Automorphisms in Gauge Theories and the Definition of CP and P
W. Grimus (U. Wien), M. N. Rebelo (U. Wien, CFIF-IST, Lisbon)
TL;DR
This paper explores how CP and parity symmetries can be defined in general gauge theories through automorphisms of the gauge Lie algebra, with focus on scalar interactions and Yukawa couplings.
Contribution
It provides a framework linking discrete symmetries to gauge algebra automorphisms, clarifying their definitions in complex gauge theories.
Findings
Automorphisms of gauge Lie algebra can define CP and P symmetries.
Conditions for CP invariance of Yukawa couplings are identified.
Implications for scalar gauge interactions are discussed.
Abstract
We study the possibilities to define 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.
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UWThPh-1995-7
June 1995
**Automorphisms in Gauge Theories and the Definition of CP and P
** W. Grimus and M.N. Rebelo*
Institut für Theoretische Physik
Universität Wien
Boltzmanngasse 5, A–1090 Wien, Austria Abstract
We study the possibilities to define 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 Förderung der wissenschaftlichen Forschung, Project No. P08955–PHY. Permanent address: Dept. de Física and CFIF-IST, Instituto Superior Técnico, Lisbon, Portugal (on leave until September 1995).
Contents
-
3 Conditions for CP–type transformations: gauge bosons and fermions
-
8.2 Real Clebsch–Gordan coefficients and the generalized CP condition
1 Introduction
The discrete transformations charge conjugation (C), parity (P) and CP played an important rôle in the development of particle physics. In particular, the hypothesis of P (C) non–conservation [1] and its subsequent experimental confirmation [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 fields as the basic fermionic degrees of freedom. In the SM, CP and P non–invariance appear in two totally different ways: parity is broken because left and right–handed fermionic multiplets belong to different 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 only affects 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 Unified 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 offers 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 defined 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 fields have the same chirality. Such a formulation is always possible since it is equivalent, e.g., to the use of the right–handed field
[TABLE]
instead of (for conventions and the definition of the matrix , see app. A). In this work we will use right–handed fields for definiteness. 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 [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 first chapters of this paper. Furthermore, gauge interactions of scalar fields, 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 definition of CP and P transformations arise for Higgs fields in real and pseudoreal irreducible representations (irreps) of . Finally, the third focus is on Yukawa interactions . In this connection not only the group structure of the couplings represented by Clebsch–Gordan coefficients is important but also the couplings pertaining to each group singlet contained in . In the case of replication of irreps of like the well–known families of quarks and leptons there is freedom of defining 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 definition of CP, and requiring CP invariance of 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 as gauge groups which entails unitary representations (reps) and the real Lie algebra of is compact (see app. B). On the other hand, requiring unitary reps on quantum theoretical grounds, it can be shown that one can confine oneself to compact groups [11]. We will not only cover simple groups but also groups of the type and with and 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 ( 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 where is the group of these additional symmetries. Multiplets identical with respect to the gauge group may be distinguished by different transformation properties under . In certain cases this can forbid operations to be discussed in what follows which involve non–trivial multiplicities of irreps of . For instance, if in a model there is a complex scalar multiplet transforming according to a real irrep of but to a complex irrep of 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 it will be tacitly assumed that these reps can be extended to reps of . 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.
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 –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 fields and fermions are derived. Since Condition A requires that a CP–type transformation on the gauge fields corresponds to an automorphism of sect. 4 is devoted to a detailed discussion of Aut . 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 are either symmetric or antisymmetric matrices. In sect. 6 the general solution of Condition B is expounded and parity transformations are defined 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:
[TABLE]
The brace signifies 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 –GUT, where all fermionic degrees of freedom are in a 16–dimensional irrep, allows to define 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 matrices and the charge conjugation matrix can be found. All properties of Lie algebras necessary for our purposes are collected in app. B. In app. C spinor reps of are defined via the Clifford algebra and app. D describes the Lie algebra isomorphisms and . 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 and in CP–type transformations. Equivalence of reps will be denoted by . The letter will be used for reps of and also of . Its use should be clear from the context. Yet refers only to the complex conjugate of the Lie algebra rep . Finally let us collect the abbreviations used in this work: CSA, GUT, ON, SM, rep and irrep denote Cartan subalgebra, Grand Unified Theory, orthonormal, Standard Model, representation and irreducible representation, respectively.
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 (strictly speaking, an irrep of , its universal covering group, see e.g. ref. [16]).
QED:
Denoting the electron field by we know that the Lagrangian is invariant under
[TABLE]
where . In terms of chiral fields eq. (2.1) reads
[TABLE]
expressing the fact that, having started with Dirac fields, CP does not mix chiralities whereas parity connects fields with different chirality. Defining
[TABLE]
we can get yet another view of eq. (2.1) [17], namely
[TABLE]
With
[TABLE]
we observe that the form of CP and P is only distinguished by the matrix in the CP–type transformation
[TABLE]
How are and characterized with respect to the gauge group ? The chiral fields transform with complex conjugate phases, i.e.
[TABLE]
and the chiral field vector consists of two irreps which are exchanged under whereas acts within the respective irreps. Thus, in the formulation where the previously mentioned structural similarity becomes obvious, CP and P can be distinguished by group theoretical properties of the matrix . Eq. (2.4) has to be supplemented by the transformation properties of the photon field
[TABLE]
where the are the signs for and for .
QCD:
We confine ourselves to a single quark flavour . 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
[TABLE]
and
[TABLE]
Eq. (2.7) now reads
[TABLE]
with and eq. (2.8) is adapted to QCD by
[TABLE]
with . The signs depend on the transposition properties of the generators where are the Gell–Mann matrices [18] and are obtained by
[TABLE]
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 being an automorphism of the Lie algebra .
– GUT:
A more involved example of CP and P is provided by a Grand Unified Theory based on [19, 20, 21, 22] or actually on its universal covering group . Here, all 15 fundamental fermions of one SM family together with an additional right–handed neutrino carrying lepton number fit 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
[TABLE]
since the classification of the states according to is well known. Here, denotes the colour or Pati–Salam– [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 by
[TABLE]
where or, abbreviated, , and
[TABLE]
with
[TABLE]
The 16–dimensional subspaces for the spinor irreps and are given by the two projectors on the 32–dimensional space where
[TABLE]
This means we are taking as basis vectors those with
[TABLE]
for the or the , respectively.
The subalgebra of is assumed to be generated by , , and by , . The CSA {} of is easily carried over to the spinor representation of by using the Clifford algebra (see app. C) and the mapping of the Gell–Mann basis into as given in app. D.111The in eq. (D.4) have to be replaced by to comply with the embedding of into chosen above. Thus we obtain
[TABLE]
Here, and are baryon and lepton number, respectively, and {} generate the CSA of . The symbol denotes the p-fold tensor product of the unit matrix and the subscript refers to the spinor rep.
By a completely analogous procedure the two subalgebras of can be obtained:
[TABLE]
Comparing eq. (2.19) with eq. (D.1), we see the correspondences and . The advantage of the choice compared with the equivalent will become clear at the end of this section. The algebra generates whereas generates .
The remaining two elements of the Cartan subalgebra in the spinor rep of are given by
[TABLE]
This completes the definition of the CSA of in the 32–dimensional spinor rep. We can now easily construct multiplets by means of the lowering operators
[TABLE]
with and find the basis vectors, omitting the part,
[TABLE]
Calling the quarks with quantum numbers , , red, yellow and blue, respectively, we can start with a right–handed red up–quark state . Then, applying we obtain . The multiplet is completed by the raising and lowering operators
[TABLE]
Thus, having embedded in we find for its multiplet of right–handed quarks
[TABLE]
The electric charges of the states can easily be checked by means of the charge operator
[TABLE]
From this and using , (see eq. (2.18)) we find that has the quantum numbers of a right–handed SM singlet neutrino, whereas is the right–handed electron state. The state has the quantum numbers of an anti–red quark belonging to the (2,1) multiplet of with eigenvalue of . This leads to the multiplets
[TABLE]
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 . All the quantum numbers correspond to the choice of right–handed fields in the irrep. The construction of the gauge theory based on and the spinor representation for the multiplet , the physical content of which has been described above, is now straightforward using the hermitian generators , with (see eqs. (C.3) and (C.4)).
Anticipating sect. 3 and trying
[TABLE]
we merely have to check that invariance of the Lagrangian is achieved by an appropriate transformation of the gauge fields. In our case it is convenient to number the 45 gauge fields by , corresponding to the generators . Analogously to the case of QCD it is easy to see that it is sufficient for CP invariance to transform the gauge fields as
[TABLE]
where the signs are obtained through
[TABLE]
It is easy to calculate the recalling the definition of in terms of the elements of the Clifford algebra (see app. C). We have
[TABLE]
therefore
[TABLE]
and
[TABLE]
In sect. 5 it will be shown that one can always choose a basis in representation space such that is valid for the generators. Eq. (2.29) tells us that
[TABLE]
is an automorphism of (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 and correspond to each other, we now have such a correspondence within one multiplet. In sect. 6 we will call the first kind of parity “external” and the second one “internal”. One can indeed formulate a parity transformation by
[TABLE]
with
[TABLE]
and
[TABLE]
where the signs are now obtained by (to be derived in sect. 3)
[TABLE]
or else, using eqs. (2.31) and (2.32), by
[TABLE]
Again, one can check that the pure gauge terms of the Lagrangian are invariant under the transformation (2.35). Note that the element of the CSA commutes with , whereas , , , anticommute. In other words, transforms eigenvectors of into states with the same eigenvalue, whereas the eigenvalues of , , and are reversed.
