basis one can say more about the matrices U and W . With the phase convention of eq. (7.6) we get the following result: where U = 1 is the convention of the (CP)0 transformation in the CP basis. CP basis ? U = 1, W = W ? = ?W T (7.12)

30

7.2

Real scalars in complex disguise and CP transformations

Scalar multiplets belonging to potentially real irreps D [16] can also be represented by complex ?elds. In general, in a CP basis, matrices of a potentially real irrep are complex. When these matrices are rotated into a real rep the scalar multiplets are also transformed and there are two possibilities – either the scalar multiplets also become real or else they remain complex in which case they can be split into two real multiplets belonging to the same real irrep. The second case will be illustrated by a familiar example in the gauge theory SU (2)L × SU (2)R × U (1)B?L . The treatment of two real multiplets collected in a complex multiplet φ can proceed along the same lines as in the case of pseudoreal scalars by the introduction of the corresponding ?eld φ, instead of splitting it into its real components. When φ represents a single real irrep it coincides with φ in the real basis so that its introduction in this case would be meaningless. For real scalars one has now three unitary matrices associated with the irrep. U is given as in eq. (7.2), W is now symmetric [16, 31] W (?D T )W ? = D, D = V DR V ? . WT = W (7.13)

and V transforms DR , an explicitly real realization of the potentially real rep D , into D : (7.14)

Let us ?rst have a look on the relationship between the (CP)0 transformation in DR and D . To do this we note that W U ?W = U (7.15) and V ?W V ? = 1 with appropriate phase factors for W and V . Proof: Eq. (7.15) is proved as before in the pseudoreal case but now we choose a phase of W such that we have a plus sign instead of minus in eq. (7.6). The reason for this choice will become clear when we later go to a CP basis. The second relation derives from inserting eq. (7.14) into eq. (7.13) giving W V ? DR V T W ? = V DR V ? or [V ? W V ? , DR ] = 0. With Schur’s lemma and a phase choice for V we obtain eq. (7.16). 2 ? Let ? = V φ be the real scalar ?eld. Then the CP transformation on ? is given by CP : ? → V ? UV ? ? and V ? UV ? real. (7.18) (7.17) (7.16)

31 Proof: To prove eq. (7.18) we rewrite eq. (7.15) as U ?W = W ?U Therefore (V ? UV ? )? = V T (U ? V V T )V ? = V T (V ? V ? U )V ? = V ? UV ? . 2 Thus the general CP formalism is consistent with real ?elds. Choosing the CP basis we now have CP basis ? U = 1, W = W? = WT. (7.19) As before W is real. This is a consequence of the phase choice in eq. (7.15). Now we come to the second topic, namely to the discussion of a complex multiplet comprising two real multiplets transforming under the same irrep. As in the pseudoreal case we de?ne φ ≡ W φ? with φ = φ. (7.20) Note that now we have a plus sign in the second relation (compare to eq. (7.4)) because W W ? = 1 for real irreps. As before φ and φ transform alike under CP, eq. (7.5). If D has multiplicity m we can form the vectors χ, χ. Then eq. (7.7) is valid and CP can be de?ned as in eq. (7.8). But now the restriction on H di?ers from eq. (7.11). Imposing the same condition as in the pseudoreal case (see eq. (7.10)) but taking into account the properties of U and W in the real case we obtain [10] H= A B B ? A? . (7.21) or U ? V V T = V ? V ? U.

It is easy to check that the matrices H of the form eq. (7.21) are exactly those unitary matrices for which 0 1 H T IH = I with I= (7.22) 1 0 is valid. Therefore they form a group. This group is identical with O (2m) up to the basis transformation Z ? HZ = H ′ ∈ O (2m) with 1 Z=√ 2 1 i1 1 ?i1 . (7.23)

This can readily be seen by inserting eq. (7.23) into eq. (7.22) where we obtain H ′ T (Z T IZ )H ′ = Z T IZ = 12m . De?ning real ?elds χ1 , χ2 via V ?χ = χ1 + iχ2 √ 2 (7.24)

(7.25)

32 the CP transformation (7.8) for the potentially real irrep D is now of the form CP : χ1 χ2 → (Z ? HZ )(V ? UV ? ) χ1 ?χ2 (7.26)

after some calculation. Both matrix products embraced by the parentheses are real. This again shows the consistency of the formalism. Clearly, both φ and φ have to be coupled in the Yukawa sector to get the most general interaction. It may look strange that real scalars are packed together in complex ?elds. But we will see now by an example from SU (2)L × SU (2)R × U (1)B?L [13, 32, 33] that such cases are not uncommon. There a multiplet φm transforming as (2,2,0) exists which gives masses to the quarks. It is commonly written as a 2 × 2 matrix of complex ?elds and transforms as φ11 φ12 ? , UL,R ∈ SU (2) (7.27) φm = → UL φm UR φ21 φ22 under the gauge group. The index m denotes the 2 × 2 matrix version of the ?eld. φm is given by ? φ? 22 ?φ21 (7.28) φm = τ2 φ? ? ? m τ2 = ?φ12 φ11

where τ2 is the second Pauli matrix. Switching to a vector notation we read o? W from eq. (7.27):
? ? ? ?

φ≡?

φ11 φ21 φ12 φ22

?

? ? ?, ?

φ = W φ?

with

W =?

? ? ? ?

0 0 0 0 0 ?1 0 ?1 0 1 0 0

1 0 0 0

?

? ? ?. ?

(7.29)

Since W T = W we suspect immediately that (2,2,0) is a real rep. This can indeed be con?rmed by recalling the following theorem. Theorem: The irreps of a direct product of groups G × G′ are exactly the tensor products of irreps D and D ′ of G and G′ , respectively [30]. If D and D ′ are both real or both pseudoreal then D ? D ′ is real [34]. In our case the de?ning irrep of SU (2) is pseudoreal having spin j = 1/2 and therefore the rep (7.27) actually decays into two real irreps. Since this is an instructive example we want to show this explicitly. With UL = a b ?b? a? , |a|2 + |b|2 = 1 and UR = c d ?d? c? , |c|2 + |d|2 = 1 (7.30)

we ?nd the corresponding transformations for the vector φ UL φm → UL 0 0 UL φ ≡ UL φ,

33
? ? ? ?

? φm UR → ?

c? 0 d? 0 c? 0 ?d 0 c 0 ?d 0

0 d? 0 c

?

? ? ?φ ?

≡ UR φ.

(7.31)

? Of course, now we have W UL,R W ? = UL,R . If we perform a basis transformation we get ′ φ = Zφ′ ? UL,R = Z ? UL,R Z

and

W ′ = Z ?W Z ?.

(7.32)

The last relation can easily be obtained from eq. (7.13). It is obvious that in the basis where W ′ = 1 we have real representation matrices. With 1 ? ? V ≡Z= √ ? 2? we have indeed W ′ = 1 and
′ ? UL → ?

?

1 i 0 0 0 0 ?i 1 0 0 ?i ?1 1 ?i 0 0
?

? ? ? ? ?

(7.33)

? ? ? ? ? ? ?

a1 ?a2 b2 a2 a1 ?b1 ?b2 b1 a1 ?b1 ?b2 ?a2

b1 b2 a2 a1

′ UR → ?

c1 c2 ?d2 ?d1 ?c2 c1 ?d1 d2 ? ? ?, d2 d1 c1 c2 ? d1 ?d2 ?c2 c1

? ? ?, ?

a1 = Re a, a2 = Im a b1 = Re b, b2 = Im b

?

c1 = Re c, c2 = Im c d1 = Re d, d2 = Im d.

(7.34)

Clearly, the new ?eld multiplet φ′ (φ′ = φ′ ? ) is still complex and we can split it into two real multiplets as in eq. (7.25). This concludes our discussion of real scalars. The left–right symmetric example will be taken up once more in sect. 9 where CP will be discussed together with C and P in this context.

7.3

The general case

In the general case the scalar multiplets may contain real, pseudoreal and complex irreps. We put them together into a vector
? ? ? ? ? ? ? ?

Φ=

φR φP φP φC φ? C

? ? ? ? ? ? ? ?

and

a D? Φ = (?? + ig Ta W? )Φ

(7.35)

34 where the indices R, P and C denote real, pseudoreal and complex, respectively. D? is the covariant derivative. Therefore we can write the scalar kinetic and gauge Lagrangian as 1 LH = (D? Φ)? (D ? Φ). (7.36) 2 In this way we summarize the gauge interactions into one formula with automatically correct factors 1/2 for φR and 1 for φP and φC since the contribution of φP and φP are equal in LH (see eq. (7.7)) and the same holds true for φC and φ? C (of course, if in the C ? C T φC sector Ta is given by Ta then for φC it has to be ?(Ta ) ). In analogy to the fermionic sector we de?ne a CP–type transformation by Φ(x) → UH Φ(x)? leading to
? = Ta . Condition BH : UH (?TbT Rab )UH

(7.37)

(7.38)

Clearly, Condition BH is connected to Condition B by the same automorphism represented by R. For the total Lagrangian a CP–type transformation is characterized by the triple (R, U, UH ). It is clear from eq. (7.37) that it is always possible to de?ne a CP transformation in LH in the same way as is done in the fermionic sector. Therefore as for CP–type transformations it remains to consider the automorphisms ψR = id and ψR = ψd in the relevant cases. When considering the algebras su(? + 1) (? ≥ 2), so(2?) (? = 5, 7, 9, . . .) and cE6 in the general discussion of a CP–type transformation the automorphism ψR = id is relevant in addition to ψ △ . Obviously, if we consider the scalars φC in complex irreps we automatically have their complex conjugate irreps in in LH eq. (7.35) so that Condition BH can be solved for ψR = id and complex reps. For real irreps and ψR = id Condition BH becomes trivial because in this case ?D T = D in a basis where the ?elds are real. In the case of pseudoreal irreps and ψR = id the CP–type transformation has to be de?ned as φ → W φ? with W speci?ed by eq. (7.1). A simple calculation reveals that now eqs. (7.5), (7.8) and (7.11) are valid with U replaced by W . This shows that for all simple algebras except so(2?) with ? = 4, 6, 8, . . . (see table 1) arbitrary CP–type transformations are always symmetries of LH and in this sense the existence of a CP–type symmetry for LH is more likely than for LF . For the simple Lie algebras so(2?) with ? = 4, 6, 8, . . . the relevant case is ψR = ψd , T a diagram automorphism. In this case we have (?DΛ ) ? ψd ? DΛ ? ψd ? DΛ if and only if n??1 = n? for the highest weight Λ (? ≥ 6) (see subsect. 4.3). In this special case a CP–type symmetry may not exist for LH since here it requires that both irreps DΛ and DΛ ? ψd occur in Φ. The more complicated structure of UH in this case and also for similar cases in so(8) is easily worked out with the methods established in this paper. If T 2 (?DΛ ) ? ψd ? DΛ and ψd = id again the considerations in subsect. 7.1 can be used with △ ψ replaced by ψd .

35 Finally, we want to mention that basis transformations are performed as in the fermionic case eqs. (3.25) and (3.26) but attention has to be paid to two points. In the real case the horizontal part of UH is real apart from situations discussed in subsect. 7.2. Therefore the corresponding basis transformations have to be performed by orthogonal matrices. Similarly, in the pseudoreal case the transformation matrix must be an element of Sp(2m). These points will be of signi?cance in subsect. 8.3.

8
8.1

Yukawa couplings and CP–type symmetries
The condition on Yukawa couplings

With the vector of scalar ?eld Φ as de?ned in eq. (7.35) and all fermionic degrees of freedom in a right–handed vector ωR the most general form of Yukawa couplings is given by 1 T ?1 C Γj ωR Φj + h.c. (8.1) LY = iωR 2 where Fermi statistics implies ΓT j = Γj ? j . A CP–type transformation characterized by (R, U, UH ) leads to Condition C:
H U T Γk UUkj = Γ? j

?j

(8.2)

1 if we require invariance of LY . As a matter of convention we have pulled out the factor 2 i from the coupling matrices Γj . The i gives plus signs on both sides of eq. (8.2) whereas ?1 ?1 the factor 1/2 is motivated by the fact that χT Γj χ′R = χ′ T Γj χR , i.e. terms with RC RC ′ di?erent multiplets χR , χR occurring in ωR appear twice in LY , eq. (8.1). Before we continue the discussion of Condition C we want to make a few general remarks on CP–type symmetries which we de?ne as those CP–type transformations which ful?ll simultaneously Conditions A, B, BH and C. First we have the following statements: T (R, U, UH ) CP–type symmetry ?? (RT , U T , UH ) CP–type symmetry.

