matrix representation of symmetries in flavor space, invariant functions of mass matrices, and...

PHYSICAL REVIEW D VOLUME 35, NUMBER 5 1 MARCH 1987

Matrix representation of symmetries in flavor space, invariant functions of mass matrices, and applications

C. Jarlskog Department of Physics, University of Stockholm, Stockholm, Sweden

(Received 3 November 1986)

A formalism suitable for studying the symmetry properties of the standard electroweak model in the flavor space is given. In this formalism the (quark) mass matrices play the leading role. It is shown that any amplitude in this model may be written in a matrix form In flavor space, whereby the amplitude contains two kinds of matrices: y matrices which stand for the space-time properties and mass matrices which represent the flavor-symmetry properties. In this matrix representation the analytic expressions obtained for the amplitudes are concise and the flavor-symmetry properties are more transparent than they are usually, where the summations over internal flavors tend to ob- scure these symmetries. Furthermore, the concept of invariant functions of mass matrices is discussed and it is shown how the measurable quantities may be expressed as invariant functions. Ex- amples given are cross sections, form factors, and measurables of the quark mixing matrix. Finally, the relationship of the present work and the commutator formalism for CP violation in the standard model with three families is discussed.

I. INTRODUCTION

In the past couple of decades, the electroweak model' has had a tremendous impact on our understanding of the structure of the forces in nature. In addition, this model has also been amazingly successful in accounting for all relevant experimental data. In fact, there is not, so far, any sign of the slightest disagreement between this model and the data. Because of these facts it is certainly very much worthwhile to devote one's attention to further studies of the structure of the electroweak model.

The goal of this study is to obtain a closer familiarity with the symmetry structure of the standard model in flavor space. The point is that we have a great deal of familiarity with the space-time structure of this model in the sense that we know how to quickly write down analytic expressions for the amplitudes in a compact form in terms of, say, y matrices. However, it is not always so easy to "see through" the flavor-symmetry properties of, for example, the transition amplitudes involving flavor changes. This is due to the fact that the amplitudes may contain sums over internal flavors. These sums tend to hide the symmetry structure (see below).

In this paper I wish to take some steps toward develop- ing the necessary "formalism" to make the flavor symmetries more transparent, for any number of families. Al- though the "formalism" is developed for the standard model there is no a priori reason why one should not be able to generalize it to other theories such as the grand unified theories or the supersymmetric theories.

The strategy chosen is as follows. Since the flavor structure of the standard model is dictated by its mass matrices it is essential to focus attention on them and bring them to the foreground. By doing so it is in fact possible to immediately write down an analytic expression for any arbitrary amplitude in terms of the mass matrices themselves. In this way the expression for the amplitude

in question will in general contain products of two kinds of matrices: (i) y matrices which summarize the space- time properties and (ii) mass matrices which represent the flavor-symmetry structure.

A useful concept turns out to be the notion of "invariant functions" of the mass matrices. These are functions which are independent of the choice of basis for the mass matrices. They are thus the flavor space analogs of the functions of the ~~ller- ande el st am^ variables, in the coordinate space, in the sense that the latter functions do not depend on the choice of the inertial frame. The measurable quantities must, of course, be invariant functions of mass matrices. In this paper I shall give examples where measurables are expressed as such functions.

The plan of this paper is as follows. In Sec. I1 the notations used in this paper are defined and in Sec. I11 the concept of invariant functions of mass matrices is discussed. Section IV gives the amplitudes as functions of mass matrices. In the next section it is shown how one may express the measurables as invariant functions of mass matrices. Several examples are provided, these being cross sections, form factors, and some measurables of the quark mixing matrix. The method and the applications are general, in the sense that they are either directly valid for any number of families or they may be trivially gen- eralized. In Sec. VI the particular case of three families is considered and the relationship between the work present- ed in this paper, the commutator formalism (see below), and the question of the degree of CP violation3 in the standard model is discussed.

11. PRELIMINARIES AND NOTATIONS

In order to be specific the discussion below is confined to the hadronic sector of the standard model.4 The lep- tonic sector may be treated analogously.