Our sign conventions are such that verifies
[TABLE]
where this notation means that is applied to all the states in the parentheses. Taking into account eq. (2.19) and for we obtain
[TABLE]
From this it follows that the fields and corresponding to (4,1,2) and , respectively, in the irrep transform with matrices which are exactly complex conjugate to each other. The same is true for the “diagonal ” generated by . Thus and transform alike under and correspond to and in QED, or and in QCD. Thus our conventions eqs. (2.19) and (2.38) were suggested to exhibit the common features of the three examples.
3 Conditions for CP–type transformations: gauge bosons and
fermions
3.1 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 be the hermitian generators in the fermionic representation such that
[TABLE]
and
[TABLE]
with totally antisymmetric coefficients (see app. B, first subsection). Defining
[TABLE]
the pure gauge Lagrangian is given by (see, e.g., ref. [23])
[TABLE]
with
[TABLE]
Under a gauge transformation, the fermionic multiplet and the fields transform as
[TABLE]
with . As a result the field strength tensor transforms according to the adjoint representation written as
[TABLE]
therefore leaving invariant.
Having fixed our notation we will now examine in detail the effect 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
[TABLE]
where is the number of gauge bosons and thus equal to the number of group generators. The fields are real and therefore is a real matrix. It will shortly become clear that the orthogonality condition (3.2) requires to be orthogonal. Let us now consider the effect of the transformation (3.9) on the field strength tensor. For the terms with derivatives we have
[TABLE]
where is the derivative with respect to . The commutator transforms according to
[TABLE]
Consequently, under a CP–type transformation behaves as
[TABLE]
with
[TABLE]
This leads us to the first condition for invariance of under a CP–type transformation:
[TABLE]
In what follows eq. (3.14) will be referred to as Condition A. If it is fulfilled we get
[TABLE]
and is clearly invariant under such a transformation. Note that eq. (3.10) together with eq. (3.2) already leads to orthogonal in order to get invariance of the quadratic part of 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 field vector transforming according to the rep (see app. A). Hence the fermionic Lagrangian is given by
[TABLE]
The general form of a CP transformation acting on the fermionic multiplet is given by
[TABLE]
where , here, is a constant unitary matrix in representation space, i.e., in the same space on which the rep operates. It can be easily checked that the kinetic part of eq. (3.16) transforms as under eq. (3.17) whilst the interaction term leads to the invariance condition
[TABLE]
which can readily be cast into the form
[TABLE]
In what follows we will refer to this equation as Condition B. It is easily verified that fulfills the commutation relations (3.1) for 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 defined above.
As we have seen is an orthogonal matrix and a unitary matrix with the dimension of . In general the rep will not be irreducible and we can therefore perform a decomposition into irreps
[TABLE]
where is an ON basis of the real compact Lie algebra (see app. B); is the multiplicity of the irrep in of dimension ; the direct sum runs over all irreps included in and its total dimension is . 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 be a solution of Conditions A and B and let be another solution. Then
[TABLE]
where the are unitary matrices.
Proof:
Inserting into Condition B we obtain
[TABLE]
Then Schur’s lemma forces to be of the form (3.20). Since and are both unitary the matrices are unitary as well.
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 with the freedom in the horizontal component simply given by eq. (3.20).
Defining a linear operator on by (see app. B)
[TABLE]
we infer from Condition A that
[TABLE]
Therefore is an automorphism of as well as since the set of automorphisms of forms a group (this can also be verified by examining Condition A). As a result Condition A can be formulated as
[TABLE]
For every irrep the complex conjugate irrep is given by . Clearly, defined by is also an irrep. Thus Condition B can be read as
[TABLE]
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.
Finally, let us write down the effect of changing the basis of the fermionic multiplet on a CP–type transformation. Let be the field vector in the new basis and
[TABLE]
As a result the new matrix in eq. (3.17) is given by
[TABLE]
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 Automorphisms of
4.1 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 , Aut , plays an important rôle 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 of we can define an automorphism of the form
[TABLE]
Automorphisms of this kind are called inner. The representation of 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 from a group denoted by Int . For a connected Lie group it is isomorphic to where is the centre of [26]. Moreover, Int is a normal subgroup of Aut since if then and therefore .
Outer automorphisms are automorphisms which are not inner. Clearly, it is sufficient to consider one representative of each coset in Aut in order to make a complete study of outer automorphisms.
Root rotations
[16]: The group of root rotations Aut is defined as a mapping of , the set of (non–zero) roots of (complexification of ), onto itself which fulfills
- a)
such that and 2. b)
.
Aut has an important normal subgroup, the Weyl group . It is the group generated by the elements which act on as
[TABLE]
One can show that .
Associated with each root rotation there is an automorphism of the corresponding algebra , which is also an automorphism of . The connection between Aut and Aut is given by the following theorem.
Theorem:
For every there is a mapping of onto itself defined by
[TABLE]
where such that
[TABLE]
[TABLE]
[TABLE]
Then is an automorphism of and, moreover, restricted to we also have . See app. B for the notation in eqs. (4.3) and (4.4).
In what follows we will need the automorphism induced by root reflexion
[TABLE]
It is clear that and . The index stands for contragredient.
Another important class of root rotations in the case of simple Lie algebras is defined by those which are symmetries of the Dynkin diagram or, equivalently, of the Cartan matrix (see app. B). The group of these rotations is thus given by Aut . From the set of Dynkin diagrams depicted in fig. 1 it is clear that Aut can only be isomorphic to or or . The latter case is only possible for , the complexification of , where the three outer simple roots , , can be permutated leading to the permutation group . For , , and all exceptional algebras except there is no trivial diagram symmetry and therefore Aut is reduced to . For there is symmetry under the inversion of the order of the simple roots, i.e. , . For exchange of and is the unique Dynkin diagram symmetry, i.e., , , . In we can reverse the order of the roots in the line containing the five roots , i.e., there the Dynkin diagram is symmetric under , . The automorphisms associated with non–trivial elements of Aut will be called diagram automorphisms and denoted by .
The essence of the above discussion is related to the facts that for a simple Lie algebra , is an inner automorphism if and only if the root rotation is an element of the Weyl groyp of and diagram automorphisms are always outer automorphisms so that we have [16, 24]
[TABLE]
For our purposes it will be sufficient to use as representations of the cosets in Aut the automorphisms , or id (see table 1) depending on the algebra 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 in more complicated cases we need two trivial observations:
- i)
Let be an ideal of and . Then is again an ideal. 2. ii)
Let , be ideals of then also is an ideal.
Let us now consider a few cases:
- i)
with , simple and non–isomorphic:
Here we have
[TABLE]
Proof:
is an ideal of . Then is an ideal of and also of . Since is simple and we have . The same reasoning is valid for the other summand .
In the following we will adopt the physical point of view that even when mathematically , if the associated gauge couplings are different we will consider both Lie algebras as being different from each other.
- ii)
, simple and associated to a group :
In analogy with the previous case we have now
[TABLE]
with the generated by
[TABLE]
Now we can imagine that the Lie algebra is actually associated to a group defined by enlarging by an element such that
[TABLE]
The physical background for this construction is that here the reps we are interested in are actually reps of with the representation of giving a symmetry reason for equal coupling constants for both group factors in . In app. E the irreps of are derived. Here we give the full list. Let denote the irreps of . Then all irreps of are given by one of the following constructions:
- is represented by and .
- is represented by and .
These irreps serve as a guideline for the construction of theories with gauge group and equal coupling constants. In specific models can be used as the automorphism associated with P (see sect. 6) or C (see sect. 9) and define in that way a transformation forcing equal gauge coupling constants. 2. iii)
, simple:
Again one can show that
[TABLE]
with generated by
[TABLE]
Actually, since is abelian any transformation with would be an automorphism but the gauge Lagrangian can only be invariant under since associated with must be orthogonal.
4.3 Irreps and automorphisms
As discussed in subsect. 3.2 composition of irreps with automorphisms plays a crucial rôle in solving Condition B. Confining ourselves to simple Lie groups we know that any can be written as or . If is an arbitrary rep then
[TABLE]
Therefore only outer automorphisms can give non–equivalent reps through composition.
To explore the effects of diagram automorphisms we consider first the action of on the fundamental weights (see app. B):
[TABLE]
For convenience we did not distinguish between the diagram symmetry and its ensuing permutation of the indices , i.e. . Next we find the weights of the irrep by comparing it with . Let be the eigenvector with the weight of and where is the multiplicity of . Then
[TABLE]
Since is a root rotation one can show that [16]
[TABLE]
Therefore, if is a weight of then is a weight of . This is valid for any root rotation . If in addition we obtain the highest weight of as (see eq. (4.14))
[TABLE]
if is the highest weight of .
Now we can list the simple complex Lie algebras with non–trivial diagram symmetries and give the conditions for where we indicate the highest weight as a subscript. In all cases except the automorphism is unique. Thus we have if and only if
[TABLE]
For there are five non–trivial diagram automorphisms and only if
[TABLE]
In subsect. 4.2 we have also defined the automorphism associated with root reflexion. Clearly, has the weight if is a weight of . Therefore and relates a rep to its contragredient rep. It is outer for , and . In these cases and thus the condition for is given by eq. (4.18). For all other simple Lie algebras is inner and for all their irreps (see table 1).
5 The existence of CP in gauge theories and Condition B
5.1 The automorphism associated with CP
In this section we will show that the fermionic and vector boson sectors of gauge theories (the terms and of the Lagrangian) are always CP symmetric. The same is true for the couplings of scalars with the gauge fields (see sect. 7). Imposing CP symmetry to the full Lagrangian containing the terms , , , , the Yukawa interactions, and , the scalar potential, will induce conditions on the only terms that may violate this symmetry — and . In sect. 8 we will discuss the conditions induced on — the simplest will be reality of Yukawa couplings but in general the situation is more complicated (see full classification in subsect. 8.3). We will not discuss invariance of in this paper.