(8.3)

If (Ri , Ui , UiH ) (i = 1, 2) are CP–type symmetries then
? ωR (x) → ?U1 U2 ωR (x) H H? Φ(x) → U1 U2 Φ(x) a b W? (x) → (R1 R2 )ab W? (x)

(8.4)

is a symmetry of LG + LF + LH + LY . We will see in sect. 9 that eq. (8.4) is a C–type symmetry. As mentioned in the introduction we will not discuss the Higgs potential the inclusion of which would give conditions additional to Conditions A – C. It is clear that theorem II also applies to Condition C. The coupling matrices Γj have two types of indices since they have to tie together irreps and their multiplicities. Therefore, considering matrices which couple irreps Dr ,

36 Dr′ and Drφ with multiplicities mr , mr′ and mrφ , respectively, where Dr , Dr′ occur in ωR and Drφ in Φ we can write Γj as a tensor product
φ Γj = ((Γa ? γ k )rr ′)

r

(8.5)

with j corresponding to the triple (a, k, rφ ). Note that a = 1, . . . , mrφ , k = 1, . . . , drφ , rφ a rφ (γ k )rr ′ are dr × dr ′ matrices and (Γ )rr ′ mr × mr ′ matrices. Of course, for pseudoreal scalars there are separate coupling matrices for χ and χ. Clearly, the individual factors of the coupling matrices (8.5) do not have to be symmetric, only the total matrix has to rφ couples the same fermion ?elds to each other and be. For r = r ′ , the diagonal of (Γa )rr therefore the irrep Drφ has to be contained in the symmetric tensor product (Dr ? Dr )sym . In the further discussion we will con?ne ourselves to generalized CP invariance and use the following approach which we think is most appropriate in the context of this work. We know already from sect. 5 that CP can be de?ned in a canonical way for each irrep in ωR and Φ, respectively, and that a general CP symmetry is composed of the canonical CP transformation (CP)0 eq. (5.5) followed by a horizontal unitary rotation (5.8). Therefore we adopt the strategy that out of the set of CP symmetries of LG + LF + LH we have given a particular one which then imposes Condition C on the Yukawa couplings, i.e., in our strategy the symmetry is primary and determines the Lagrangian. We will see in subsect. 8.2 that in the CP basis the Clebsch–Gordan coe?cients are rφ real. Therefore, (γ k )rr ′ , the group–theoretical part of Γj , couples the ?elds in a (CP)0 – invariant way and it remains a purely horizontal condition which will be solved in general in subsect. 8.3.

8.2

Real Clebsch–Gordan coe?cients and the generalized CP condition

It is easy to see that in a tensor product of two irreps D , D ′ the Clebsch–Gordan coe?cients can be chosen real if we take the tensor product of the respective CP bases. One only has to remember the general procedure for deducing the Clebsch–Gordan series. We additionally imagine that we are in a CP basis where all D (Hj ) are diagonal. This is possible because the representation matrices of the CSA can simultaneously be diagonalized by an orthogonal matrix (D (Hj ) is symmetric and real) without disturbing symmetry or antisymmetry of the D (Xa ) (a = 1, . . . , nG ). Therefore the basis vectors of D ? D ′ given by {ei ? e′j | i = 1, . . . , d; j = 1, . . . , d′ } are all eigenvectors of D (Hj ) ? 1d′ + 1d ? D ′ (Hj ) ′ (ei , e′j are the canonical basis vectors of Cd , Cd , respectively). We can e.g. choose e1 ? e′1 to be the unique vector with highest weight Λ + Λ′ in D ? D ′ , if Λ, Λ′ are the highest weights of D , D ′ , respectively. Applying all D (e?α ), α ∈ ? or, equivalently, all D (e?αj ) (j = 1, . . . , ?) with αj simple to e1 ? e′1 gives the representation space associated with Λ + Λ′ . Because of eq. (5.13) the basis of this space is real and therefore also the basis of its orthogonal complement can be chosen real. The highest weight in the orthogonal complement can have multiplicity larger than one. We choose a (real) basis vector and make the same procedure as before. We continue along these lines until we have ex′ hausted the whole space Cd ? Cd [16]. Since all D (e?α ) are real and we started with a

37 real basis we ?nally arrived at a real basis in the Clebsch–Gordan series. Therefore the Clebsch–Gordan coe?cients associated with this basis are real. For the rest of this section we assume that we consider ?xed irreps Dr , Dr′ , Drφ with ?eld multiplets χR , χ′R , φ, respectively. Then one can readily check that
′ ′? ? (CP )0 : χR → ?Cχ? R , χR → ?Cχ R , φ → φ

(8.6) (8.7)

is a symmetry of
?1 k ′ iχT γ χR φk + h.c. RC

for real Clebsch–Gordan matrices γ k since the invariance condition is just
?1 k ′ kT ? ? T ?1 k ′ iχT γ χR φk ?→ ?iχ′ ? γ χR φk )? R Cγ χR φk = (iχR C RC (CP )0

(8.8)

or γ kT = γ k? . Consequently, taking into account the multiplicities mr , mr′ , mrφ of the respective irreps a generalized CP transformation given by
′ ′ ′? ? CP : χR → ?UCχ? R , χR → ?U Cχ R , φ → Hφ

(8.9) (8.10)

leads to the condition U T Γb U ′ Hba = Γa? in the horizontal spaces on which U , U ′ , H act. Since CP does not connect di?erent irreps the above discussion is also fully general. Of course, the canonical CP symmetry U = 1, U ′ = 1′ , H = 1φ would simply require real matrices Γa . Thus we have achieved to separate group indices and horizontal indices in the case of generalized CP invariance. The classes of solutions of eq. (8.10) will be discussed in the following subsection.

8.3

Solutions of the generalized CP condition

The discussion of eq. (8.10) is greatly simpli?ed by the freedom of choosing suitable bases in the horizontal spaces. As in the previous subsection we stick to ?xed irreps Dr , Dr′ , Drφ and therefore we have three independent basis transformations (3.26) represented by the unitary matrices Z , Z ′ , Zφ . According to the theorem proved in ref. [29] one can ?nd Z and Z ′ such that π Z ? UZ ? = diag (O (Θ1 ), . . . , O (Θk ), 1p ), 0 < Θν ≤ 2 π ′ ′? ′ ′? ′ ′ 0 < Θν ≤ Z UZ = diag (O (Θ1 ), . . . , O (Θk′ ), 1p′ ), (8.11) 2 with O (?) ≡ cos ? sin ? ? sin ? cos ? . (8.12)

In the Higgs sector we have to distinguish the cases as in sect. 7. For a complex scalar multiplet the above theorem applies yielding π ? ? H . (8.13) Zφ HZφ = diag (O (ΘH 0 < ΘH 1 ), . . . , O (ΘkH ), 1pH ), ν ≤ 2

38 For a real scalar H is real and Zφ has to be orthogonal. Then the real version of the spectral theorem for normal operators tells that one can achieve
T H Zφ HZφ = diag (O (ΘH 1 ), . . . , O (ΘkH ), ?1p? , 1p+ ),

0 < ΘH ν < π.

(8.14)

The case of a pseudoreal scalar requires Zφ ∈ Sp(2m). In app. G we prove that for any H ∈ Sp(2m) there is such a Zφ giving
? ? Zφ HZφ =?

? ? ? ?

0 0 D 0 0 1pH 0 0 ? ?D 0 0 0 0 0 0 1pH

? ? ? ? ?

(8.15)

with D = diag (d1 , . . . , dkH ), |dν | = 1. These basis choices allow to solve eq. (8.10) in a piecewise manner with submatrices of Γa of maximal size 2 × 2 and at most two di?erent Γa involved at a time. We will denote these submatrices by A or A1 , A2 and indicate by arrows which part of Z ? UZ ? , ? ? Z ′? U ′ Z ′? , Zφ HZφ is under discussion. The number of di?erent cases to be discussed is reduced by the following observations. Solutions of the cases with H → ?1 are obtained from those with H → 1 by multiplying A or A1 , A2 by i. The solutions of the cases U → 1, U ′ → O (Θ′ ) are obtained from those with U → O (Θ), U ′ → 1 by transposition of A or A1 , A2 . This leaves the following nine generic cases to be investigated: 1) U → 1, U ′ → 1 ? A, A1 , A2 are 1 × 1 matrices 1a) H → 1: A = A? 1b) H → O (ΘH ): 1c) H → h(d): A1 cos ΘH ? A2 sin ΘH = A? 1 A1 sin ΘH + A2 cos ΘH = A? 2

A1 d = A? 2 ?A2 d? = A? 1

2) U → O (Θ), U ′ → 1 ? A, A1 , A2 are 2 × 1 matrices 2a) H → 1: O (Θ)T A = A? 2b) H → O (ΘH ): 2c) H → h(d): O (Θ)T (A1 cos ΘH ? A2 sin ΘH ) = A? 1 O (Θ)T (A1 sin ΘH + A2 cos ΘH ) = A? 2

O (Θ)T A1 d = A? 2 ?O (Θ)T A2 d? = A? 1

3) U → O (Θ), U ′ → O (Θ′) ? A, A1 , A2 are 2 × 2 matrices 3a) H → 1: O (Θ)T AO (Θ′) = A?

39 O (Θ)T (A1 cos ΘH ? A2 sin ΘH )O (Θ′) = A? 1 O (Θ)T (A1 sin ΘH + A2 cos ΘH )O (Θ′) = A? 2

3b) H → O (ΘH ): 3c) H → h(d):

O (Θ)T A1 O (Θ′)d = A? 2 ?O (Θ)T A2 O (Θ′ )d? = A? 1. (8.16)

In all the cases we have 0 < Θ, Θ′ ≤ π , 2 0 < ΘH < π, h(d) = 0 d ?d? 0 with |d| = 1.

In the following the solutions for A or A1 , A2 of 1a) – 3c) are given as functions of Θ, Θ′ , ΘH or d according to the spirit of our strategy outlined at the end of subsect. 8.1. The methods used thereby can be found in app. H. Some cases in the above list correspond to each other like 1b) – 2a) and 2b) – 3a). They are, however, kept apart for the sake of clearness. In the following list of solutions ε denotes the two options ±1. Solutions: 1a) A ∈ R 1b) A1 = A2 = 0 1c) d = iε ? A2 = ?dA? 1 , A1 ∈ C d2 = ?1 ? A1 = A2 = 0 π ? A2 = 2 π ? A2 = 2 2a) A = 0 2b) ΘH = Θ = ΘH = Θ = 0 ?1 1 0 0 ?1 1 0
2 A? 1 , A1 ∈ C

A1 , A1 ∈ R2 A1 , iA1 ∈ R2

ΘH = π ? Θ =

π ? A2 = 2

ΘH ∈ {Θ, π ? Θ} ? A1 = A2 = 0 2c) Θ = Θ= π , d = ε ? A2 = ε 2 0 ?1 1 0

0 1 ?1 0

2 A? 1 , A1 ∈ C

π , d = iεe±iΘ ? A1 = a 2

d2 = ?e±2iΘ ? A1 = A2 = 0

1 ±i

, A2 = ?iεA? 1, a ∈ C

40 3a) Θ = Θ′ = Θ = Θ′ = π ?A= 2 π ?A= 2 a b ?b? a? a b ?b a

, a, b ∈ C , a, b ∈ R
?

Θ = Θ′ ? A = 0


π 3b) ΘH = Θ + Θ = ? A1 = 2 cos ΘH = ε sin Θ′ , Θ = a, b ∈ C cos ΘH = ε sin Θ, Θ′ = a, b ∈ C

a b b ?a

, A2 =

?b a a b

, a, b ∈ C , A2 = εb ?εa a? b? ?b? a? ?εa ?εb

π π , Θ′ = ? A1 = 2 2 π π ,Θ= ? A1 = 2 2

a b ?εb? εa? a b ?εb? εa?