In the standard model, assuming the number of families

35 1685 - @ 1987 The American Physical Society

1686 C. JARLSKOG 3 5 -

to be n, there are n up-type and n down-type quarks. Since each can be both left handed and right handed, there are 4n flavor degrees of freedom. Thus the one- quark transitions, which are the building blocks of the theory, define a 4n-dimensional flavor space. However, there is a great deal of simplification due to the fact that there are no right-handed charged currents, and also because the lasho ow-~lio~oulos-~aiani~ (GIM) mechanism makes the structure of the neutral-current transitions trivial, in the sense that they are all flavor conserving. Thus we need only consider n by n matrices.

The flavor-symmetry structure in the electroweak model is dictated by the mass matrices m and m ' for the up-type and down-type quarks, respectively. These are, both, n x n matrices. In fact, because of simplifications mentioned in the previous paragraph, only the "squares of mass matrices," S and S', defined by

enter. These matrices are Hermitian and have non- negative eigenvalues. As usual they are diagonalized via

where U and U' are unitary matrices. They are not separately measurables. However the product

which is the quark mixing matrix enters in physics. Note that S and Sf represent 2 n 2 real parameters. However, not all of these parameters are relevant to physics. The independent measurable quantities are the 2n quark masses and ( n - 1 )' parameters associated with the quark mixing matrix. Thus the number of independent measurables is n 2 + 1 which implies that n - 1 parameters in S and S' are irrelevant to physics.

111. INVARIANT FUNCTIONS OF MASS MATRICES

The "squares" of mass matrices S and Sf dictate the flavor structure of the standard model. However, it is important to keep in mind that these are by no means unique. The reason is that the measurable quantities are not affected if S and S' are both rotated with the same unitary matrix. Such a rotation leaves the eigenvalues (masses) as well as the quark mixing matrix V invariant. In order to keep this fact in mind, it is useful to introduce the concept of invariant functions of mass matrices as fol- l o w ~ : ~ A function f (S,S') is called an invariant function of mass matrices if it is invariant under the transformation

where X is an arbitrary unitary matrix. The above unitary rotation corresponds to choosing a particular basis, or frame of reference, in flavor space. Since measurables cannot depend on the choice of the coordinate frame they must be invariant functions. One can convince oneself

that a complete set of invariants, in flavor space, is given by (see below)

where k goes from 1 to n, whereas r and t go from 1 to n - 1. The quantities Ik and I; determine the masses of the up-type and the down-type quarks, respectively. Hav- ing determined the quark masses, the quantities I,, yield the ( n - 1 )2 measurables of the quark mixing matrix. The reader may wonder why I,, with r =n or t =n do not enter. The reason is that S and S' satisfy the secular equations, which for S reads

For the down-type quarks we must take the primed ver- sion of Eq. (6). Thus I,, with r or t equal to n may be expressed in terms of the variables introduced in Eq. (5).

It is important to keep in mind that although all measurables are invariant functions of mass matrices the different flavors are distinguishable from each other because the values of all these invariant functions are not the same. For example, the eigenvalues of the secular equation (the squares of the masses) are all invariants but they are not equal to each other, e.g., mU2#mc2, etc.

IV. AMPLITUDES AS FUNCTION OF MASS MATRICES

In order to investigate the flavor-symmetry properties of the amplitudes it is convenient to express them as functions of S and Sf. This can be done for any number of families. Below, the method is first demonstrated for n = 3 , and afterwards the generalization to an arbitrary n is discussed.

Consider the transition between two down-type quarks j and k through the intermediary of the up-type quarks a , a = u,c, t , as shown in Fig. 1. We get the flavor-dependent quantity

where Q is the four-momentum of the quark a and the sum runs over u, c , and t . In order to write this quantity in a matrix form we must get rid of the sum. This can be done by exploiting the unitarity of the quark mixing matrix. A simple computation yields

FIG. 1 . The transition between two down-type quarks ( q ' ) via the flow of the up-type quarks q, between two W vertices.

35 - MATRIX REPRESENTATION OF SYMMETRIES IN FLAVOR . . . 1687

where P ( Q ) is an invariant function of mass matrices, F k j ( Q ) = p ( Q ) [ u l L ( Q , s ) u " ] ~ ~ , which in addition is flavor independent:

I where L (Q,S) is the link in the "weak basis," obtained P ( Q ) = [ ( Q ~ - ~ ~ ~ ) ( ~ ~ - ~ ~ ~ ) ( ~ ~ ~ mf (9 ) from Eq. (1 1) by the replacement D2-S The expression