In our formulation a CP transformation of the fermionic and gauge boson multiplets is completely specified by the two matrices and defined in sect. 3. Requiring that CP reverses all quantum numbers of each field we see that corresponds to an automorphism which induces a reflexion of all the roots or, in other words, acts as on the CSA (as required by its physical interpretation). Hence we identify with the matrix representation of the contragredient automorphism defined by eq. (4.5) [7, 8]:
[TABLE]
In what concerns the pure gauge term 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 with is too restrictive. This is, however, not the case because any two automorphisms that act in the same way on the CSA will only differ by an inner automorphism generated by an element of the CSA itself [24]. As we shall see in subsect. 6.1 using instead of leads to equivalent conditions for CP invariance.
5.2 Canonical and generalized CP transformations
We consider now how to define a CP transformation on the fermion fields. In sect. 3, eq. (3.19) shows how the representation can be decomposed into irreps . We know from the discussion of sect. 4 that
[TABLE]
because the weights of both irreps are the same. This follows immediately from
[TABLE]
Therefore there exist unitary matrices such that
[TABLE]
Thus we define a “canonical CP transformation” by [7, 8]
[TABLE]
Since solves Conditions A and B we have a symmetry of . This means that a gauge theory without Yukawa interactions and a scalar potential is automatically CP invariant (the scalar couplings to the gauge fields, , 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 fields into those with opposite quantum numbers. In order to see that this is indeed the case let and be the operator implementations of CP and , respectively, on the Hilbert space of states and be a second quantized irreducible field multiplet. If CP is conserved we have
[TABLE]
and, together with eq. (5.4), we obtain
[TABLE]
As we know, in the irrep the CSA is represented with opposite signs compared to thus verifying the above statement.
Theorem I (sect. 3) tells us that given the most general solution of Condition B is obtained by where is a horizontal transformation:
[TABLE]
In the following we will call such transformations associated with “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 can be regarded as the identity matrix. Clearly this result can be generalized to the defining rep of for arbitrary by generalizing the Gell–Mann matrices. The reason for is that these matrices are either symmetric or antisymmetric, therefore and the signs are compensated by the transformation of the gauge bosons . Thus we have . 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 (see eq. (B.23)) and a suitable basis in representation space. To be specific, in app. F the following theorem is proved.
Theorem:
For every irrep of there is an ON basis of () such that
[TABLE]
for the antihermitian generators of in . Furthermore the generators are those of the compact normal form of and therefore the root reflexion is given by
[TABLE]
In other words, the matrices are either imaginary and symmetric or real and antisymmetric and therefore a generalization of (Pauli matrices) for and (Gell–Mann matrices) for 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
[TABLE]
To be more explicit (see app. F) we have the following situation in a CP basis:
[TABLE]
(imaginary and symmetric)
[TABLE]
(real and antisymmetric).
Therefore we get
[TABLE]
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 , the reason for invariance is due to the fact that the fermionic multiplet transforms in such a way that the CP transformed fermion fields are associated to the complex conjugate rep whilst the same transformation in the gauge boson sector will reverse this effect through (this is obvious from the derivation of Condition B in sect. 3). This is the reason why invariance under a canonical CP transformation is verified irrespective of whether the automorphism is inner or outer. In all the examples of sect. 2, QED, QCD and the –GUT, is outer (see table 1).
6 On the general solution of Condition B
6.1 Introduction
In sect. 5 we discussed the solution of Condition B together with the requirement that 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 fields of a single chirality so that the difference between CP and P lies in the properties of and . This justifies that we study now Condition B with a general . Therefore in this section we will discuss invariance of , the fermionic part of the Lagrangian of a gauge theory, under a CP–type transformation without constraining to be the contragredient automorphism. The following theorem will simplify our discussion.
Theorem II:
Let be a pair of matrices that verify Conditions A and B and let represent an inner automorphism of . Then the pairs
[TABLE]
where
[TABLE]
are also solutions of Conditions A and B.
Proof:
and are solutions of Condition A because the set of automorphisms form a group. Since is inner there are real numbers such that eq. (6.2) is fulfilled. Translating eq. (6.2) into the fermion representation we get
[TABLE]
Writing Condition B in terms of the second pair of eq. (6.1) we get
[TABLE]
The other pair can be dealt with in an analogous way.
Theorem II means for the discussion of invariance of under CP–type transformations that if for a given automorphism there is a solution of Condition B (i.e., there is a matrix that verifies the equality) then for any other automorphism differing 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 to the initial one. In mathematical terms this means that it is only the quotient group Aut that needs to be considered so that we can confine ourselves to a particular representative of each coset.
Let us consider now the simple Lie algebras in the light of Theorem II and distinguish three classes of Lie algebras (see table 1) [7]:
- a)
, , , , , ,
Here no outer automorphisms exist and we take as representative. Since is inner is valid for all irreps and therefore there is a matrix such that
[TABLE]
Consequently Condition B is solvable without restrictions on the irrep content of . 2. b)
, ,
Here we have both inner and outer automorphisms. For the inner automorphisms we can consider again as representative, for the outer automorphisms we can choose the contragredient one, , since in this case is outer. In these Lie algebras any outer automorphism will be a composition of with an inner automorphism
: In general an irrep is not equivalent to its complex conjugate. As a result Condition B requires that for any irrep also its complex conjugate must be contained in .
: Choosing a basis where the CSA in the irrep is diagonal it is clear that the weights of are identical with those of . Therefore and Condition B is solvable without restriction on . 3. c)
In this case is an inner automorphism, therefore all irreps are equivalent to . Hence it is appropriate to represent Aut by the two cases and , the unique diagram automorphism for . As we know for there are five non–trivial diagram automorphisms.
: Analogously to case a) Condition B is solvable without restriction on .
: Condition B is solvable only if for any irrep also is contained in . The relationship between and 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)
, simple: We know that there is the automorphism eq. (4.12) on in addition to Aut . Thus we can represent Aut by or where . Each irrep of is characterized by the generator , . Denoting by an irrep of then the irreps of are given by . For also must be contained in whereas for it is sufficient to consider . 2. ii)
, , simple: This case leads to the discussion of each simple summand. Note that the irreps of 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 different gauge coupling constants for and even for and therefore no automorphisms other than Aut exist. 3. iii)
with defined in subsect. 4.2: The only additional automorphism is given by eq. (4.9). It is easy to check that
[TABLE]
for all irreps . Consequently is as good as inner and one only has to consider a single summand.
6.2 On the definition of parity
In the definition of a CP transformation we had the automorphism (changing the sign of all elements of the CSA ) with . This suggests a mathematical definition of a parity transformation via an involutive automorphism (i.e., verifying ) which maps the CSA into itself and does not change the sign of the whole algebra. Therefore we can find an ON basis of where [7]
[TABLE]
From the discussion in subsect. 6.1 it is obvious that if parity is a symmetry of the theory the matrix associated with this transformation either maps a given irrep into itself or connects the irrep to . Both irreps have to be present in with the same multiplicity to fulfill Condition B. In the first case we call such a parity transformation internal with respect to and in the second external. In sect. 2 we had the external cases of QED and QCD and the of as an internal case.
Let us for the time being consider the internal case with a single irrep with dimension . Then we can define a mapping:
[TABLE]
Given an ON basis of weight vectors (see app. B) in the Hilbert space associated with the irrep , where and denotes the multiplicity of , we have
[TABLE]
To derive this relation we have used eq. (6.6) together with the fact that is hermitian and that Condition B can be rewritten as
[TABLE]
As a result eq. (6.8) shows that relates states characterized in general by different weights and given by
[TABLE]
Thus weight vectors associated with are transformed into weight vectors associated with . Only vectors with have no partner. Since is hermitian is orthogonal to for . Obviously the contragredient automorphism does not establish such a relationship between weight vectors because for we would have for all weights.
In the external case restricting oneself to single copies of and , respectively, the above discussion goes through without change, but now and lie in different spaces, namely in the spaces associated with and , 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. Defining
[TABLE]
we get
[TABLE]
and
[TABLE]
This allows to define
[TABLE]
for which has the same quantum numbers as . Then P transforms right into left and vice versa:
[TABLE]
and, similarly,
[TABLE]
To derive this equation we have taken into account eq. (6.8) and
[TABLE]
for internal and external parity, respectively222 is the phase of in or for the internal or external case, respectively.. Eq. (6.15) can be cast into a more general form:
If is a CP–type transformation with then
[TABLE]
Proof:
[TABLE]
In contrast to CP there is no canonical way to define P in general. With eq. (6.6) a mathematical definition was given but what is considered as parity in a model also 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 . 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 , is the type of parity we are used to (QED, QCD). In a basis where the matrices of are just the complex conjugate matrices of we simply form the Dirac field where and transform according to and , respectively. In the example (sect. 2) is an inner automorphism.
7 CP–type transformations in the Higgs sector
7.1 Pseudoreal scalars and CP transformations
Considering a pseudoreal irrep [16] there are two unitary matrices associated with it which are important in our context. The first one is given by the equivalence of the irrep and its complex conjugate. In Lie algebra form it is written as
[TABLE]
where antisymmetry of follows from pseudoreality [16, 31]. The other unitary matrix is associated with the (CP)0 symmetry:
[TABLE]
The matrix is symmetric:
[TABLE]
Proof:
The symmetry of is actually contained in the proof of app. F. There it is first demonstrated that . Then antisymmetry is excluded for irreps of semisimple compact Lie algebras.
Eq. (7.3) is, of course, valid for arbitrary irreps. Clearly, in a CP basis and part of the discussion here would be superfluous. 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 allows to define
[TABLE]
which transforms in exactly the same way as the scalar multiplet under the gauge group. It is more difficult to see that both fields also transform alike under CP:
[TABLE]
This follows from the fact that and are related by
[TABLE]
with an appropriate choice of phase, e.g., for .