, A2 =

cos ΘH = ?ε cos(Θ + Θ′ ), Θ =

π π π , Θ′ = , Θ + Θ′ = ? 2 2 2 ? ? a b ?b a ? ? A1 = i , A2 = ?i for ε = 1 ? ? a b b ?a a, b ∈ R ? a b ?b a ? ? A1 = , A2 = for ε = ?1 ? ? b ?a a b π π π cos ΘH = ε cos(Θ ? Θ′ ), Θ = , Θ′ = , Θ + Θ′ = ? 2 2 2 ? ? a b ?b a ? A1 = , A2 = η for ε = 1 ? ? ? a, b ∈ R and ?b a ?a ?b ′ ? a b ?b a ? η = sgn(Θ ? Θ) A1 = i , A2 = ?iη for ε = ?1 ? ? ? ?b a ?a ?b All other choices of Θ, Θ′ , ΘH lead to A1 = A2 = 0. π , d = iε ? A1 = 2 π , d = iε ? A1 = 2 π , d = ε ? A1 = 2 a b c d a b ?b a a b b ?a , A2 = ?iε , A2 = ?iε , A2 = ε a ib ?ia b a ?ia ib b d? ?c? ?b? a? , a, b, c, d ∈ C , a, b ∈ C

3c) Θ = Θ′ = Θ = Θ′ = Θ + Θ′ = Θ′ = Θ=

?b? a? a? b? , A2 = ε , A2 = ε

a? b? ?b? a?

, a, b ∈ C ib? a? ?b? ia? , a, b ∈ C , a, b ∈ C

π π , Θ = , d = εe?iΘ ? A1 = 2 2 π ′ π ′ , Θ = , d = εe?iΘ ? A1 = 2 2

ib? ?b? a? ia?

41 π ′ π ′ , Θ = , d = εeiΘ ? A1 = 2 2 π π , Θ = , d = εeiΘ ? A1 = 2 2 a ia ?ib b a ?ib ia b ?ib? ?b? a? ?ia?

Θ= Θ′ =

, A2 = ε , A2 = ε

, a, b ∈ C , a, b ∈ C

The remaining four non–trivial cases can be uniformly described in the following way: Θ= π , 2 Θ′ = π , 2 1 ?i
T d = iεe?i(rΘ+sΘ ) ? A1 = cvr vs ,


?ib? a? ?b? ?ia?

A2 = ?iεA? 1

with r, s = ±, v± =

, c ∈ C.

For all other choices of Θ, Θ′ and d we have A1 = A2 = 0. This concludes the complete discussion of the generalized CP condition (8.10). It shows which solutions apart from the trivial one with real couplings 1a) one can expect. These solutions might be helpful for model building. Of course, they are bound to the bases introduced in the beginning of this subsection. If one has additional conditions on the Yukawa couplings from further symmetries it might not be useful to work in these bases. An example for such a case is given in subsect. 9.3.

9
9.1

C–type transformations
Charge conjugation

We have seen in eq. (8.4) that if we carry out two CP–type transformations one after the other we obtain a transformation of the type [7]
b a W? (x) → Rab W? (x) ωR (x) → UωR (x) Φ(x) → UH Φ(x).

(9.1)

It is reasonable to call eq. (9.1) a C–type transformation since composing CP with P, both of CP–type, should give charge conjugation. Though the discussion of CP and P in the previous sections contains implicitly also C–type transformations it is nevertheless useful to consider their features in a separate section. If LG + LF + LH + LY is invariant under the transformation (9.1) we have UTb Rba U ? = Ta ? UH Tb Rba UH = Ta T H U Γk UUkj = Γj .

(9.2)

42 As we have learned in subsect. 3.2 the ?rst relation can be interpreted as {Ta } composed with the automorphism ψR being equivalent to {Ta }. The second relation has the analogous interpretation for the Higgs ?elds. It is clear from Schur’s lemma that for R = 1 the matrices U and UH only act horizontally. To de?ne a charge conjugation we require as for P and CP that ψR = ψC is a non– trivial involution. Having ?xed the CSA H which determines the quantum numbers of the states in the irreps it is reasonable to require ψC (H) = H. As in the case of P the CSA can be split into H = H+ ⊕ H? with ψC (X ) = ±X for X ∈ H± . We assume that c the CSA of the unbroken part of the SM group is contained in H? , i.e. Qem , F3 , Yc ∈ H? as any physically viable model must have U (1)em × SU (3)c ? G. Thus ψC has to ?ip the sign of at least three generators in H. To every ψC there is associated a subgroup of G generated by the elements of LC with ψC (X ) = X . In refs. [7, 35] there is a complete list of all such “symmetric subgroups” S for simple groups G. Since in this paper we are mainly concerned with the symmetry aspect of the Lagrangians and not with the embedding of the SM group in G and future symmetry breaking we refer the reader to the extensive discussion in ref. [7] of these questions. Let the automorphism de?ning charge conjugation be given by ψC (Hj ) = ρj Hj = ?Hj , j = 1, . . . , p Hj , j = p + 1, . . . , ?. (9.3)

Then as for P we can distinguish an internal and external case with respect to an irrep D if D ? D ? ψC or D ? D ? ψC , respectively. As before we consider now the representation space of D in the internal case and the direct sum of the spaces of D and D ? ψC in the external case. Then D (Hj )UC e(λ, q ) = ρj λ(Hj )UC e(λ, q ) (9.4) for the ON basis {e(λ, q )} with weights λ of D . The situation is analogous to P in eqs. (6.8) and (6.10). All states with (λ(H1 ), . . . , λ(Hp )) = (0, . . . , 0) have a counterpart with opposite quantum numbers with respect to H1 , . . . , Hp . De?ning
λC ,q λ,q T (x))? (x) ≡ Cγ0 (ω R ωL

(9.5)

with λC given as in eq. (6.10) we can perform an analysis as in subsect. 6.2 and obtain
λ,q λ,q T ?1 (ω L (x))? . = eiδ Cγ0 UC ωR (x)UC

(9.6)

2 Here we have used that UC is a phase in each irrep of {Ta } just as in the analogous 3 situation in eq. (6.15) . In eq. (9.6) charge conjugation has the familiar form. As mentioned before there is a close relationship between C and P via CP. It is clear that invariance under CP and P is equivalent to invariance under CP and C=CP?P. This ? relationship is given by the identi?cations ψC = ψ △ ? ψP , λP = λC and UC = ?UP UCP (see eq. (8.4)). Once we ?x ψCP = ψ △ and choose a CP basis one can therefore identify UP with UC in the internal case of parity (see, e.g., the SO (10) example in sect.2).
3 2 δ is the phase of UC in D or D ? ψC for the internal or external case, respectively.

43 Like parity, charge conjugation cannot be de?ned in the SM where ψC would either reverse the sign of the electric charge and colour charges or of all the four elements of the CSA. In any case the right–handed singlets have no partners with opposite charges as required by the presence of D ? ψC in the case of C invariance. In the light of the discussion in this section we also learn that in the irreps of G? , eq. (4.10), D (E ) would be a candidate for UC if ψC = ψE .

9.2

Compatibility of CP and C

In this subsection we will show that one can construct Yukawa couplings which are not only invariant under the canonical CP transformation but also under a “canonical” charge conjugation at the same time. We will make use of the advantages of the CP basis where (CP)0 is simply given by eq. (8.6) and where the Yukawa couplings are real. To de?ne a canonical charge conjugation we con?ne ourselves to D and D ? ψC for every irrep D contained in {Ta }. Given the involutive automorphism ψC we will assume in the following that either D ? ψC ? D or D ? ψC is included in {Ta } to allow for a de?nition of C. The whole discussion will be con?ned to simple Lie algebras. It is clear that for ψC ≡ ψY inner (see eq. (4.1)) the C transformation χR → W χR , with W = e?D(Y ) , χ′R → W ′ χ′R , W ′ = e?D (Y ) ,


φ → Wφ φ Wφ = e?Dφ (Y )

(9.7) (9.8)

where the irreps D , D ′ , Dφ are coupled together in LY is just a gauge transformation and LY is obviously invariant under transformation (9.7). This allows to restrict the further discussion to ψ = ψ △ or ψd . In the following the matrices acting on the minimal number of irreps involved in the Yukawa couplings for a de?nition of C will always be called W , H W ′ , Wφ . They are part of UC and UC in eq. (9.1). T △ In a CP basis ?D = D ? ψ is valid and therefore, if D ? D ? ψ △ , we obtain W real, W T = λW with λ = ?1 for D pseudoreal and λ = 1 for D real (see eqs. (7.12) and (7.19), respectively). For W ′ , Wφ the parameters analogous to λ will be denoted by λ′ , λφ , respectively. If D ? D ? ψd it is shown in app. I that W is real and symmetric, i.e. λ = 1. As discussed in sect. 4 we choose ψd as the relevant outer automorphism for so(2?) (? = 4, 6, 8, . . .) whereas in all other cases of non–trivial outer involutive automorphisms, i.e., the Lie algebras su(? + 1) (? ≥ 2), so(2?) (? = 5, 7, 9, . . .) and cE6 , we take ψ △ as the relevant automorphism. We have to distinguish four cases according to the types of irreps with respect to ψC coupled together in LY : a) D ? D ? ψC , D ′ ? D ′ ? ψC , Dφ ? Dφ ? ψC The LY of eq. (8.7) transforms as
?1 ?1 k ′ W T γ ? W ′ χ′R Wφ?k φk γ χR φk → iχT iχT RC RC

(9.9)

44 under the C transformation (9.7). In case a) we always have λλ′ λφ = 1. (9.10)

This is so for two reasons. First, for ψd the W ’s are symmetric and all λ’s are 1. Second, as for ψ △ one can show that in a tensor product of real or pseudoreal irreps of connected semisimple Lie groups pseudoreal irreps do not occur. Also if one factor is real and the other one pseudoreal then real irreps do not occur [34] and thus eq. (9.10) is valid. Consequently, we can replace γ k by
k T γ k → γ± ≡ γ k ± W γ ? W ′ T (Wφ )?k

?k

(9.11)

with the C transformation given by aW, a′ W ′ , aφ Wφ with aa′ aφ = 1 ?1 for + for ? . (9.12)

k given Clearly, eq. (9.10) is necessary to have invariance of LY eq. (8.7) with γ± by eq. (9.11). Since the coupling matrices (9.11) are real (CP)0 invariance is not spoiled. We do not know if one can take a speci?c sign for all simple groups and all k k irreps. In any case, in any situation the two sets of coupling matrices {γ+ } and {γ? } cannot consist of only zero matrices at the same time and therefore it is possible to have (CP)0 invariance and invariance under the “canonical” C transformation k k given by eq. (9.12) for LY rede?ned with {γ+ } or {γ? } in case a).

b) D ? D ? ψC , D ′ ? D ′ ? ψC , Dφ ? Dφ ? ψC In this case W, W ′ , Wφ = Then
?1 k ′ T ?1 k ′ i(χT γ χR1 φ1 γ χR2 φ2 R1 C k + χR2 C k)

0 1 1 0

,

χR =

χR1 χR2

etc.

(9.13)

(9.14)

is clearly invariant under (CP)0 and C given by eq. (9.13). c) D ? D ? ψC , D ′ ? D ′ ? ψC , Dφ ? Dφ ? ψC With and Wφ = invariance is obtained.
?1 ?1 k ′ 1 W γ k W ′ T χ′R φ2 γ χR φk + χT i(χT k) RC RC

(9.15) (9.16)

0 λλ′ 1 1 0

,

φ=

φ1 φ2

45 c’) D ′ ? D ′ ? ψC , Dφ ? Dφ ? ψC , D ? D ? ψC As before we have
?1 k ′ ?1 ? T i(χT γ χR φk + χT γ W ′ T χ′R (Wφ )?k φk ) R1 C R2 C

(9.17)

with W =

0 λ′ λφ 1 1 0

,

χR =

χR1 χR2

.

(9.18)

d) D ? D ? ψC , D ′ ? D ′ ? ψC , Dφ ? Dφ ? ψC Now
?1 k ′ ?1 ? T i(χT γ χR1 + χT γ (Wφ )?k χ′R2 )φk R1 C R2 C

(9.19)

is invariant with (a, a′ = ±1) W = 0 a1 1 0 , W′ = 0 a′ 1 1 0 , λφ = aa′ . (9.20)

All remaining cases are to be discussed analogously. Thus we have seen that for every C transformation such that D ? ψC is contained in {Ta } for every irrep D of {Ta } one can write down Yukawa couplings invariant under canonical C and CP with “canonical C” de?ned in this subsection. This invariance is achieved by a rede?nition of the γ k . It is not clear to us whether one could circumvent this rede?nition by directly exploring properties of the Clebsch–Gordan coe?cients.