Proof of eq. (7.6):
Putting on the right–hand side of eq. (7.2) and using eq. (7.1) one obtains
[TABLE]
Taking advantage of one further derives
[TABLE]
Choosing the phase of such that and making some algebraic manipulations one arrives at eq. (7.6).
Proof of eq. (7.5):
Under CP transforms as
[TABLE]
where in the last step eq. (7.6) has been taken into account.
Let us assume now that there are copies all in the same irrep . Putting the multiplets into a vector then , the scalar kinetic and gauge Lagrangian, can be rewritten as [9, 10]
[TABLE]
This allows for a wider class of generalized CP transformations for pseudoreal scalars leaving invariant defined by
[TABLE]
where is a unitary “horizontal” matrix. One has to take into account, however, that is related to via definition (7.4) which gives a restriction on . With
[TABLE]
this restriction is derived by the requirement
[TABLE]
Exploiting this condition by using eq. (7.6) in the form we finally obtain , and therefore
[TABLE]
where is the unitary symplectic group [16]. is defined by
[TABLE]
In a CP basis one can say more about the matrices and . With the phase convention of eq. (7.6) we get the following result:
[TABLE]
where is the convention of the (CP)0 transformation in the CP basis.
7.2 Real scalars in complex disguise and CP transformations
Scalar multiplets belonging to potentially real irreps [16] can also be represented by complex fields. 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 . The treatment of two real multiplets collected in a complex multiplet can proceed along the same lines as in the case of pseudoreal scalars by the introduction of the corresponding field , instead of splitting it into its real components. When represents a single real irrep it coincides with 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. is given as in eq. (7.2), is now symmetric [16, 31]
[TABLE]
and transforms , an explicitly real realization of the potentially real rep , into :
[TABLE]
Let us first have a look on the relationship between the (CP)0 transformation in and . To do this we note that
[TABLE]
and
[TABLE]
with appropriate phase factors for and .
Proof:
Eq. (7.15) is proved as before in the pseudoreal case but now we choose a phase of 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
[TABLE]
or
[TABLE]
With Schur’s lemma and a phase choice for we obtain eq. (7.16).
Let be the real scalar field. Then the CP transformation on is given by
[TABLE]
and
[TABLE]
Proof:
To prove eq. (7.18) we rewrite eq. (7.15) as
[TABLE]
Therefore
[TABLE]
Thus the general CP formalism is consistent with real fields. Choosing the CP basis we now have
[TABLE]
As before 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 define
[TABLE]
Note that now we have a plus sign in the second relation (compare to eq. (7.4)) because for real irreps. As before and transform alike under CP, eq. (7.5). If has multiplicity we can form the vectors , . Then eq. (7.7) is valid and CP can be defined as in eq. (7.8). But now the restriction on differs from eq. (7.11). Imposing the same condition as in the pseudoreal case (see eq. (7.10)) but taking into account the properties of and in the real case we obtain [10]
[TABLE]
It is easy to check that the matrices of the form eq. (7.21) are exactly those unitary matrices for which
[TABLE]
is valid. Therefore they form a group. This group is identical with up to the basis transformation
[TABLE]
This can readily be seen by inserting eq. (7.23) into eq. (7.22) where we obtain
[TABLE]
Defining real fields , via
[TABLE]
the CP transformation (7.8) for the potentially real irrep is now of the form
[TABLE]
after some calculation. Both matrix products embraced by the parentheses are real. This again shows the consistency of the formalism. Clearly, both and 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 fields. But we will see now by an example from [13, 32, 33] that such cases are not uncommon. There a multiplet transforming as (2,2,0) exists which gives masses to the quarks. It is commonly written as a matrix of complex fields and transforms as
[TABLE]
under the gauge group. The index denotes the matrix version of the field. is given by
[TABLE]
where is the second Pauli matrix. Switching to a vector notation we read off from eq. (7.27):
[TABLE]
Since we suspect immediately that (2,2,0) is a real rep. This can indeed be confirmed by recalling the following theorem.
Theorem:
The irreps of a direct product of groups are exactly the tensor products of irreps and of and , respectively [30]. If and are both real or both pseudoreal then is real [34].
In our case the defining irrep of is pseudoreal having spin 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
[TABLE]
we find the corresponding transformations for the vector
[TABLE]
Of course, now we have . If we perform a basis transformation we get
[TABLE]
The last relation can easily be obtained from eq. (7.13). It is obvious that in the basis where we have real representation matrices. With
[TABLE]
we have indeed and
[TABLE]
Clearly, the new field multiplet 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
[TABLE]
where the indices , and denote real, pseudoreal and complex, respectively. is the covariant derivative. Therefore we can write the scalar kinetic and gauge Lagrangian as
[TABLE]
In this way we summarize the gauge interactions into one formula with automatically correct factors 1/2 for and 1 for and since the contribution of and are equal in (see eq. (7.7)) and the same holds true for and (of course, if in the sector is given by then for it has to be .
In analogy to the fermionic sector we define a CP–type transformation by
[TABLE]
leading to
[TABLE]
Clearly, Condition BH is connected to Condition B by the same automorphism represented by . For the total Lagrangian a CP–type transformation is characterized by the triple .
It is clear from eq. (7.37) that it is always possible to define a CP transformation in in the same way as is done in the fermionic sector. Therefore as for CP–type transformations it remains to consider the automorphisms and in the relevant cases.
When considering the algebras , and in the general discussion of a CP–type transformation the automorphism is relevant in addition to . Obviously, if we consider the scalars in complex irreps we automatically have their complex conjugate irreps in in eq. (7.35) so that Condition BH can be solved for and complex reps. For real irreps and Condition BH becomes trivial because in this case in a basis where the fields are real. In the case of pseudoreal irreps and the CP–type transformation has to be defined as with specified by eq. (7.1). A simple calculation reveals that now eqs. (7.5), (7.8) and (7.11) are valid with replaced by . This shows that for all simple algebras except with (see table 1) arbitrary CP–type transformations are always symmetries of and in this sense the existence of a CP–type symmetry for is more likely than for .
For the simple Lie algebras with the relevant case is , a diagram automorphism. In this case we have if and only if for the highest weight (see subsect. 4.3). In this special case a CP–type symmetry may not exist for since here it requires that both irreps and occur in . The more complicated structure of in this case and also for similar cases in is easily worked out with the methods established in this paper. If and again the considerations in subsect. 7.1 can be used with replaced by .
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 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 . These points will be of significance in subsect. 8.3.
8 Yukawa couplings and CP–type symmetries
8.1 The condition on Yukawa couplings
With the vector of scalar field as defined in eq. (7.35) and all fermionic degrees of freedom in a right–handed vector the most general form of Yukawa couplings is given by
[TABLE]
where Fermi statistics implies . A CP–type transformation characterized by leads to
[TABLE]
if we require invariance of . As a matter of convention we have pulled out the factor from the coupling matrices . The gives plus signs on both sides of eq. (8.2) whereas the factor 1/2 is motivated by the fact that , i.e. terms with different multiplets , occurring in appear twice in , 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 define as those CP–type transformations which fulfill simultaneously Conditions A, B, BH and C. First we have the following statements:
[TABLE]
If are CP–type symmetries then
[TABLE]
is a symmetry of . 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 have two types of indices since they have to tie together irreps and their multiplicities. Therefore, considering matrices which couple irreps , and with multiplicities , and , respectively, where , occur in and in we can write as a tensor product
[TABLE]
with corresponding to the triple . Note that , , are matrices and matrices. Of course, for pseudoreal scalars there are separate coupling matrices for and . Clearly, the individual factors of the coupling matrices (8.5) do not have to be symmetric, only the total matrix has to be. For , the diagonal of couples the same fermion fields to each other and therefore the irrep has to be contained in the symmetric tensor product .
In the further discussion we will confine 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 defined in a canonical way for each irrep in and , 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 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 coefficients are real. Therefore, , the group–theoretical part of , couples the fields 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 coefficients and the generalized CP
condition
It is easy to see that in a tensor product of two irreps , the Clebsch–Gordan coefficients 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 are diagonal. This is possible because the representation matrices of the CSA can simultaneously be diagonalized by an orthogonal matrix is symmetric and real) without disturbing symmetry or antisymmetry of the . Therefore the basis vectors of given by are all eigenvectors of are the canonical basis vectors of , , respectively). We can e.g. choose to be the unique vector with highest weight in , if , are the highest weights of , , respectively. Applying all , or, equivalently, all with simple to gives the representation space associated with . 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 exhausted the whole space [16]. Since all are real and we started with a real basis we finally arrived at a real basis in the Clebsch–Gordan series. Therefore the Clebsch–Gordan coefficients associated with this basis are real.
For the rest of this section we assume that we consider fixed irreps , , with field multiplets , , , respectively. Then one can readily check that
[TABLE]
is a symmetry of
[TABLE]
for real Clebsch–Gordan matrices since the invariance condition is just
[TABLE]
or . Consequently, taking into account the multiplicities , , of the respective irreps a generalized CP transformation given by
[TABLE]
leads to the condition
[TABLE]
in the horizontal spaces on which , , act. Since CP does not connect different irreps the above discussion is also fully general. Of course, the canonical CP symmetry , , would simply require real matrices . 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 simplified by the freedom of choosing suitable bases in the horizontal spaces. As in the previous subsection we stick to fixed irreps , , and therefore we have three independent basis transformations (3.26) represented by the unitary matrices , , . According to the theorem proved in ref. [29] one can find and such that
[TABLE]
with
[TABLE]
In the Higgs sector we have to distinguish the cases as in sect. 7. For a complex scalar multiplet the above theorem applies yielding
[TABLE]
For a real scalar is real and has to be orthogonal. Then the real version of the spectral theorem for normal operators tells that one can achieve
[TABLE]
The case of a pseudoreal scalar requires . In app. G we prove that for any there is such a giving
[TABLE]
with , .
These basis choices allow to solve eq. (8.10) in a piecewise manner with submatrices of of maximal size and at most two different involved at a time. We will denote these submatrices by or , and indicate by arrows which part of , , is under discussion.
The number of different cases to be discussed is reduced by the following observations. Solutions of the cases with are obtained from those with by multiplying or , by . The solutions of the cases , are obtained from those with , by transposition of or , . This leaves the following nine generic cases to be investigated:
, are matrices 2. 1a)
: 3. 1b)
: \begin{array}[]{l}A_{1}\cos\Theta_{H}-A_{2}\sin\Theta_{H}=A_{1}^{*}\\ A_{1}\sin\Theta_{H}+A_{2}\cos\Theta_{H}=A_{2}^{*}\end{array} 4. 1c)
: \begin{array}[]{r}A_{1}d=A_{2}^{*}\\ -A_{2}d^{*}=A_{1}^{*}\end{array} 5. 2)
, are matrices 6. 2a)
: 7. 2b)
: \begin{array}[]{l}O(\Theta)^{T}(A_{1}\cos\Theta_{H}-A_{2}\sin\Theta_{H})=A_{1}^{*}\\ O(\Theta)^{T}(A_{1}\sin\Theta_{H}+A_{2}\cos\Theta_{H})=A_{2}^{*}\end{array} 8. 2c)
: \begin{array}[]{r}O(\Theta)^{T}A_{1}d=A_{2}^{*}\\ -O(\Theta)^{T}A_{2}d^{*}=A_{1}^{*}\end{array} 9. 3)
, are matrices 10. 3a)
: 11. 3b)
: \begin{array}[]{l}O(\Theta)^{T}(A_{1}\cos\Theta_{H}-A_{2}\sin\Theta_{H})O(\Theta^{\prime})=A_{1}^{*}\\ O(\Theta)^{T}(A_{1}\sin\Theta_{H}+A_{2}\cos\Theta_{H})O(\Theta^{\prime})=A_{2}^{*}\end{array} 12. 3c)
: \begin{array}[]{r}O(\Theta)^{T}A_{1}O(\Theta^{\prime})d=A_{2}^{*}\\ -O(\Theta)^{T}A_{2}O(\Theta^{\prime})d^{*}=A_{1}^{*}.\end{array}
[TABLE]
In all the cases we have
[TABLE]
In the following the solutions for or , of 1a) – 3c) are given as functions of , , or 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 denotes the two options .
Solutions:
- 1a)
2. 1b)
3. 1c)
,
4. 2a)
5. 2b)
\Theta_{H}=\Theta=\displaystyle\frac{\pi}{2}\Rightarrow A_{2}=\left(\begin{array}[]{cr}0&-1\\ 1&0\end{array}\right)A_{1}^{*},
\Theta_{H}=\Theta\neq\displaystyle\frac{\pi}{2}\Rightarrow A_{2}=\left(\begin{array}[]{cr}0&-1\\ 1&0\end{array}\right)A_{1},
\Theta_{H}=\pi-\Theta\neq\displaystyle\frac{\pi}{2}\Rightarrow A_{2}=\left(\begin{array}[]{rc}0&1\\ -1&0\end{array}\right)A_{1},
6. 2c)
, d=\varepsilon\Rightarrow A_{2}=\varepsilon\left(\begin{array}[]{cr}0&-1\\ 1&0\end{array}\right)A_{1}^{*},
, d=i\varepsilon e^{\pm i\Theta}\Rightarrow A_{1}=a\left(\begin{array}[]{r}1\\ \pm i\end{array}\right), ,
7. 3a)
\Theta=\Theta^{\prime}=\displaystyle\frac{\pi}{2}\Rightarrow A=\left(\begin{array}[]{cl}a&b\\ -b^{*}&a^{*}\end{array}\right),
\Theta=\Theta^{\prime}\neq\displaystyle\frac{\pi}{2}\Rightarrow A=\left(\begin{array}[]{rc}a&b\\ -b&a\end{array}\right),
8. 3b)
\Theta_{H}=\Theta+\Theta^{\prime}=\displaystyle\frac{\pi}{2}\Rightarrow A_{1}=\left(\begin{array}[]{cr}a&b\\ b&-a\end{array}\right), A_{2}=\left(\begin{array}[]{rc}-b&a\\ a&b\end{array}\right)^{*},
, , \Theta^{\prime}\not=\displaystyle\frac{\pi}{2}\Rightarrow A_{1}=\left(\begin{array}[]{cc}a&b\\ -\varepsilon b^{*}&\varepsilon a^{*}\end{array}\right), A_{2}=\left(\begin{array}[]{cc}\varepsilon b&-\varepsilon a\\ a^{*}&b^{*}\end{array}\right)
, , \Theta\not=\displaystyle\frac{\pi}{2}\Rightarrow A_{1}=\left(\begin{array}[]{cc}a&b\\ -\varepsilon b^{*}&\varepsilon a^{*}\end{array}\right), A_{2}=\left(\begin{array}[]{lc}-b^{*}&a^{*}\\ -\varepsilon a&-\varepsilon b\end{array}\right)
, , ,
\left.\begin{array}[]{lll}A_{1}=i\left(\begin{array}[]{cr}a&b\\ b&-a\end{array}\right),&A_{2}=-i\left(\begin{array}[]{rc}-b&a\\ a&b\end{array}\right)&\mbox{ for }\varepsilon=1\\ A_{1}=\left(\begin{array}[]{cr}a&b\\ b&-a\end{array}\right),&A_{2}=\left(\begin{array}[]{rc}-b&a\\ a&b\end{array}\right)&\mbox{ for }\varepsilon=-1\end{array}\right\}a,b\in{\bf R}
, , ,
\left.\begin{array}[]{lll}A_{1}=\left(\begin{array}[]{rc}a&b\\ -b&a\end{array}\right),&A_{2}=\eta\left(\begin{array}[]{rr}-b&a\\ -a&-b\end{array}\right)&\mbox{ for }\varepsilon=1\\ A_{1}=i\left(\begin{array}[]{rc}a&b\\ -b&a\end{array}\right),&A_{2}=-i\eta\left(\begin{array}[]{rr}-b&a\\ -a&-b\end{array}\right)&\mbox{ for }\varepsilon=-1\end{array}\right\}\begin{array}[]{l}a,b\in{\bf R}\mbox{ and}\\ \eta=\mbox{sgn}(\Theta^{\prime}-\Theta)\end{array}
All other choices of , , lead to . 9. 3c)
, d=i\varepsilon\Rightarrow A_{1}=\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right), A_{2}=-i\varepsilon\left(\begin{array}[]{rr}d^{*}&-c^{*}\\ -b^{*}&a^{*}\end{array}\right),
, d=i\varepsilon\Rightarrow A_{1}=\left(\begin{array}[]{cc}a&b\\ -b&a\end{array}\right), A_{2}=-i\varepsilon\left(\begin{array}[]{rr}a^{*}&b^{*}\\ -b^{*}&a^{*}\end{array}\right),
, d=\varepsilon\Rightarrow A_{1}=\left(\begin{array}[]{cr}a&b\\ b&-a\end{array}\right), A_{2}=\varepsilon\left(\begin{array}[]{rr}-b^{*}&a^{*}\\ a^{*}&b^{*}\end{array}\right),
, , d=\varepsilon e^{-i\Theta}\Rightarrow A_{1}=\left(\begin{array}[]{rr}a&ib\\ -ia&b\end{array}\right), A_{2}=\varepsilon\left(\begin{array}[]{rr}ib^{*}&a^{*}\\ -b^{*}&ia^{*}\end{array}\right),
, , d=\varepsilon e^{-i\Theta^{\prime}}\Rightarrow A_{1}=\left(\begin{array}[]{rr}a&-ia\\ ib&b\end{array}\right), A_{2}=\varepsilon\left(\begin{array}[]{rr}ib^{*}&-b^{*}\\ a^{*}&ia^{*}\end{array}\right),
, , d=\varepsilon e^{i\Theta^{\prime}}\Rightarrow A_{1}=\left(\begin{array}[]{rr}a&ia\\ -ib&b\end{array}\right), A_{2}=\varepsilon\left(\begin{array}[]{rr}-ib^{*}&-b^{*}\\ a^{*}&-ia^{*}\end{array}\right),
, , d=\varepsilon e^{i\Theta}\Rightarrow A_{1}=\left(\begin{array}[]{rr}a&-ib\\ ia&b\end{array}\right), A_{2}=\varepsilon\left(\begin{array}[]{rr}-ib^{*}&a^{*}\\ -b^{*}&-ia^{*}\end{array}\right),
The remaining four non–trivial cases can be uniformly described in the following way:
[TABLE]
with , v_{\pm}=\left(\begin{array}[]{r}1\\ \mp i\end{array}\right), .
For all other choices of and we have .
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 C–type transformations
9.1 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]
[TABLE]
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 is invariant under the transformation (9.1) we have
[TABLE]
As we have learned in subsect. 3.2 the first relation can be interpreted as composed with the automorphism being equivalent to . The second relation has the analogous interpretation for the Higgs fields. It is clear from Schur’s lemma that for the matrices and only act horizontally.
To define a charge conjugation we require as for P and CP that is a non–trivial involution. Having fixed the CSA which determines the quantum numbers of the states in the irreps it is reasonable to require . As in the case of P the CSA can be split into with for . We assume that the CSA of the unbroken part of the SM group is contained in , i.e. , , as any physically viable model must have . Thus has to flip the sign of at least three generators in . To every there is associated a subgroup of generated by the elements of with . In refs. [7, 35] there is a complete list of all such “symmetric subgroups” for simple groups . 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 and future symmetry breaking we refer the reader to the extensive discussion in ref. [7] of these questions.