9.3

We want to close this section with a left–right symmetric example [13, 32, 33] which exhibits a lot of interesting features in the light of the discussion of discrete symmetries. We consider fermions χL , χR transforming as (2, 1, B ? L), (1, 2, B ? L), respectively, where the numbers indicate the dimension 2j + 1 of an irrep Dj (j = 0, 1/2, 1, . . .) of SU (2). The scalar φm transforming as (2,2,0) was already introduced in subsect. 7.2 as an example of two real irreps in complex disguise. For quarks the U (1) charge is 1/3 whereas for leptons it is ?1. Therefore it is given by baryon number minus lepton number (B ? L). Assuming nf families in χL,R we have ωR = and ? LY = χ ?L Γφm χR + χ ?L ?φm χR + h.c. (9.22) with φm given by eq. (7.28). Γ and ? are nf × nf matrices in family (?avour) space. Note that in eq. (9.22) both φm and φm have to be coupled to get the most general Yukawa couplings. (χL )c χR (9.21)

An example with G = SU (2)L × SU (2)R × U (1)B ?L

46 Let us now investigate the e?ects of the following generalized CP and C transformations [27]: CP : χL → ?Cχ? χR → ?iCχ? φm → ?iφ? L, R, m c c C : χL → ?(χR ) , χR → ?i(χL ) , φm → ?iφT m. Examining the invariance conditions we obtain CP ? Γ real, ? imaginary, C ? ΓT = Γ, ?T = ??. (9.25) (9.26) (9.23) (9.24)

Taking eqs. (9.25) and (9.26) together we see that Γ has to be real and symmetric and ? imaginary and antisymmetric. Here we have not used the bases in the horizontal space introduced in subsect. 8.3 for CP because there is the additional C invariance (9.24) whose simple form would be spoiled in these bases. Furthermore, eqs. (9.25) and (9.26) together are rather restrictive. It was shown in ref. [27] that the e?ect of these conditions is similar to the e?ect of a horizontal symmetry allowing for relations between ?avour mixing and masses but for nf = 3 the top quark mass comes out too low. Interpreting the transformation properties of fermions and φm in the light of the irreps of G? in subsect. 4.2 and ignoring the U (1), ωR eq. (9.21) is in the irrep D1/2,0 + of SU (2)? and φm contains two D1 /2 . Thus φm can be considered as a tensor product D1/2 ? D1/2 with respect to SU (2) × SU (2) with D (E ) exchanging the vectors in the product. Since D (E ) is related to the C in eq. (9.24) it becomes clear that C has the e?ect of transposition on φm . Combining CP and C we obtain P = C ? CP : χL → ?γ0 χR , χR → ?γ0 χL , φm → φ? m. (9.27)

Now we want to switch to the formalism discussed in this paper, namely using ωR eq. (9.21) and φ eq. (7.29) instead of φm . Then the fermion rep has the form ?12 0 0 12 (9.28) with Pauli matrices τa . The generators Ta acting on φ are obtained from the transformation property of φm eq. (7.27), i.e., Ta φ (a = 1, 2, 3) corresponds to ?φm τa /2 and T3+a φ to τa φm /2. Therefore we obtain Ta = , T3+a = , a = 1, 2 , 3 , 1? ? T1 = ? ? 2? 1? ? T3 = ? ? 2?
? ?

0 0 0 τa /2

T ?τa /2 0 0 0

1 T7 = (B ? L) 2

0 0 1 0 1 0 0 0

0 0 0 1 0 1 0 0

1 0 0 1 ? ? ?, 0 0 ? 0 0 0 0 0 0 ?1 0 0 ?1

?

?

1? ? T2 = ? ? 2? T3+a =
1 2

?

0 0 0 0 ?i 0 0 ?i τa 0 0 τa

i 0 0 0 ,

0 i 0 0

?

? ? ?, ?

(9.29) T7 = 0.

? ? ?, ?

47 Now CP eq. (9.23) is quickly treated. Acting on ωR , eq. (3.17) determines UCP = ?1nf 0 0 i1nf . (9.30)

Then the automorphism is ?xed by Condition B: RCP = diag (?1, 1, ?1, ?1, 1 ? 1, ?1) (9.31)

H is just ψ △ for SU (2) × SU (2) × U (1). For φ we have UCP = ?i14 and a quick look at eq. T (9.29) con?rms that ?Ta RCP aa = Ta is ful?lled. As for charge conjugation eq. (9.24) ?xes

?

UC = ?

0 1nf i1nf 0

and

RC =

? ? ? ? ? ? ? ? ? ? ? ?

0 0 0 ?1 0 0 0

0 0 ?1 0 0 0 0 0 0 0 0 0 1 0 0 0 ?1 0 0 0 0

0 0 0 1 0 0 0 ?1 0 0 0 0 0 0 0 0 0 0 0 0 ?1

? ? ? ? ? ? ? ? ? ? ? ? ?

(9.32)

is then given by eq. (9.2). Now we have to check consistency in the Higgs sector where from eq. (9.24) we calculate
? ? ? ?

H UC = ?i ?

1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 1

?

? ? ?. ?

(9.33)

It is then tedious but easy to verify that
H? H = Ta . UC (Tb RCba )UC

This was a somewhat unorthodox look at the discrete symmetries CP, C and P in left–right symmetric models. In this example P has the simple form coming from the idea of left–right symmetry whereas CP and C are “generalized” in the sense of subsect. 5.2 because there is a phase in one part of the horizontal space. The general discussion in this paper is quite nicely illustrated here and though the gauge group is rather small the structure of the discrete symmetries is more complex than in the cases of QED and QCD.

10

Comments and conclusions

In this work we have discussed the possibility of de?ning CP, P and C invariance in gauge theories before spontaneous symmetry breaking. As mentioned in the introduction the

48 breaking of the gauge group and of the discrete symmetries could be performed at the same time. For CP this possibility was ?rst envisaged in ref. [36] and for P in refs. [37, 32]. Our ?rst comment concerns the question of when the above scenario really happens. Let us suppose that the Lagrangian is invariant under a CP transformation where UH is the unitary matrix appearing in the transformation of the scalar ?eld vector Φ eq. (7.37). Then, if the vacuum expectation value Φ 0 ful?lls UH Φ
? 0

= Φ

0

(10.1)

the CP symmetry is not broken [38, 39, 40]. Furthermore, if eq. (10.1) is not ful?lled there are two possibilities. On the one hand, there might exist an element of the total symmetry group G × H of the Lagrangian (G is the gauge group and H the group of all other internal symmetry transformations commuting with G) such that its action on Φ is given by SH and SH UH Φ ? (10.2) 0 = Φ 0. Then one can de?ne a new CP symmetry Φ(x) → SH UH Φ(x)? (10.3)

with adequate rede?nitions in the fermion and gauge boson sectors and eq. (10.1) is ′ ful?lled with UH replaced by UH = SH UH . On the other hand, only if for a given UH one ′ complies with eq. (10.1) the CP symmetry associated cannot ?nd an SH such that UH with the UH of eq. (10.1) is spontaneously broken together with the gauge group. The second comment refers to CPT invariance. One might be tempted to de?ne a CPT–type transformation in analogy to CP–type transformations de?ned in sect. 3 by
a b W? (x) → ?Rab W? (?x)

T ωR (x) → ?U T γ5 ωR (?x)? T Φ(x) → ?UH Φ(?x)? .

(10.4)

It can easily be checked that the invariance conditions ensuing from eq. (10.4) and from its antiunitary operator implementation are identical with the conditions (9.2) following from the C–type transformations (9.1) with the matrices R, U , UH of eq. (10.4).4 Clearly, if R, U , UH are identity matrices, eq. (9.2) is always ful?lled suggesting that CPT should be de?ned by5
a a W? (x) → ?W? (?x)

CP T :

4 We use the transposed matrices in eq. (10.4) to get exactly the form of eq. (9.2) for the invariance conditions. 5 The minus in the transformation of ωR is convention, however, the other two minus signs are required for invariance of the Lagrangian.

T ωR (x) → ?γ5 ωR (?x)? Φ(x) → ?Φ(?x)? .

(10.5)

49 In other words, invariance of the Lagrangian under the transformation (10.4) leads to invariance under the corresponding C–type transformation (9.1) yet the Lagrangian is anyway invariant under the transformation (10.5). Thus the concept of CPT–type transformations is void and there is just one canonical form (10.5) of CPT. This together with the de?nition of CP eq. (5.1) via the contragredient automorphism ψ △ ?xes the de?nition of time reversal in the generalized sense analogous to CP in sect. 5. It is interesting to note that with R = 1, but U , UH non–trivial, conditions (9.2) tell us that U , UH act on ωR , Φ, respectively, as representations of an element of G × H . If U , UH belong to H alone then the e?ect of the CPT–type transformation (10.4) or the corresponding C–type transformation (9.1) is a horizontal symmetry [39]. The comment on CPT explains why in the discussion of the discrete symmetries C, P, T we could concentrate on CP and P in this work and being fully general at the same time. Let us now summarize the main points of this work. The starting point of our discussion was the observation that CP and P transformations have the same structure when formulated with fermion ?elds of one chirality. General transformations of that type we have called CP–type transformations. The crucial point is that if a CP–type transformation is a symmetry of the Lagrangian its action on the gauge bosons can be described in terms of automorphisms of the Lie algebra Lc of the gauge group. Consequently, the invariance conditions in the fermion and scalar sectors also have a straightforward interpretation in terms of Lie algebra representations. In addition, all possible automorphisms of simple Lie algebras are known and classi?ed in the literature. Now what distinguishes CP and P from each other? The automorphism associated with CP is given by the contragredient automorphism ψ △ which has the property ψ △ (h) = ?h for all elements h of the CSA whereas P is associated with an involutive automorphism ψP which reverses the signs of at most part of the CSA. These abstract mathematical de?nitions were substantiated by physical considerations. CP is always a symmetry of the gauge Lagrangian Lgauge , the Lagrangian without Yukawa couplings and the Higgs potential. This statement formulated in the language of representations is expressed by ?D T ? ψ △ ? D , saying that for every irrep D of Lc its complex conjugate irrep ?D T connected with the automorphism ψ △ is equivalent to D . (This follows immediately from the fact that the weights of both irreps are identical.) Consequently, CP does not impose conditions on the irrep content of the fermion or scalar representation. If irreps occur with non–trivial multiplicities a CP transformation not acting in these “horizontal” spaces is called “canonical CP” or (CP)0 , a special case of the “generalized” CP transformations. Since P is not uniquely associated with an automorphism the choice of ψP is subject to physical boundary conditions. For instance, one would require that ψP does not change the sign of those elements in the CSA which are associated with the electric and colour charges in order to get a reasonable de?nition of parity. Also one could imagine cases with large gauge groups where several automorphisms lead to viable de?nitions of P. In contrast to CP, the gauge interactions of fermions and scalars are not automatically invariant under parity but, in general, parity invariance introduces a condition on the irrep

50 content of the fermion and scalar representations or on irreps themselves, depending on the gauge group. For simple gauge groups a common condition is that with every irrep in the fermionic sector also its complex conjugate irrep occurs. This condition is always trivially satis?ed in the scalar sector. We have also worked out the connection between parity and the de?nition of Dirac ?elds. In the scalar sector a complication arises for pseudoreal irreps where the horizontal part of a CP–type transformation in general mixes the pseudoreal ?elds φ and φ = W φ? (7.4) which transform alike under the gauge group. This leads to unitary symplectic matrices acting horizontally. Though generalized CP transformations are automatically symmetries of Lgauge this is not the case for the Yukawa interactions. Choosing a particular transformation one obtains conditions on the Yukawa couplings. For (CP)0 this amounts to real couplings in some phase conventions. In the general case horizontal transformations are involved. With suitable basis choices all possible solutions of these conditions for all possible generalized CP transformations can be derived. This might be of interest for models relating fermion masses and mixing angles. We have also considered C–type transformations de?ned as the composition of two CP–type transformations. Such symmetry transformations associated with the trivial automorphism ψC = id are just horizontal symmetries. Finally we have shown that at least for simple gauge groups but arbitrary reps CP and P are compatible in the following sense: given fermion and scalar reps, real Yukawa couplings, i.e. LY invariant under (CP)0 , and an automorphism ψC associated with C such that D ? ψC is also contained in the reps for every irrep D occurring there, then one can construct Yukawa couplings which are invariant under the “simplest” C transformation pertaining to ψC . In conclusion, in this work we have tried to understand CP and P symmetries in gauge theories by pointing out their intimate connection with automorphisms of the Lie algebra of the gauge group. We have furthermore studied in detail the action of such symmetries in the multiplicity spaces of the irreps in the fermionic and scalar sectors and how these rotations imply conditions on the Yukawa couplings. Finally, we have made extensive use of certain basis transformations: in the representation spaces of the irreps they lead to symmetric or antisymmetric generators of Lc in arbitrary irreps and real Clebsch Gordan coe?cients in the Yukawa couplings and in the horizontal spaces they give simple forms of generalized CP transformations. All these considerations might be useful for the construction of models beyond the SM.

Acknowledgements
We would like to thank H. Stremnitzer for thorough and helpful discussions on so(10). Furthermore, W. G. acknowledges numerous enlightening conversations on group theory with H. Urbantke.

51

Appendices A Notation and conventions
(g?ν ) = diag (1, ?1, ?1, ?1) and thus the Dirac algebra is given by {γ? , γν } = 2g?ν 14 . Furthermore, we assume that the γ matrices ful?ll the hermiticity conditions
? γ? = γ 0 γ ? γ 0 = ε (? )γ ?

We follow the conventions of ref. [15]. Therefore we use the metric (A.1)

(A.2)

with ε(?) =

1, ? = 0 ?1, ? = 1, 2, 3.

(A.3)

Then γ5 ≡ iγ 0 γ 1 γ 2 γ 3 is hermitian. Left and right–handed fermion ?elds are given by the conditions 1 + γ5 χR = χR , 2 1 ? γ5 χL = χL , 2 (A.5) (A.4)

respectively. The charge conjugation matrix C is de?ned by
T C ?1 γ? C = ?γ? .

(A.6)

As a consequence of eqs. (A.3) and (A.6) we have6 C ? = C ?1 , C T = ?C and
T C ?1 γ 5 C = γ 5 .

(A.7)

A time reversal transformation requires the de?nition of a matrix T verifying
T T γ ? T ?1 = γ ? .

(A.8)

From the properties of C it is clear that T = eiβ C ?1 γ5 = ?T T (A.9)

with an arbitrary phase β . Then, given a solution χ(x) of the Dirac equation we obtain its time–re?ected solution χT (x0 , x) = T ? χ(?x0 , x)? . (A.10)
6

Actually, it can only be derived that C ? is proportional to C ?1 . For convenience and without loss of generality we assume that these matrices are equal.

52 In the second quantized version this translates into T χ(x0 , x)T ?1 = T χ(?x0 , x) (A.11)

with the time reversal operator T acting on the Hilbert space of states as an antiunitary operator. In this paper we only use right–handed fermion ?elds. If one starts in a theory with ?elds of both chirality fL , fR then the fermionic Lagrangian
? R a L a ? ? ? LF = f R iγ (?? + igTa W? )fR + fL iγ (?? + igTa W? )fL ,

(A.12)

R L where {Ta }, {Ta } are arbitrary, in general di?erent reps of the gauge group, can easily be rewritten as a LF = ω ? R iγ ? (?? + igTa W? )ω R (A.13)

with ωR = (fL )c fR ,
T ? (fL )c ≡ Cγ0 fL

and

Ta =

L T ?(Ta ) 0 R 0 Ta

.

B

Facts about semisimple Lie groups

In this appendix we collect all the facts about Lie algebras (in particular, semisimple Lie algebras) which are used in this work. Extensive expositions of this subject can e.g. be found in the books by Cornwell [16], Georgi [20], Samelson [24], Jacobson [25], Varadarajan [26], Wybourne [41], Cahn [42] and others. The Killing form: Every element X of a Lie algebra (always assumed to be over a ?eld K = R or C) allows to de?ne a linear mapping ad X : L → L Y → [X, Y ]. (B.1)

Then the symmetric bilinear form κ obtained by κ(X, Y ) = Tr (ad X ad Y ) (B.2)

is called Killing form. Automorphisms ψ of L are de?ned as linear mappings which respect the Lie algebra product, i.e. ψ ([X, Y ]) = [ψ (X ), ψ (Y )] ? X, Y ∈ L. (B.3)

The set of automorphisms form a group denoted by Aut (L). Note that the Killing form is invariant under automorphisms: κ(ψ (X ), ψ (Y )) = κ(X, Y ) ? X, Y ∈ L. (B.4)

53 Semisimple Lie algebras are those which have no non–zero Abelian ideals. Cartan’s second criterion states that a Lie algebra L is semisimple if and only if its dimension is positive and its Killing form non–degenerate. Given a basis {Xa } of L one can de?ne structure constants by
c [Xa , Xb ] = Cab Xc .

(B.5)

One can show that a semisimple Lie group G (a group whose (real) Lie algebra is semisimple) is compact if and only if its Lie algebra Lc has a negative de?nite Killing form. Therefore on such a Lie algebra a scalar product is given by ?κ. This allows the de?nition of ON bases {Xa } with κ(Xa , Xb ) = ?δab . In such a case the structure constants c Cab are usually denoted by fabc . One can prove that fabc is totally antisymmetric in a, b, c c c whereas in general only Cab = ?Cba is valid. Automorphisms and bases of L: Fixing a basis {Xa } of L allows to associate a matrix with every linear operator on L and vice versa. Denoting such a matrix by A and the corresponding operator by ψA we can thus write ψA : L → L Xa → Aba Xb . (B.6)

It is then easy to check by analysing eqs. (B.3) and (B.5) that matrices A corresponding to automorphisms ψA ful?ll the following condition:
c c ψA ∈ Aut (L) ?? Aa′ a Ab′ b (A?1 )cc′ Ca ′ b′ = Cab .


(B.7)

Furthermore, invariance of the Killing form is expressed by Aa′ a Ab′ b κ(Xa′ , Xb′ ) = κ(Xa , Xb ). (B.8)

For compact Lie algebras and ON bases this immediately translates into the conditions that A is an orthogonal matrix and fabc an invariant tensor with respect to A: Aa′ a Ab′ b Ac′ c fa′ b′ c′ = fabc . (B.9)

The structure of semisimple Lie algebras: With every Lie group G a real Lie algebra L is associated and by complexi?cation of L a complex Lie algebra L. The transition from L to L conserves semisimplicity. Complex semisimple Lie algebras are fully classi?ed. Conversely, given a complex semisimple Lie algebra L there is a well de?ned procedure to go back to the real Lie algebras L associated with L, i.e. to ?nd all inequivalent L’s whose complexi?cation is L. The structure of L is closely connected with L. This motivates the consideration of complex semisimple Lie algebras though in gauge theories only real Lie algebras occur. Furthermore, since here we are concerned only with compact Lie groups it is su?cient to consider the construction of the unique compact Lie

54 algebra Lc associated with L (see eq. (B.23)) and refer the reader to the books quoted at the beginning of this appendix for the general procedure of “reali?cation”. A CSA can be de?ned and its existence proved for general Lie algebras (see refs. [24, 25, 26]). In the case of a semisimple complex Lie algebra a CSA H is a maximal abelian subalgebra of L such that ad h is completely reducible (i.e. diagonalizable) ? h ∈ H. All CSAs are mutually conjugate which means that given two CSAs H, H′ of L then there is an inner automorphism (see eq. (4.1)) ψ such that ψ (H) = H′ . In this sense a CSA is unique for semisimple complex Lie algebras L. The dimension of the CSA, dim H = ?, is called the rank of L. As a vector space L can be decomposed into L=
α∈?

Lα ⊕ H

(B.10)

which corresponds to the decomposition of L into common eigenstates of ad h (h ∈ H): (ad h)(X ) = [h, X ] = α(h)X with X ∈ Lα . One can show that dim Lα = 1 ?α∈? (B.12) (B.11)

for L semisimple. The eigenvalues α(h) depend linearly on h and can therefore be regarded as linear functionals on H. These non–zero functionals are called roots and the set of roots is denoted by ?. Since the spaces Lα are one–dimensional it su?ces to choose a non–zero vector eα ∈ Lα as a basis ? α ∈ ?. Then one can show that κ(eα , h) = 0 ? α ∈ ?, h ∈ H and κ(eα , eβ ) = 0 ? α, β ∈ ? with β = ?α. (B.13) Therefore, non–degeneracy of κ requires α ∈ ? ?? ?α ∈ ?. (B.14)

As a matter of fact already κ|H is non–degenerate. Thus ? α ∈ ? there exists a unique hα ∈ H such that α(h) = κ(hα , h) ? h ∈ H. (B.15) The elements hα are called root vectors. If α, β, α + β ∈ ? then hα+β = hα + hβ , h?α = hα . (B.16)

The real linear span of {hα |α ∈ ?} is denoted by HR and H is the complexi?cation of HR . κ|HR is a positive de?nite scalar product and therefore HR is an ?–dimensional euclidean space. By α, β ≡ κ(hα , hβ ) ?α, β ∈ ? (B.17)

55 length of roots and the angle between two roots are de?ned. The basis elements eα and the root vectors are connected through [eα , e?α ] = κ(eα , e?α )hα . A crucial point of the theory is that the numbers aβα ≡ 2 β, α α, α (B.19) (B.18)

are integers (the Cartan integers) and that only aβα = 0, ±1, ±2, ±3 is allowed. On ? a weak order can be de?ned by choosing an element h0 ∈ HR such that α(h0 ) = 0 ? α ∈ ?. Then for functionals ?, ?′ on HR the relation ? > ?′ (? ≥ ?′ ) is de?ned by ?(h0 ) > ?′ (h0 ) (?(h0 ) ≥ ?′ (h0 )). Let ?± = {α ∈ ?|α(h0 ) > < 0}. Then ? = ?+ ∪ ?? and ?? = {?α|α ∈ ?+ }. A root is called simple if it is positive but not the sum of two positive roots. The set of simple roots consists of exactly ? linearly independent elements {α1 , . . . , α? }. The Weyl canonical form of L: By choosing suitable bases the commutation relations characterizing semisimple complex Lie algebras can be brought to certain standard forms one of which is the Weyl canonical form (for other standard forms see ref. [16]). There the basis elements eα are normalized to κ(eα , e?α ) = ?1. Then one has the following commutators: [eα , e?α ] [h, eα ] [h, h′ ] [eα , eβ ] [eα , eβ ] with Nαβ ∈ R \ {0}, Nαβ = N?α?β . Given a weak order on ? every root can be written as a linear combination of simple roots such that ? α ∈ ?+
?

(B.20)

= = = = =

?hα α(h)eα or [hβ , eα ] = β, α eα ′ 0 ? h, h ∈ H 0 if α + β ∈ / ?, α + β = 0 Nαβ eα+β for α + β ∈ ?

(B.21)

α=
j =1

α kj αj

with

α kj ∈ N0

(B.22)

for rank ? = dim H = dimR HR .

56 The compact real form Lc of L: Given L we can go back to the compact real Lie algebra Lc by choosing the following basis elements {Xa }: ?iHj (j = 1, . . . , ?) eα + e?α √ 2 with and Hj ∈ HR , eα ? e?α √ 2i κ(Hj , Hk ) = δjk , ? α ∈ ?+ . (B.23)

To a given semisimple complex Lie algebra L there is a unique compact real form L, up to isomorphisms. One can easily check that the above basis elements ful?ll κ(Xa , Xb ) = ?δab . Dynkin diagrams: With the simple roots the Cartan matrix Ajk = 2 αj , αk αk , αk (B.24)

is associated. Clearly, Ajj = 2 and one can show that Ajk can only be 0, ?1, ?2 or ?3. Considering the case j = k in more detail we ?nd (no sum implied here) Ajk Akj = 4 cos2 Θ (B.25)

where Θ is the angle between αj and αk . Note that for Θ = π/2 either Ajk or Akj have to be ?1. According to the possible angles Θ we have to distinguish four cases (assuming 2 that Ajk = ?1 for Θ = π/2 and ωj ≡ αj , αj ): a) cos Θ = 0 ? Θ = π/2 or 900 , ωk /ωj undetermined b) cos Θ = ?1/2 ? Θ = 2π/3 or 1200 , ωk /ωj = 1 √ √ c) cos Θ = ?1/ 2 ? Θ = 3π/4 or 1350 , ωk /ωj = 2 √ √ d) cos Θ = ? 3/2 ? Θ = 5π/6 or 1500 , ωk /ωj = 3. For Θ = 0 and Ajk = ?1 the ratio of the lengths of the simple roots is given by ωk /ωj = ?2 cos Θ. A Dynkin diagram of a semisimple complex Lie algebra is de?ned in the following way: i) To each simple root is associated a point (or vertex) of the diagram. ii) The points associated with αj and αk are connected by Ajk Akj lines, i.e. there are zero, one, two or three lines for the above cases a, b, c, d, respectively. iii) If there are two or three lines connecting two points then a black dot denotes the shorter root. Thus a Dynkin diagram is the graphical representation of the Cartan matrix.