Let the automorphism defining charge conjugation be given by
[TABLE]
Then as for P we can distinguish an internal and external case with respect to an irrep if or , respectively. As before we consider now the representation space of in the internal case and the direct sum of the spaces of and in the external case. Then
[TABLE]
for the ON basis with weights of . The situation is analogous to P in eqs. (6.8) and (6.10). All states with have a counterpart with opposite quantum numbers with respect to . Defining
[TABLE]
with given as in eq. (6.10) we can perform an analysis as in subsect. 6.2 and obtain
[TABLE]
Here we have used that is a phase in each irrep of just as in the analogous situation in eq. (6.15)333 is the phase of in or for the internal or external case, respectively.. 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 CCPP. This relationship is given by the identifications , and (see eq. (8.4)). Once we fix and choose a CP basis one can therefore identify with in the internal case of parity (see, e.g., the example in sect.2).
Like parity, charge conjugation cannot be defined in the SM where 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 in the case of C invariance.
In the light of the discussion in this section we also learn that in the irreps of , eq. (4.10), would be a candidate for if .
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 define a canonical charge conjugation we confine ourselves to and for every irrep contained in . Given the involutive automorphism we will assume in the following that either or is included in to allow for a definition of C. The whole discussion will be confined to simple Lie algebras.
It is clear that for inner (see eq. (4.1)) the C transformation
[TABLE]
with
[TABLE]
where the irreps , , are coupled together in is just a gauge transformation and is obviously invariant under transformation (9.7). This allows to restrict the further discussion to or . In the following the matrices acting on the minimal number of irreps involved in the Yukawa couplings for a definition of C will always be called , , . They are part of and in eq. (9.1).
In a CP basis is valid and therefore, if , we obtain real, with for pseudoreal and for real (see eqs. (7.12) and (7.19), respectively). For , the parameters analogous to will be denoted by , , respectively. If it is shown in app. I that is real and symmetric, i.e. . As discussed in sect. 4 we choose as the relevant outer automorphism for whereas in all other cases of non–trivial outer involutive automorphisms, i.e., the Lie algebras , and , we take as the relevant automorphism.
We have to distinguish four cases according to the types of irreps with respect to coupled together in :
- a)
, ,
The of eq. (8.7) transforms as
[TABLE]
under the C transformation (9.7). In case a) we always have
[TABLE]
This is so for two reasons. First, for the ’s are symmetric and all ’s are 1. Second, as for 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 by
[TABLE]
with the C transformation given by
[TABLE]
Clearly, eq. (9.10) is necessary to have invariance of eq. (8.7) with given 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 specific sign for all simple groups and all irreps. In any case, in any situation the two sets of coupling matrices and 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 given by eq. (9.12) for redefined with or in case a). 2. b)
, ,
In this case
[TABLE]
Then
[TABLE]
is clearly invariant under (CP)0 and C given by eq. (9.13). 3. c)
, ,
With
[TABLE]
and
[TABLE]
invariance is obtained. 4. c’)
, ,
As before we have
[TABLE]
with
[TABLE] 5. d)
, ,
Now
[TABLE]
is invariant with
[TABLE]
All remaining cases are to be discussed analogously. Thus we have seen that for every C transformation such that is contained in for every irrep of one can write down Yukawa couplings invariant under canonical C and CP with “canonical C” defined in this subsection. This invariance is achieved by a redefinition of the . It is not clear to us whether one could circumvent this redefinition by directly exploring properties of the Clebsch–Gordan coefficients.
9.3 An example with
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 , transforming as , , respectively, where the numbers indicate the dimension of an irrep of . The scalar 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 charge is 1/3 whereas for leptons it is . Therefore it is given by baryon number minus lepton number . Assuming families in we have
[TABLE]
and
[TABLE]
with given by eq. (7.28). and are matrices in family (flavour) space. Note that in eq. (9.22) both and have to be coupled to get the most general Yukawa couplings.
Let us now investigate the effects of the following generalized CP and C transformations [27]:
[TABLE]
Examining the invariance conditions we obtain
[TABLE]
Taking eqs. (9.25) and (9.26) together we see that 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 effect of these conditions is similar to the effect of a horizontal symmetry allowing for relations between flavour mixing and masses but for the top quark mass comes out too low.
Interpreting the transformation properties of fermions and in the light of the irreps of in subsect. 4.2 and ignoring the , eq. (9.21) is in the irrep of and contains two . Thus can be considered as a tensor product with respect to with exchanging the vectors in the product. Since is related to the C in eq. (9.24) it becomes clear that C has the effect of transposition on . Combining CP and C we obtain
[TABLE]
Now we want to switch to the formalism discussed in this paper, namely using eq. (9.21) and eq. (7.29) instead of . Then the fermion rep has the form
[TABLE]
with Pauli matrices . The generators acting on are obtained from the transformation property of eq. (7.27), i.e., corresponds to and to . Therefore we obtain
[TABLE]
Now CP eq. (9.23) is quickly treated. Acting on , eq. (3.17) determines
[TABLE]
Then the automorphism is fixed by Condition B:
[TABLE]
is just for . For we have and a quick look at eq. (9.29) confirms that is fulfilled.
As for charge conjugation eq. (9.24) fixes
[TABLE]
is then given by eq. (9.2). Now we have to check consistency in the Higgs sector where from eq. (9.24) we calculate
[TABLE]
It is then tedious but easy to verify that
[TABLE]
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 defining CP, P and C invariance in gauge theories before spontaneous symmetry breaking. As mentioned in the introduction the breaking of the gauge group and of the discrete symmetries could be performed at the same time. For CP this possibility was first envisaged in ref. [36] and for P in refs. [37, 32].
Our first comment concerns the question of when the above scenario really happens. Let us suppose that the Lagrangian is invariant under a CP transformation where is the unitary matrix appearing in the transformation of the scalar field vector eq. (7.37). Then, if the vacuum expectation value fulfills
[TABLE]
the CP symmetry is not broken [38, 39, 40]. Furthermore, if eq. (10.1) is not fulfilled there are two possibilities. On the one hand, there might exist an element of the total symmetry group of the Lagrangian ( is the gauge group and the group of all other internal symmetry transformations commuting with ) such that its action on is given by and
[TABLE]
Then one can define a new CP symmetry
[TABLE]
with adequate redefinitions in the fermion and gauge boson sectors and eq. (10.1) is fulfilled with replaced by . On the other hand, only if for a given one cannot find an such that complies with eq. (10.1) the CP symmetry associated with the of eq. (10.1) is spontaneously broken together with the gauge group.
The second comment refers to CPT invariance. One might be tempted to define a CPT–type transformation in analogy to CP–type transformations defined in sect. 3 by
[TABLE]
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 , , of eq. (10.4).444We use the transposed matrices in eq. (10.4) to get exactly the form of eq. (9.2) for the invariance conditions. Clearly, if , , are identity matrices, eq. (9.2) is always fulfilled suggesting that CPT should be defined by555The minus in the transformation of is convention, however, the other two minus signs are required for invariance of the Lagrangian.
[TABLE]
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 definition of CP eq. (5.1) via the contragredient automorphism fixes the definition of time reversal in the generalized sense analogous to CP in sect. 5.
It is interesting to note that with , but , non–trivial, conditions (9.2) tell us that , act on , , respectively, as representations of an element of . If , belong to alone then the effect 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 fields 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 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 classified in the literature.
Now what distinguishes CP and P from each other? The automorphism associated with CP is given by the contragredient automorphism which has the property for all elements of the CSA whereas P is associated with an involutive automorphism which reverses the signs of at most part of the CSA. These abstract mathematical definitions were substantiated by physical considerations.
CP is always a symmetry of the gauge Lagrangian , the Lagrangian without Yukawa couplings and the Higgs potential. This statement formulated in the language of representations is expressed by , saying that for every irrep of its complex conjugate irrep connected with the automorphism is equivalent to . (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 is subject to physical boundary conditions. For instance, one would require that 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 definition of parity. Also one could imagine cases with large gauge groups where several automorphisms lead to viable definitions 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 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 satisfied in the scalar sector. We have also worked out the connection between parity and the definition of Dirac fields.
In the scalar sector a complication arises for pseudoreal irreps where the horizontal part of a CP–type transformation in general mixes the pseudoreal fields and (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 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 defined as the composition of two CP–type transformations. Such symmetry transformations associated with the trivial automorphism 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. invariant under (CP)0, and an automorphism associated with C such that is also contained in the reps for every irrep occurring there, then one can construct Yukawa couplings which are invariant under the “simplest” C transformation pertaining to .
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 in arbitrary irreps and real Clebsch Gordan coefficients 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 . Furthermore, W. G. acknowledges numerous enlightening conversations on group theory with H. Urbantke.
Appendices
A Notation and conventions
We follow the conventions of ref. [15]. Therefore we use the metric
[TABLE]
and thus the Dirac algebra is given by
[TABLE]
Furthermore, we assume that the matrices fulfill the hermiticity conditions
[TABLE]
Then
[TABLE]
is hermitian. Left and right–handed fermion fields are given by the conditions
[TABLE]
respectively. The charge conjugation matrix is defined by
[TABLE]
As a consequence of eqs. (A.3) and (A.6) we have666Actually, it can only be derived that is proportional to . For convenience and without loss of generality we assume that these matrices are equal.
[TABLE]
A time reversal transformation requires the definition of a matrix verifying
[TABLE]
From the properties of it is clear that
[TABLE]
with an arbitrary phase . Then, given a solution of the Dirac equation we obtain its time–reflected solution
[TABLE]
In the second quantized version this translates into
[TABLE]
with the time reversal operator acting on the Hilbert space of states as an antiunitary operator.
In this paper we only use right–handed fermion fields. If one starts in a theory with fields of both chirality , then the fermionic Lagrangian
[TABLE]
where , are arbitrary, in general different reps of the gauge group, can easily be rewritten as
[TABLE]
with
[TABLE]
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 of a Lie algebra (always assumed to be over a field or C) allows to define a linear mapping
[TABLE]
Then the symmetric bilinear form obtained by
[TABLE]
is called Killing form. Automorphisms of are defined as linear mappings which respect the Lie algebra product, i.e.
[TABLE]
The set of automorphisms form a group denoted by Aut . Note that the Killing form is invariant under automorphisms:
[TABLE]
Semisimple Lie algebras are those which have no non–zero Abelian ideals. Cartan’s second criterion states that a Lie algebra is semisimple if and only if its dimension is positive and its Killing form non–degenerate.
Given a basis of one can define structure constants by
[TABLE]
One can show that a semisimple Lie group (a group whose (real) Lie algebra is semisimple) is compact if and only if its Lie algebra has a negative definite Killing form. Therefore on such a Lie algebra a scalar product is given by . This allows the definition of ON bases with . In such a case the structure constants are usually denoted by . One can prove that is totally antisymmetric in , , whereas in general only is valid.
Automorphisms and bases of :
Fixing a basis of allows to associate a matrix with every linear operator on and vice versa. Denoting such a matrix by and the corresponding operator by we can thus write
[TABLE]
It is then easy to check by analysing eqs. (B.3) and (B.5) that matrices corresponding to automorphisms fulfill the following condition:
[TABLE]
Furthermore, invariance of the Killing form is expressed by
[TABLE]
For compact Lie algebras and ON bases this immediately translates into the conditions that is an orthogonal matrix and an invariant tensor with respect to :
[TABLE]
The structure of semisimple Lie algebras:
With every Lie group a real Lie algebra is associated and by complexification of a complex Lie algebra . The transition from to conserves semisimplicity. Complex semisimple Lie algebras are fully classified. Conversely, given a complex semisimple Lie algebra there is a well defined procedure to go back to the real Lie algebras associated with , i.e. to find all inequivalent ’s whose complexification is . The structure of is closely connected with . 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 sufficient to consider the construction of the unique compact Lie algebra associated with (see eq. (B.23)) and refer the reader to the books quoted at the beginning of this appendix for the general procedure of “realification”.
A CSA can be defined and its existence proved for general Lie algebras (see refs. [24, 25, 26]). In the case of a semisimple complex Lie algebra a CSA is a maximal abelian subalgebra of such that is completely reducible (i.e. diagonalizable) . All CSAs are mutually conjugate which means that given two CSAs , of then there is an inner automorphism (see eq. (4.1)) such that . In this sense a CSA is unique for semisimple complex Lie algebras . The dimension of the CSA, , is called the rank of .
As a vector space can be decomposed into
[TABLE]
which corresponds to the decomposition of into common eigenstates of :
[TABLE]
with . One can show that
[TABLE]
for semisimple. The eigenvalues depend linearly on and can therefore be regarded as linear functionals on . These non–zero functionals are called roots and the set of roots is denoted by . Since the spaces are one–dimensional it suffices to choose a non–zero vector as a basis . Then one can show that
[TABLE]
Therefore, non–degeneracy of requires
[TABLE]
As a matter of fact already is non–degenerate. Thus there exists a unique such that
[TABLE]
The elements are called root vectors. If then
[TABLE]
The real linear span of is denoted by and is the complexification of . is a positive definite scalar product and therefore is an –dimensional euclidean space. By
[TABLE]
length of roots and the angle between two roots are defined. The basis elements and the root vectors are connected through
[TABLE]
A crucial point of the theory is that the numbers
[TABLE]
are integers (the Cartan integers) and that only is allowed.
On a weak order can be defined by choosing an element such that . Then for functionals on the relation is defined by . Let . Then and . 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 .
The Weyl canonical form of :
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 are normalized to
[TABLE]
Then one has the following commutators:
[TABLE]
with
[TABLE]
Given a weak order on every root can be written as a linear combination of simple roots such that
[TABLE]
for rank .
The compact real form of :
Given we can go back to the compact real Lie algebra by choosing the following basis elements :
[TABLE]
[TABLE]
To a given semisimple complex Lie algebra there is a unique compact real form , up to isomorphisms. One can easily check that the above basis elements fulfill .
Dynkin diagrams:
With the simple roots the Cartan matrix
[TABLE]
is associated. Clearly, and one can show that can only be or .
Considering the case in more detail we find (no sum implied here)
[TABLE]
where is the angle between and . Note that for either or have to be . According to the possible angles we have to distinguish four cases (assuming that for and ):
- a)
or , undetermined 2. b)
or , 3. c)
or , 4. d)
or , .
For and the ratio of the lengths of the simple roots is given by .
A Dynkin diagram of a semisimple complex Lie algebra is defined in the following way:
- i)
To each simple root is associated a point (or vertex) of the diagram. 2. ii)
The points associated with and are connected by lines, i.e. there are zero, one, two or three lines for the above cases a, b, c, d, respectively. 3. 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.
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 fig. 1 all possible connected Dynkin diagrams are depicted with the names of the ’s associated with them. In table 1 all ’s are listed with their respective compact real forms . Table 2 contains all isomorphisms of the low–dimensional classical ’s explaining thus why the series , , start with , respectively.
Some facts about irreps of semisimple complex Lie algebras:
All facts mentioned here are also valid for .
In a rep of weight vectors are those elements of the vector space which are eigenvectors of the CSA, i.e.
[TABLE]
where is the multiplicity of the weight which, like a root, can be conceived as a functional on . One can show that for any weight of a rep of and for any
[TABLE]
The fundamental weights are defined by
[TABLE]
and it follows that
[TABLE]
For any irrep of a semisimple complex Lie algebra there is a unique highest weight (a weight is called highest if is not a weight ). It can be written as
[TABLE]
where are non–negative integers. Conversely, given with all there is an irrep of unique up to equivalence such that is its highest weight. Thus there is a one–to–one correspondence between functionals of the form (B.30) and irreps of . All irreps are faithful except the trivial irrep with .
No real form of admits unitary non–trivial irreps except the compact real form where all irreps are equivalent to unitary ones. On unitarity is expressed by
[TABLE]
corresponding to unitary operators . Then from the basis (B.23) we derive that
[TABLE]
Finally we want to mention that in physics one usually employs hermitian generators of a unitary rep
[TABLE]
which are obtained by
[TABLE]
from the ON generators of . In this paper we often switch between the two forms and .
C and the spinor representations
A convenient choice of basis in the space of antisymmetric real matrices is given by
[TABLE]
with commutation relations
[TABLE]
A Clifford algebra with basis elements is defined by the anticommutators
[TABLE]
It is easy to show that the elements
[TABLE]
also verify the commutation relations given by eq. (C.2). Thus a rep of the Clifford algebra automatically gives also a rep of . It is straightforward to check that for a rep of the in the dimensional space (–fold tensor product) is given, in terms of the Pauli matrices and the two–dimensional identity matrix, by
[TABLE]
Then the of eq. (C.4) define the irreducible spinor irrep of which has dimension .
For one simply has to take of the Clifford algebra of . However, the –dimensional rep of is not irreducible because
[TABLE]
It decays into two inequivalent irreps of dimension given by the projectors .
D On the isomorphisms
and
In sect. 2 the above isomorphisms are exploited for the example of the spinor representation of . Since there one assumes that it is known how to classify the fields according to an explicit realization of the above isomorphisms has to be established to transfer this classification to .
Starting from the basis as given in app. C we can define the following new basis of :
[TABLE]
Then it is easy to check with eqs. (C.1) and (C.2) that
[TABLE]
thus proving the first of the isomorphisms.
In order to prove the second isomorphism we will specify a basis of such that there is a one–to–one correspondence between the matrices and the matrices of obtained by generalizing the Gell–Mann basis of , i.e.
[TABLE]
such that the matrices obey the same commutation relations as . The result which is unique up to orthogonal basis transformations is given by777The first eight matrices listed are generators of the subalgebra and can be obtained by a general procedure to derive the generators of the subalgebra of as given in ref. [20].
[TABLE]
E Irreps of
For the definition of see eq. (4.10). Given a rep of it can be decomposed according to the subgroup . Since irreps of are given by with , being irreps of [30] we infer that an irrep of decays into under . Using
[TABLE]
we obtain
[TABLE]
Therefore, in the irrep only summands of the type
[TABLE]
appear and thus
[TABLE]
with and , being the multiplicities of the reps (E.3). The associated vector spaces will be denoted by and , respectively. Now we can define an operator by
[TABLE]
One can easily prove that
[TABLE]
and therefore
[TABLE]
Schur’s lemma guarantees that operates within and (the superscript denotes the –fold direct sum).
Discussing first we note that
[TABLE]
since . Therefore we can make a basis transformation in diagonalizing with eigenvalues . Consequently we have a type of irreps given by
[TABLE]
This defines the irreps according to the sign in eq. (E.9).
Writing according to the two inequivalent irreps of on and then has the form
[TABLE]
A small calculation reveals that requires
[TABLE]
Performing a basis transformation with
[TABLE]
we obtain
[TABLE]
on . Therefore the rep of on this space decays into equivalent copies of an irrep denoted by defined via as given in the second line of eq. (E.5). Thus we have found all irreps of as written down in subsect. 4.2.
F On the existence of a CP basis
Let be a complex semisimple Lie algebra with a –dimensional irrep and where is an ON basis of the compact real form of . Thus the are matrices and since is unitary we have in addition.