57 The connected Dynkin diagrams are exactly those associated with simple complex Lie algebras and there is a one–to–one correspondence between connected Dynkin diagrams and simple complex Lie algebras. In ?g. 1 all possible connected Dynkin diagrams are depicted with the names of the L’s associated with them. In table 1 all L’s are listed with their respective compact real forms Lc . Table 2 contains all isomorphisms of the low– dimensional classical Lc ’s explaining thus why the series B? , C? , D? start with ? = 2, 3, 4, respectively. Some facts about irreps of semisimple complex Lie algebras: All facts mentioned here are also valid for Lc . In a rep of L weight vectors are those elements of the vector space which are eigenvectors of the CSA, i.e. D (h)e(λ, q ) = λ(h)e(λ, q ) (q = 1, . . . , mλ ) (B.26)

where mλ is the multiplicity of the weight λ which, like a root, can be conceived as a functional on H. One can show that for any weight λ of a rep of L and for any α ∈ ?
? 2 λ, α ?j αj with ?j real and rational. ∈ Z and λ = α, α j =1

(B.27)

The fundamental weights are de?ned by
?

Λj = and it follows that

(A?1 )jk αk
k =1

(B.28)

2 Λj , αk = δjk . αk , αk

(B.29)

For any irrep of a semisimple complex Lie algebra L there is a unique highest weight Λ (a weight Λ is called highest if Λ + α is not a weight ?α ∈ ?+ ). It can be written as Λ = n1 Λ1 + . . . + n? Λ? (B.30)

where n1 , . . . , n? are non–negative integers. Conversely, given n1 , . . . , n? with all nj ∈ N0 there is an irrep DΛ of L unique up to equivalence such that Λ = n1 Λ1 + . . . + n? Λ? is its highest weight. Thus there is a one–to–one correspondence between functionals Λ of the form (B.30) and irreps DΛ of L. All irreps are faithful except the trivial irrep with Λ = 0. No real form L of L admits unitary non–trivial irreps except the compact real form Lc where all irreps are equivalent to unitary ones. On DΛ |Lc unitarity is expressed by D (X )? = ?D (X ) (B.31)

58 corresponding to unitary operators exp D (X ). Then from the basis (B.23) we derive that D (H )? = D (H ) ? H ∈ HR , D (eα )? = ?D (e?α ) ? α ∈ ?. (B.32)

Finally we want to mention that in physics one usually employs hermitian generators of a unitary rep ? Ta = Ta , [Ta , Tb ] = ifabc Tc (B.33) which are obtained by Ta = iD (Xa ) with [D (Xa ), D (Xb )] = fabc D (Xc ) (B.34)

from the ON generators Xa of Lc . In this paper we often switch between the two forms {Ta } and {D (Xa )}.

C

so(N ) and the spinor representations

A convenient choice of basis in the space of antisymmetric real N × N matrices is given by (Mpq )jk = δpj δqk ? δqj δpk , 1≤p<q≤N (C.1) with commutation relations [Mpq , Mrs ] = δps Mqr + δqr Mps ? δpr Mqs ? δqs Mpr . A Cli?ord algebra with N basis elements is de?ned by the anticommutators {Γp , Γq } = 2δpq 1. It is easy to show that the elements 1 1 σpq ≡ [Γp , Γq ] 2 4 (C.4) (C.3) (C.2)

also verify the commutation relations given by eq. (C.2). Thus a rep of the Cli?ord algebra automatically gives also a rep of so(N ). It is straightforward to check that for N = 2? + 1 a rep of the {Γp } in the 2? dimensional space C2 ? . . . ? C2 (?–fold tensor product) is given, in terms of the Pauli matrices and the two–dimensional identity matrix, by Γ1 = σ3 ? . . . ? σ3 ? σ3 ? σ1 Γ2??3 = σ3 ? σ1 ? 1 ? . . . ? 1 (C.5)

Γ4 = σ3 ? . . . ? σ3 ? σ2 ? 1 . . .

Γ3 = σ3 ? . . . ? σ3 ? σ1 ? 1

Γ2 = σ3 ? . . . ? σ3 ? σ3 ? σ2

Γ2?

Γ2??1 = σ1 ? 1 ? 1 ? . . . ? 1 = σ2 ? 1 ? 1 ? . . . ? 1

Γ2??2 = σ3 ? σ2 ? 1 ? . . . ? 1

Γ2?+1 = σ3 ? σ3 ? σ3 ? . . . ? σ3 .

59
1 Then the { 2 σpq } of eq. (C.4) de?ne the irreducible spinor irrep of so(2? + 1) which has ? dimension 2 . For so(2?) one simply has to take Γ1 , . . . , Γ2? of the Cli?ord algebra of so(2? + 1). However, the 2? –dimensional rep of so(2?) is not irreducible because

[σpq , Γ2?+1 ] = 0

? p, q = 1, . . . , ?.

(C.6)

It decays into two inequivalent irreps of dimension 2??1 given by the projectors (1 ± Γ2?+1 )/2.

D

? On the isomorphisms so(4) = su(2) ⊕ su(2) and ? so(6) = su(4)

In sect. 2 the above isomorphisms are exploited for the example of the spinor representation of so(10). Since there one assumes that it is known how to classify the ?elds according to su(4) ⊕ su(2) ⊕ su(2) an explicit realization of the above isomorphisms has to be established to transfer this classi?cation to so(6) ⊕ so(4) ? so(10). Starting from the basis {Mij } as given in app. C we can de?ne the following new basis of so(4): 1 1 (M23 ? M14 ) B1 = (M23 + M14 ) A1 = 2 2 1 1 (D.1) A2 = (M13 ? M42 ) B2 = (M13 + M42 ) 2 2 1 1 A3 = (M12 ? M34 ) B3 = (M12 + M34 ). 2 2 Then it is easy to check with eqs. (C.1) and (C.2) that [Ai , Aj ] = εijk Ak , [Bi , Bj ] = εijk Bk , [Ai , Bj ] = 0 (D.2)

thus proving the ?rst of the isomorphisms. In order to prove the second isomorphism we will specify a basis {Ea } of so(6) such that there is a one–to–one correspondence between the matrices {Ea } and the matrices {λa } of su(4) obtained by generalizing the Gell–Mann basis of su(3), i.e. ?i λa ←→ Ea 2 (a = 1, . . . , 15), (D.3)

such that the matrices Ea obey the same commutation relations as ?iλa /2. The result

60 which is unique up to orthogonal basis transformations is given by7 E1 = E2 = E3 = E4 = E5 = E6 = E7 = E8 = 1 (M23 ? M14 ) 2 1 (M13 ? M42 ) 2 1 (M12 ? M34 ) 2 1 (M16 ? M25 ) 2 1 (M62 ? M15 ) 2 1 (M45 ? M36 ) 2 1 (M35 ? M64 ) 2 1 √ (M12 + M34 ? 2M56 ). 12 E9 = 1 (M45 + M36 ) 2 1 ? (M35 + M64 ) 2 1 (M16 + M25 ) 2 1 ? (M15 + M62 ) 2 1 (M23 + M14 ) 2 1 ? (M13 + M42 ) 2 1 ? √ (M12 + M34 + M56 ) 6

E10 = E11 = E12 = E13 = E14 = E15 =

(D.4)

E

Irreps of G?

For the de?nition of G? see eq. (4.10). Given a rep of G? it can be decomposed according to the subgroup G × G. Since irreps of G × G are given by Dr (g1 ) ? Dr′ (g2 ) ((g1 , g2 ) ∈ G × G) with Dr , Dr′ being irreps of G [30] we infer that an irrep D of G? decays into (r,r ′ ) (Dr (g1 ) ? Dr ′ (g2 )) under G × G. Using D (E )D ((g1 , g2))D (E ) = D ((g2 , g1 )) we obtain
(r,r ′ )

(E.1)

(Dr (g1 ) ? Dr′ (g2 )) ?

(r,r ′ )

(Dr (g2 ) ? Dr′ (g1 )) ?

(r,r ′ )

(Dr′ (g1 ) ? Dr (g2 )).

(E.2)

Therefore, in the irrep D only summands of the type Dr (g1 ) ? Dr (g2 ) and (Dr (g1 ) ? Dr′ (g2 )) ⊕ (Dr′ (g1 ) ? Dr (g2 )) (r = r ′ ) appear and thus D ((g1 , g2 )) ? mr (Dr (g1 ) ? Dr (g2 )) (E.3)

The ?rst eight matrices listed are generators of the su(3) subalgebra and can be obtained by a general procedure to derive the generators of the su(N ) subalgebra of so(2N ) as given in ref. [20].

7

⊕ ?

?

r

(r ′ ,r ′′ )

mr′ ,r′′ ((Dr′ (g1 ) ? Dr′′ (g2 )) ⊕ (Dr′′ (g1 ) ? Dr′ (g2 )))?

?

(E.4)

61 with r ′ = r ′′ and mr , mr′ ,r′′ being the multiplicities of the reps (E.3). The associated vector spaces will be denoted by Vr and Vr′ ,r′′ , respectively. Now we can de?ne an operator S by x?y → y?x on Vr , (E.5) S: (v ? w, x ? y ) → (y ? x, w ? v ) on Vr′ ,r′′ . One can easily prove that and therefore [D (E )S, D ((g1, g2 ))] = 0.
⊕m
′ ′′

SD ((g1 , g2))S = D ((g2 , g1 ))

(E.6) (E.7)

r ,r ⊕mr Schur’s lemma guarantees that D (E )S operates within Vr and Vr′ ,r′′ (the superscript ⊕m denotes the m–fold direct sum). ⊕mr ? Discussing ?rst Vr = Cmr ? Vr we note that

D (E ) = A ? S

with

A2 = 1mr

(E.8)

since S 2 = id. Therefore we can make a basis transformation in Cmr diagonalizing A with eigenvalues ±1. Consequently we have a type of irreps given by Vr and D (E ) = ±S. (E.9)

± This de?nes the irreps Dr according to the sign in eq. (E.9). Writing Vr′ ,r′′ = Wr′ ,r′′ ⊕ Wr′′ ,r′ according to the two inequivalent irreps of G × G on ⊕mr ′ ,r ′′ ? Vr′ ,r′′ and Vr′ ,r′′ = (Cmr′ ,r′′ ? Wr′ ,r′′ ) ⊕ (Cmr′ ,r′′ ? Wr′′ ,r′ ) then D (E ) has the form

D (E ) = (A ? idr′ ,r′′ , B ? idr′′ ,r′ )S. A small calculation reveals that D (E )2 = id requires B = A?1 . Performing a basis transformation with Z = (1mr′ ,r′′ ? idr′ ,r′′ , A?1 ? idr′′ ,r′ ) we obtain Z ?1 D ((g1 , g2 ))Z = D ((g1, g2 ))
⊕m
′ ′′

(E.10)

(E.11)

(E.12) (E.13)

and

Z ? 1 D (E )Z = S

r ,r on Vr′ ,r′′ . Therefore the rep of G? on this space decays into mr′ ,r′′ equivalent copies of an irrep denoted by Dr′ ,r′′ de?ned via D (E ) = S as given in the second line of eq. (E.5). Thus we have found all irreps of G? as written down in subsect. 4.2.

62

F

On the existence of a CP basis

Let L be a complex semisimple Lie algebra with a d–dimensional irrep D and D (Xa ) ≡ Ya (a = 1, . . . , nG ) where {Xa } is an ON basis of the compact real form Lc of L. Thus the Ya are d × d matrices and since D is unitary we have Ya? = ?Ya in addition. In this appendix we will show that one can always choose a basis of Cd such that YaT = ?ηa Ya (F.1)

in this basis and where the signs ηa are given by ψ △ (Xa ) = ηa Xa with ψ △ being the contragredient automorphism de?ned in eq. (4.5). For the connection between the signs ηa and the ON basis {Xa } de?ned in eq. (B.23) see eq. (5.12). Eq. (F.1) de?nes the (CP)0 basis used in subsect. 5.2. Proof: We have because the weights of both irreps are identical. This means that there is a unitary matrix U such that U (?YaT ηa )U ? = Ya (F.3) and, consequently, [UU ? , Ya ] = 0 U T = λU ? a = 1, . . . , nG . λ = ±1. (F.4) ? DT ? ψ△ ? D (F.2)

Using Schur’s lemma we obtain UU ? = λ1 and therefore with

(F.5)

Thus we have shown that the matrix U must be either symmetric or antisymmetric. (Exactly the same reasoning is valid for the matrix in the equivalence ?D T ? D in which case λ = 1 corresponds to real and λ = ?1 to pseudoreal irreps.) Case 1: U = U T We use the following lemma: For every unitary symmetric matrix U there is a unitary symmetric matrix U such that U = U 2 . Its proof given at the end of this appendix. Inserting U 2 into eq. (F.3) we obtain ? ηa U YaT U ? = ?ηa (U ? Ya U )T = U ? Ya U which is just the desired result (F.1). Case 2: U T = ?U It remains to show that this case is impossible for irreps of semisimple Lie algebras Lc . The irrep D is characterized by its highest weight Λ which is simple. We get from eq. (F.3) particularized for D (?iHj ) U D (?iHj )T U ? eΛ = Λ(?iHj )eΛ (F.7) (F.6)

63 where eΛ is the weight vector corresponding to Λ. Using antisymmetry of U and the fact that D (?iHj ) is antihermitian eq. (F.7) can be rewritten as
? D (?iHj )Ue? Λ = Λ(?iHj )UeΛ .

(F.8)

T Since Λ is simple we must have Ue? Λ = aeΛ with a = 0. Finally, using U = ?U we derive the contradiction ? ? (F.9) 0 = e? Λ UeΛ = aeΛ eΛ = 0.

Thus U is symmetric and therefore the existence of a CP basis for any irrep of any compact semisimple Lie algebra is proved. 2 Proof of the lemma: Let A be a normal symmetric matrix and A1 ≡ Re A, A2 ≡ Im A. Then it follows from A? A = AA? and AT = A that [A1 , A2 ] = 0. Thus A can be diagonalized by an orthogonal matrix O . Let now U be unitary and symmetric. Then U = O dO T and U =O d OT with d = diag (ei?1 /2 , . . . , ei?d /2 ). 2 with d = diag (ei?1 , . . . , ei?d )

G

Basis transformations for pseudoreal scalars

In a situation with m scalar multiplets transforming under the group according to a pseudoreal irrep D we are led to matrices H ∈ Sp(2m) in the CP–type transformation (7.8). Basis transformations of the form (3.26) involve again Sp(2m) matrices. This suggests the question whether for a given H ∈ Sp(2m) one can ?nd a Z ∈ Sp(2m) such that H ′ = Z ? HZ ? (G.1) is as “simple” as possible. In eq. (8.15) we have written down such a normal form. The proof of its existence will be given here. It is useful to reformulate the problem in terms of antilinear operators and then take advantage of their properties. Let us de?ne the antilinear operator K : x → x? , the matrix Jm = and the antilinear operators A ≡ HK, 0 1m ?1m 0 B ≡ Jm K (G.2)

(G.3)

(G.4)

64 on C2m . Then the symplectic property of H is expressed as [A, B ] = 0. Further properties of A, B are A? A = 1, B?B = 1 and B 2 = ?1. (G.6) (G.5)

This allows the following reformulation of the problem posed above (see eq. (G.1)): Given two antilinear operators A, B on a unitary space V with properties (G.5) and (G.6) can one ?nd canonical forms for A, B ? The answer is the following. Theorem: One can always ?nd an ON basis such that B is given by Jm K and A by eq. (8.15) in that basis. Since basis transformations for matrices corresponding to antilinear operators are performed according to eq. (G.1) the original problem is solved as well. Proof: A is antiunitary and therefore A2 is unitary. Let λ be an eigenvalue of A2 with eigenvector v . Then Av has eigenvalue λ? . Therefore the eigenvalues of A2 can be denoted by
? λ1 , . . . , λν , λ? 1 , . . . , λν , 1, ?1

with

|λi| = 1, λi = ±1 (i = 1, . . . , ν )

(G.7)

with degeneracies m1 , . . . , mν and m± , respectively. The decomposition of V into eigenspaces of A2 is given by
ν

V= with AV (λi ) = V (λ? i ),

i=1

(V (λ i ) ⊕ V (λ ? i )) ⊕ V+ ⊕ V?

B V (λ i ) = V (λ ? i ),

AV± = V± ,

B V± = V± .

(G.8)

Therefore the proof of the above theorem can be devided into three parts. a) V = V (λ) ⊕ V (λ? ), λ = ±1, dim V (λ) = dim V (λ? ) = mλ We choose an ON basis {e1 , . . . , emλ } in V (λ). Then {fi = ?Bei | i = 1, . . . , mλ } de?nes an ON basis in V (λ? ). This allows to write Aei = Mji fj , Antiunitarity of A results in
? Mkj δij = Aei |Aej = Mki

Afi = Mji ej .

(G.9)

(G.10)

with an analogous consideration for M . Consequently, M , M are unitary mλ × mλ matrices. Furthermore, we infer from eq. (G.9) that A2 ei = λei = (M M ? )ki ek . (G.11)

65 Exploiting eq. (G.5) we obtain
? ABei = ?Afi = ?Mji ej = BAei = Mji ej

(G.12) (G.13) (G.14)

and therefore Now we perform a unitary basis transformation on V (λ) e′i = zji ej and explore its e?ect on M : Ae′i = (z T Mz ? )ji fj′ ,
′ Afi′ = ?(z T Mz ? )? ji ej .

M = ?M ? ,

M 2 = ?λ? 1mλ .
? fi′ ≡ ?Be′i = zji fj

and therefore

(G.15)

Unitarity of M allows to choose a z such that z T Mz ? = diag (eiΘ1 , . . . , eiΘmλ ). (G.16)

With ?λ? ≡ ?2 = e2iΘi eq. (G.13), eiΘi = εi ?, εi = ±1 and ε = diag (ε1 , . . . , εmλ ) we ?nally see that A and B are represented by A→ 0 ??? ε ?ε 0 K, B → Jmλ K (G.17)

′ ′ }. , . . . , fm in the basis {e′1 , . . . , e′mλ , f1 λ

b) V = V+

One can easily prove that it is possible to ?nd a vector e1 ∈ V+ with Ae1 = e1 as a consequence of A2 = 1m+ . Then f1 ≡ ?Be1 is orthogonal to e1 , Af1 = f1 and the space orthogonal to {e1 , f1 } is invariant under A, B . Therefore we can repeat the previous steps in {e1 , f1 }⊥ and continue the process until no dimension is left in V+ . Thus we ?nd that m+ is even and A → 1m+ K, B → Jm+ /2 K (G.18)

in the basis {e1 , . . . , em+ /2 , f1 , . . . , fm+ /2 }. c) V = V? Now we have A2 = ?1m? and AB is a unitary operator with (AB )2 = 1m? . Therefore we can ?nd an eigenvector e1 of AB with eigenvalue η1 = ±1. We de?ne f1 ≡ ?Be1 . Then Ae1 |f1 = e1 |ABe1 ? = η1 (G.19) and therefore Ae1 = η1 f1 . With the analogous basis as for eq. (G.18) we ?nd that A→ 0 ?η η 0 K, η = diag (η1 , . . . , ηm? /2 ), B → Jm? /2 K. (G.21) (G.20)

66 We have seen that the matrices for B in eqs. (G.17), (G.18) and (G.21) all have the form Jm eq. (G.3) and A is represented by matrices of the type (8.15). In general, with all cases a, b, c involved, trivial basis permutations lead to eq. (8.15). 2

H

How to solve the generalized CP conditions

The methods and strategies to solve eq. (8.10) in the bases (8.11), (8.13), (8.14) and (8.15) will be discussed in this appendix. Thereby the ranges of the angles 0 < Θ, Θ′ ≤ π/2 and 0 < ΘH < π have to be kept in mind because they will play a crucial r? ole in the following. The ?rst observation is that the matrix O (?) (8.12) has eigenvalues exp(±i?): O (?) 1 ±i = e±i? 1 ±i . (H.1)

If 0 < ? < π , there are no real eigenvalues. In the cases 1b) and 2a) eigenvectors of such a rotation matrix with eigenvalues ±1 are required which do not exist in the ranges of Θ and ΘH . Therefore, we have only the zero solutions. Case 1c) is trivially solved. Expressing A2 by A1 in case 2b) gives A2 = A1 cos ΘH ? O (Θ)A? 1 . sin ΘH (H.2)

For the second equality gives the condition cos ΘA1 = cos ΘH A? 1. (H.3)

Therefore, A1 = A2 = 0 for Θ ∈ {ΘH , π ? ΘH }. Taking Θ = ΘH = π/2, we necessarily have A1 ∈ R2 , Θ = π ? ΘH = π/2 requires A1 to be imaginary. For Θ = ΘH = π/2 eq. (H.3) gives no restriction and A1 ∈ C2 . A2 is determined by eq. (H.2). Case 2c) requires A2 = ?dO (Θ)A? 1 and O (2Θ)A1 = ?d2 A1 . (H.4)

Thus ?d2 must be an eigenvalue of O (2Θ) of the form e±2iΘ or A1 = A2 = 0 if d2 = ?e±2iΘ . For Θ = π/2 and d = ε the matrix A1 is arbitrary. On the other hand, Θ < π/2 requires A1 to be an eigenvector of O (2Θ) from which the solution follows. For the discussion of the remaining three cases we make the ansatz A1 (or A) =
r,s=± T Ars vr vs

with

v± =

1 ?i


and O (Θ)T v± = e±iΘ v± ,

T T v± O (Θ′) = e±iΘ v± .

(H.5)

67 Furthermore, after summation we obtain A1 (or A) = A++ + A?? + A+? + A?+ i(?A++ + A?? + A+? ? A?+ ) i(?A++ + A?? ? A+? + A?+ ) ?A++ ? A?? + A+? + A?+ . (H.6) (H.7)

Application of eq. (H.5) in case 3a) readily gives
r,s T (Ars ei(rΘ+sΘ ) ? A? ?r ?s )vr vs = 0.


Evaluation of this equation leads to A++ (e2i(Θ+Θ ) ? 1) = A+? (e2i(Θ?Θ ) ? 1) = 0.
′ ′

(H.8)

The only possibility for A++ · A+? = 0 requires Θ = Θ′ = π/2 and, consequently, ? A?? = ?A? ++ and A?+ = A+? . Inserting this into eq. (H.6) gives the ?rst subcase of the solutions 3a). Going on to Θ = Θ′ < π/2 one gets A++ = A?? = 0 and A?+ = A? +? . Inspecting again eq. (H.6) one sees that A has the same form as before but now its elements are real. Finally, Θ = Θ′ gives A+? = 0 and Θ + Θ′ < π . Therefore also A++ = 0 and thus A = 0 results. The last two cases are more complicated. In 3b) we get the following equation for A1 : O (Θ)T A1 O (Θ′ ) + O (Θ)A1O (Θ′ )T = 2 cos ΘH A? 1. Using the ansatz (H.5) we straightforwardly arrive at
r,s T (Ars cos(r Θ + sΘ′ ) ? A? ?r ?s cos ΘH )vr vs = 0.

(H.9)

(H.10)

Furthermore, A2 is expressed in terms of A1 by A2 =
′ T A1 cos ΘH ? O (Θ)A? 1 O (Θ ) . sin ΘH

(H.11)

Let us ?rst discuss ΘH = π/2. Then A+? = A?+ = 0 because cos(Θ ? Θ′ ) = 0. On the other hand, it is clear that A++ , A?? = 0 is only possible if Θ + Θ′ = π/2. Thus with the help of eq. (H.6) we obtain the A1 of the ?rst solution of 3b). Applying eq. (H.11) we get with Σ = diag (1, ?1) A2 = ?O (Θ) = ?Σ a b b ?a
? ?

O (Θ′ )T = ?ΣO (?Θ)


a b ?b a

O (?Θ ? Θ ) =

a b b ?a

?

a b ?b a

?

O (Θ′)T = ?b a a b
?