In this appendix we will show that one can always choose a basis of such that
[TABLE]
in this basis and where the signs are given by with being the contragredient automorphism defined in eq. (4.5). For the connection between the signs and the ON basis defined in eq. (B.23) see eq. (5.12). Eq. (F.1) defines the (CP)0 basis used in subsect. 5.2.
Proof:
We have
[TABLE]
because the weights of both irreps are identical. This means that there is a unitary matrix such that
[TABLE]
and, consequently,
[TABLE]
Using Schur’s lemma we obtain and therefore
[TABLE]
Thus we have shown that the matrix must be either symmetric or antisymmetric. (Exactly the same reasoning is valid for the matrix in the equivalence in which case corresponds to real and to pseudoreal irreps.)
Case 1:
We use the following lemma: For every unitary symmetric matrix there is a unitary symmetric matrix such that . Its proof given at the end of this appendix.
Inserting into eq. (F.3) we obtain
[TABLE]
which is just the desired result (F.1).
Case 2:
It remains to show that this case is impossible for irreps of semisimple Lie algebras .
The irrep is characterized by its highest weight which is simple. We get from eq. (F.3) particularized for
[TABLE]
where is the weight vector corresponding to . Using antisymmetry of and the fact that is antihermitian eq. (F.7) can be rewritten as
[TABLE]
Since is simple we must have with . Finally, using we derive the contradiction
[TABLE]
Thus is symmetric and therefore the existence of a CP basis for any irrep of any compact semisimple Lie algebra is proved.
Proof of the lemma:
Let be a normal symmetric matrix and , . Then it follows from and that . Thus can be diagonalized by an orthogonal matrix .
Let now be unitary and symmetric. Then
[TABLE]
and
[TABLE]
G Basis transformations for pseudoreal scalars
In a situation with scalar multiplets transforming under the group according to a pseudoreal irrep we are led to matrices in the CP–type transformation (7.8). Basis transformations of the form (3.26) involve again matrices. This suggests the question whether for a given one can find a such that
[TABLE]
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 define the antilinear operator
[TABLE]
the matrix
[TABLE]
and the antilinear operators
[TABLE]
on . Then the symplectic property of is expressed as
[TABLE]
Further properties of , are
[TABLE]
This allows the following reformulation of the problem posed above (see eq. (G.1)):
Given two antilinear operators , on a unitary space with properties (G.5) and (G.6) can one find canonical forms for , ? The answer is the following.
Theorem:
One can always find an ON basis such that is given by and 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:
is antiunitary and therefore is unitary. Let be an eigenvalue of with eigenvector . Then has eigenvalue . Therefore the eigenvalues of can be denoted by
[TABLE]
with degeneracies and , respectively. The decomposition of into eigenspaces of is given by
[TABLE]
with
[TABLE]
Therefore the proof of the above theorem can be devided into three parts.
- a)
, ,
We choose an ON basis in . Then defines an ON basis in . This allows to write
[TABLE]
Antiunitarity of results in
[TABLE]
with an analogous consideration for . Consequently, , are unitary matrices. Furthermore, we infer from eq. (G.9) that
[TABLE]
Exploiting eq. (G.5) we obtain
[TABLE]
and therefore
[TABLE]
Now we perform a unitary basis transformation on
[TABLE]
and explore its effect on :
[TABLE]
Unitarity of allows to choose a such that
[TABLE]
With eq. (G.13), , and we finally see that and are represented by
[TABLE]
in the basis . 2. b)
One can easily prove that it is possible to find a vector with as a consequence of . Then is orthogonal to , and the space orthogonal to is invariant under , . Therefore we can repeat the previous steps in and continue the process until no dimension is left in . Thus we find that is even and
[TABLE]
in the basis . 3. c)
Now we have and is a unitary operator with . Therefore we can find an eigenvector of with eigenvalue . We define . Then
[TABLE]
and therefore
[TABLE]
With the analogous basis as for eq. (G.18) we find that
[TABLE]
We have seen that the matrices for in eqs. (G.17), (G.18) and (G.21) all have the form eq. (G.3) and 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).
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 , and have to be kept in mind because they will play a crucial rôle in the following.
The first observation is that the matrix (8.12) has eigenvalues :
[TABLE]
If , there are no real eigenvalues. In the cases 1b) and 2a) eigenvectors of such a rotation matrix with eigenvalues are required which do not exist in the ranges of and . Therefore, we have only the zero solutions.
Case 1c) is trivially solved.
Expressing by in case 2b) gives
[TABLE]
For the second equality gives the condition
[TABLE]
Therefore, for . Taking , we necessarily have , requires to be imaginary. For eq. (H.3) gives no restriction and . is determined by eq. (H.2).
Case 2c) requires
[TABLE]
Thus must be an eigenvalue of of the form or if . For and the matrix is arbitrary. On the other hand, requires to be an eigenvector of from which the solution follows.
For the discussion of the remaining three cases we make the ansatz
[TABLE]
and
[TABLE]
Furthermore, after summation we obtain
[TABLE]
Application of eq. (H.5) in case 3a) readily gives
[TABLE]
Evaluation of this equation leads to
[TABLE]
The only possibility for requires and, consequently, and . Inserting this into eq. (H.6) gives the first subcase of the solutions 3a). Going on to one gets and . Inspecting again eq. (H.6) one sees that has the same form as before but now its elements are real. Finally, gives and . Therefore also and thus results.
The last two cases are more complicated. In 3b) we get the following equation for :
[TABLE]
Using the ansatz (H.5) we straightforwardly arrive at
[TABLE]
Furthermore, is expressed in terms of by
[TABLE]
Let us first discuss . Then because . On the other hand, it is clear that , is only possible if . Thus with the help of eq. (H.6) we obtain the of the first solution of 3b). Applying eq. (H.11) we get with
[TABLE]
Now we turn to the discussion of . Our starting point is
[TABLE]
We can only have both and non–zero at the same time if . It is easy to check that in such a case either or (in addition we have to prevent ). Let us first discuss and . Then , and , . With eq. (H.6) we recover of the second solution of 3b). Using eq. (H.9) to replace in eq. (H.11) we obtain
[TABLE]
This equation is generally valid in case 3b). Taking eq. (H.14) simplifies to
[TABLE]
The subcase , is dealt with in an analogous way.
Finally, we consider . Under this condition and cannot be non–zero at the same time. Thus assuming first we have and . This gives of the fourth solution of 3b). Now we apply eq. (H.14) and derive
[TABLE]
In this way we have obtained the complete fourth solution. For one can apply the same procedure.
Finally, let us discuss case 3c). is determined by
[TABLE]
Thus with the decomposition (H.5) we get
[TABLE]
There is exactly one possibility to have an arbitrary , namely , . is always computed by
[TABLE]
in case 3c).
Next we observe that it is possible to get by , . In this case . The is given by
[TABLE]
There is an analogous case with and characterized by and .
Allowing for the conditions
[TABLE]
must be fulfilled. In this case and . Then eqs. (H.6) and (H.19) determine and , respectively. There are three similar cases characterized by , and .
Finally we are left with cases where only one is non–zero. The discussion goes along the line presented here.
I On the case in
The non–trivial diagram automorphism of corresponds to exchanging and in the Dynkin diagram (fig. 1), for there are other ones in addition (see sect. 4). Then for irreps with highest weight is valid if and only if . Here we want to show that if this is the case and if is given in a CP basis (see subsect. 5.2) then the unitary matrix establishing the equivalence
[TABLE]
can be chosen real and symmetric (see subsect. 9.2 for the use of this result).
To proceed with the proof we first describe how irreps of the kind can be built up as tensor products of the defining fundamental irrep . For simplicity of notation we write . Then one can show that [24]
[TABLE]
The symbol denotes the –fold antisymmetric tensor product. Thus with can be obtained as the irrep with highest weight in
[TABLE]
where the superscript indicates the –fold tensor product of . can therefore be constructed from tensor products of .
The résumé of this consideration is that if one can show that is achieved with a real symmetric matrix for in a CP basis the for with is obtained by suitable tensor products of conserving reality and symmetry. Also in these tensor products will be real and imaginary and symmetric if this is so in . Then by reversing the argument leading to eq. (5.13) and by using we easily see that is given in a CP basis.
Let us now consider and show first that there is a real and symmetric matrix with where is given in the natural basis of with real antisymmetric matrices. We will closely follow the discussion in app. G of ref. [16]. Using the matrices (see eq. (C.1)) as a basis of or of its complexification then defines a basis of the CSA. It is shown in ref. [16] that with the functionals on the CSA defined by
[TABLE]
all basis elements with positive roots are given by
[TABLE]
with . Therefore
[TABLE]
and the simple roots may be defined as
[TABLE]
Then one can calculate
[TABLE]
and
[TABLE]
With
[TABLE]
we have
[TABLE]
In eq. (I.10) the symbol diag means arranging the matrices along the diagonal and denotes the zero matrix. The basis elements corresponding to simple roots are represented by
[TABLE]
with
[TABLE]
and, for ,
[TABLE]
with
[TABLE]
The diagram automorphism is already uniquely determined by its action on and :
[TABLE]
Then one quickly confirms with eqs. (I.10), (I.11) and (I.12) that eq. (I.13) is reproduced by
[TABLE]
for or its complexification . Since we have obtained
[TABLE]
with real and symmetric.
In the last step we have to switch to a CP basis by conserving the above properties of . Defining
[TABLE]
we get
[TABLE]
with the Pauli matrix . Furthermore, one readily verifies that
[TABLE]
This allows to check easily that
[TABLE]
and
[TABLE]
Therefore, looking at eq. (I.5) we immediately see that we have
[TABLE]
Thus the matrix achieves a basis transformation into a CP basis. Finally, with the Pauli matrix we obtain
[TABLE]
which is real and symmetric. This proves the claim made in the beginning of this appendix.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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