0 1 ?1 0

. (H.12)

Now we turn to the discussion of ΘH = π/2. Our starting point is A++ (cos2 (Θ + Θ′ ) ? cos2 ΘH ) = A+? (cos2 (Θ ? Θ′ ) ? cos2 ΘH ) = 0. (H.13)

68 We can only have both A++ and A+? non–zero at the same time if cos2 (Θ + Θ′ ) = cos2 (Θ ? Θ′ ). It is easy to check that in such a case either Θ = π/2 or Θ′ = π/2 (in addition we have Θ = Θ′ to prevent ΘH = π/2). Let us ?rst discuss Θ = π/2 and Θ′ < π/2. Then cos(Θ ± Θ′ ) = ? sin Θ′ , cos ΘH = ε sin Θ′ (ε = ±1) and A?? = ?εA? ++ , ? A?+ = εA+? . With eq. (H.6) we recover A1 of the second solution of 3b). Using eq. (H.9) to replace A1 cos ΘH in eq. (H.11) we obtain A2 = 1 ′ ? ′ T (O (Θ)T A? 1 O (Θ ) ? O (Θ)A1 O (Θ ) ). 2 sin ΘH (H.14)

This equation is generally valid in case 3b). Taking Θ = π/2 eq. (H.14) simpli?es to A2 = 0 ?1 1 0 A? 1. (H.15)

The subcase Θ′ = π/2, Θ < π/2 is dealt with in an analogous way. Finally, we consider Θ, Θ′ < π/2. Under this condition A++ and A+? cannot be non– zero at the same time. Thus assuming ?rst cos ΘH = ?ε cos(Θ + Θ′ ) we have A+? = A?+ = 0 and A?? = ?εA? ++ . This gives A1 of the fourth solution of 3b). Now we apply eq. (H.14) and derive A2 = 1 ′ ? ′ (ΣO (Θ)(ΣA? 1 )O (Θ ) ? ΣO (?Θ)(ΣA1 )O (?Θ )) 2 sin ΘH 1 0 1 ′ ′ ? = A? . 1 (O (Θ + Θ ) ? O (?Θ ? Θ )) = A1 ? 1 0 2 sin ΘH

(H.16)

In this way we have obtained the complete fourth solution. For cos ΘH = ε cos(Θ ? Θ′ ) one can apply the same procedure. Finally, let us discuss case 3c). A1 is determined by A1 = ?d2 O (2Θ)T A1 O (2Θ′). Thus with the decomposition (H.5) we get Ars (d2 e2i(rΘ+sΘ ) + 1) = 0


(H.17)

? r, s = ±.

(H.18)

There is exactly one possibility to have an arbitrary A1 , namely d = iε, Θ = Θ′ = π/2. A2 is always computed by ′ A2 = d? O (Θ)T A? (H.19) 1 O (Θ ) in case 3c). Next we observe that it is possible to get A++ , A?? = 0 by Θ + Θ′ = π/2, d = ε. In this case A+? = A?+ = 0. The A2 is given by
′ ? A2 = εΣO (Θ)ΣA? 1 O (Θ ) = εA1

0 1 ?1 0

.

(H.20)

69 There is an analogous case with A++ = A?? = 0 and A+? , A?+ = 0 characterized by Θ = Θ′ = π/2 and d = iε. Allowing for A++ , A+? = 0 the conditions ? d2 e2i(Θ+Θ ) = ?d2 e2i(Θ?Θ ) = 1
′ ′

(H.21)

must be ful?lled. In this case Θ′ = π/2 and d = εe?iΘ . Then eqs. (H.6) and (H.19) determine A1 and A2 , respectively. There are three similar cases characterized by A++ , A?+ = 0, A?? , A+? = 0 and A?? , A?+ = 0. Finally we are left with cases where only one Ars is non–zero. The discussion goes along the line presented here.

I

On the case DΛ ? DΛ ? ψd in so(2?)

The non–trivial diagram automorphism of D? (? ≥ 5) corresponds to exchanging α??1 and α? in the Dynkin diagram (?g. 1), for D4 there are other ones in addition (see sect. 4). Then DΛ ? DΛ ? ψd for irreps with highest weight Λ is valid if and only if n??1 = n? . Here we want to show that if this is the case and if DΛ is given in a CP basis (see subsect. 5.2) then the unitary matrix W establishing the equivalence W DΛ W ? = DΛ ? ψd (I.1)

can be chosen real and symmetric (see subsect. 9.2 for the use of this result). To proceed with the proof we ?rst describe how irreps of the kind n??1 = n? can be built up as tensor products of the de?ning fundamental irrep DΛ1 . For simplicity of notation we write Dj ≡ DΛj . Then one can show that [24] ∧k D1 ? = Dk (k = 1, . . . , ? ? 2) and ∧??1 D1 ? = DΛ ′ with Λ′ = Λ??1 + Λ? . (I.2)

The symbol ∧k denotes the k –fold antisymmetric tensor product. Thus DΛ with Λ = n1 Λ1 + . . . + n??2 Λ??2 + n? (Λ??1 + Λ? ) can be obtained as the irrep with highest weight in
?n? ?n1 D1 ? . . . ? D??2??2 ? DΛ ′ ?n

(I.3)

where the superscript ?nj indicates the nj –fold tensor product of Dj . DΛ can therefore be constructed from tensor products of D1 . The r? esum? e of this consideration is that if one can show that AD1 A? = D1 ? ψd is achieved with a real symmetric matrix A for D1 in a CP basis the W for DΛ with n??1 = n? is obtained by suitable tensor products of A conserving reality and symmetry. Also in these tensor products DΛ (eα ) will be real and DΛ (?iHj ) imaginary and symmetric if this is so in D1 . Then by reversing the argument leading to eq. (5.13) and by using DΛ (eα )? = ?DΛ (e?α ) we easily see that DΛ is given in a CP basis. Let us now consider D1 and show ?rst that there is a real and symmetric matrix A with AD1 A? = D1 ? ψd where D1 is given in the natural basis of so(2?) with real antisymmetric

70 matrices. We will closely follow the discussion in app. G of ref. [16]. Using the matrices {Mpq | 1 ≤ p < q ≤ 2?} (see eq. (C.1)) as a basis of so(2?) or of its complexi?cation D? then {hj = M2j ?1,2j | j = 1, . . . , ?} de?nes a basis of the CSA. It is shown in ref. [16] that with the functionals εj on the CSA de?ned by εj (hk ) = iδjk 1 16(? ? 1) 1 16(? ? 1) (j, k = 1, . . . , ?) (I.4)

all basis elements eα with positive roots are given by eεj +εk = eεj ?εk = (M2j,2k ? iM2j,2k?1 ? iM2j ?1,2k ? M2j ?1,2k?1 ) (M2j,2k + iM2j,2k?1 ? iM2j ?1,2k + M2j ?1,2k?1 ) (I.5)

with 1 ≤ j < k ≤ ?. Therefore ?+ = {εj + εk , εj ? εk | 1 ≤ j < k ≤ ?} and the simple roots may be de?ned as αj = Then one can calculate hαj = ? and hα? = ? With i (M2j ?1,2j ? M2j +1,2j +2) 4(? ? 1) (j = 1, 2, . . . , ? ? 1) (I.8) εj ? εj +1, j = 1, 2, . . . , ? ? 1 ε??1 + ε? , j = ?. (I.7) (I.6)

i (M2??3,2??2 + M2??1,2? ). 4(? ? 1) t≡ 0 1 ?1 0

(I.9)

we have hα? = ? hα??1 hα??2

i diag (02??4 , t, t), 4(? ? 1) i diag (02??4 , t, ?t), = ? 4(? ? 1) i diag (02??6 , t, ?t, 02 ), etc. = ? 4(? ? 1)

(I.10)

In eq. (I.10) the symbol diag means arranging the matrices along the diagonal and 0m denotes the m × m zero matrix. The basis elements corresponding to simple roots are represented by 1 diag (02??4 , F ′) eα? = 16(? ? 1)

71 with
? ? ? ? ?

F′ = ? and, for j = ?, eαj = with 1

0 0 ?1 ?i 0 0 ?i 1 ? ? ? 1 i 0 0 ? i ?1 0 0 diag (02(j ?1) , F, 02(??1?j ))
?

(I.11)

16(? ? 1)
? ? ? ?

F =? The diagram automorphism ψd is and eαj : ? ? ? hαj , ψd (hαj ) = hα? , ? ? hα??1 ,

0 0 0 0 ?1 ?i i ?1

already uniquely determined by its action on hαj

1 ?i i 1 ? ? ?. 0 0 ? 0 0

(I.12)

j ≤??2 j = ? ? 1, ψd (eαj ) analogously j=?

(I.13)

Then one quickly con?rms with eqs. (I.10), (I.11) and (I.12) that eq. (I.13) is reproduced by ψd (X ) = AXA with A = diag (12??2 , ?1, 1) (I.14) for X ∈ so(2?) or its complexi?cation D? . Since D1 (X ) = X we have obtained (D1 ? ψd )(X ) = ψd (X ) = AXA = AD1 (X )A? (I.15)

with A real and symmetric. In the last step we have to switch to a CP basis by conserving the above properties of A. De?ning T = diag (s, . . . , s) we get T ? hj T = i diag (02(j ?1) , τ3 , 02(??j ) ) with the Pauli matrix τ3 . Furthermore, one readily veri?es that
? ? (T ? Mpq T )rs = Tpr Tqs ? Tqr Tps .

with

1 s= √ 2

1 1 i ?i

(I.16)

(I.17)

(I.18)

This allows to check easily that (T ? M2j,2k T )rs and (T ? M2j,2k?1 T )rs and (T ? M2j ?1,2k T )rs are imaginary ? r, s. and (T ? M2j ?1,2k?1 T )rs are real ? r, s

72 Therefore, looking at eq. (I.5) we immediately see that we have T ? eεj ±εk T real ? j, k = 1, . . . , ? with j < k. (I.19)

Thus the matrix T achieves a basis transformation into a CP basis. Finally, with the Pauli matrix τ1 we obtain A = T ? AT = diag (12??2 , ?τ1 ) (I.20)

which is real and symmetric. This proves the claim made in the beginning of this appendix.

73 L A1 A? (? ≥ 2) B? (? ≥ 2) C? (? ≥ 3) D4 D? (? = 5, 7, 9, . . .) D? (? = 6, 8, 10, . . .) E6 E7 E8 F4 G2 Lc su(2) su(? + 1) so(2? + 1) sp(2?) so(8) so(2?) so(2?) cE6 cE7 cE8 cF4 cG2 Aut (Lc )/Int (Lc ) {e} Z2 {e} {e} S3 Z2 Z2 Z2 {e} {e} {e} {e} ψ△ inner outer inner inner inner outer inner outer inner inner inner inner

Table 1: The complete list of simple complex Lie algebras with their compact real forms and the structure of their automorphism groups. We have also indicated in which algebras the contragredient automorphism ψ △ is inner (ψ △ ∈ Int (Lc )) or outer (ψ △ ∈ / Int (Lc )). S3 denotes the group of permutations of {1, 2, 3} and sp(2?) is the Lie algebra corresponding to the compact symplectic group Sp(2?) = {U ∈ U (2?)| U T JU = J } with J= 0 1? ?1? 0 .

74

so(3) sp(2) sp(4) so(2) so(4) so(6)

? = ? = ? = ? = ? = ? =

su(2) su(2) so(5) u(1) su(2) ⊕ su(2) su(4)

Table 2: Complete list of isomorphisms within the four series of classical compact algebras su(N ) (N ≥ 2), so(N ) (N ≥ 2) and sp(N ) (N = 2, 4, 6, . . .). Of all these algebras only so(2) and so(4) are not simple. The last two isomorphisms are proved in apps. C and D, respectively.

References
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77

A? (? = 1, 2, 3, . . .)

α1

f

α2

f

α3

f

··· f α??1

α?

f

B? (? = 2, 3, 4, . . .)

α1

f

α2

f

α3

f

··· f α??1

α?

v

C? (? = 3, 4, 5, . . .)

α1

v

α2

v

α3

v

··· v α??1
 

α?
fα?

f

D? (? = 4, 5, 6, . . .)

α1

f

α2

f

 ··· f α3 α??2 T f fα6

T T fα

??1

E6

α1

f

α2

f

α3

f fα7

α4

f

α5

f

E7

α1

f

α2

f

α3

f fα8

α4

f

α5

f

α6

f

E8

α1

f

α2

f

α3

f

α4

f

α5

f

α6

f

α7

f

F4

α1

f

α2

f

α3

v

α4

v

G2

α1

f

α2

v

Figure 1: The Dynkin diagrams of all simple complex Lie algebras.



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