In other words, P ( Q ) may be expressed in terms of the invariants introduced in Eq. ( 5 ) . Putting z =Q2, we have

where detS is, of course, a function of the Ik introduced in Eq. ( 5 ) . Note that P ( Q ) is simply the inverse of the left-hand side (LHS) of the secular equation. We may call P ( Q ) the flavor-invariant propagator. The quantity L (Q,D2) in Eq. (81, "links" the two vertices specified by V and V' to each other. This "link" is given by

where again z = Q 2 and ~ ~ = d i a ~ ( m , ~ , m , ~ , m , ~ ) . Note that the link is Hermitian and has two pieces, Lo which is trivial in the flavor space, in the sense that it is propor- tional to the unit matrix. L on the other hand, is non- trivial. Using Eqs. (2) and (3) one may rewrite the link in terms of the mass matrix (more precisely in terms of S ) :

(12) has a simple pictorial interpretation. The transition shown in Fig. 1 is determined by the "hooks" U' and U", which denote the identity of the external down-type quarks, as well as the "link" L ( Q , S ) of the up-type quarks which joins the two vertices. Finally, we have the invariant propagator P ( Q ) of the up-type quarks running between the two vertices. The appearance of the hooks U' and U' ' may look somewhat strange. After all these are not measurables. How can they then show up in physical amplitudes? The answer is that an amplitude by itself is not a measurable quantity. The measurables are related to the square of the modulus of the amplitudes, etc. As we shall see in the next section, indeed the measurables are independent of these hooks. The essential feature of Eq. (12) is that F is now a matrix in the flavor space. It certainly did not look like a matrix when we started out with Eq. (4).

One may generalize the above result to any arbitrary amplitude in the standard model and for any number of families. In other words, any arbitrary diagram may be written in a matrix form. The sums over the internal flavors may be removed by using the identity

Here r and t run from 1 to n and f is an arbitrary function (see below). Furthermore, P ( Q ) and L (Q,S) denote the invariant propagator and the link, respectively, but now for the case of n families. The derivation of the above identity uses the definition of the quark mixing matrix Eq. (3) and the fact that it is unitary. Of course, the specific forms of the invariant propagators and the links depend on the number of families. In all cases the link is given by

For example, for n = 2 we have

Whereas for n = 4 we find

The invariant propagator is again, in each case, the inverse of the LHS of the appropriate secular equation [see Eq. (611.

The identity (13) has, again, a very simple pictorial interpretation. We are computing a transition (between two down-type quarks j and k ) similar to the one in Fig. 1, but now the internal up-type quark line has been modified by attaching any number of photons, gluons, Z bosons, or

Higgs particles to it. These do not change the flavor a but may give mass-dependent factors, depicted by the arbitrary functibn f ( m a 2 ) and several internal propagators with momenta Q,. Equation (13) tells us that all such transitions may be represented by matrices in flavor space. For each internal quark line we obtain an appropriate link L and a flavor-invariant propagator, P. The mass- dependent factor f ( ma2) turns to the matrix f ( S ) . Note that the factors appearing between the hooks in Eq. (13) are all functions of a single matrix S and thus commute with each other.

Using Eq. (13) and its counterpart for the down-type quarks, we may immediately write down an analytic expression for any arbitrary diagram of the standard model, in terms of the mass matrices (more precisely in terms of S and S ' ) . Every diagram may then be written as a sum of terms, where each term is a direct product of two types of matrices: the y matrices and the mass matrices. In flavor space, we get chains of the type

where L ' is the link for the down-type quarks. It is obtained from the link for the up-type quarks by just replac- ing S by S'. In addition, for each external quark there is an appropriate hook which keeps track of the identity of that external quark. As mentioned before, the hooks will not appear in the expressions for the measurable quantities (see Sec. V). An important point to keep in mind is

1688 C. JARLSKOG 35 -

that it is indeed trivial to transform the above results to the physical basis for the masses at any stage o f the computations by using Eq. (2) . Thus one may rewrite every- thing in terms o f the matrices D 2 and V D t 2 v t (or D l 2 and V ' D ~ V ) . However, written in terms o f S and S' the flavor symmetries are more transparent. Another essential feature o f Eq. (15) is that every factor in it is Hermi- tian. The presence o f a possible non-Hermitian piece in Eq. (15) is due to the fact that Sand S ' do not commute.

V. MEASURABLES AS INVARIANT FUNCTIONS IN FLAVOR SPACE

Consider any transition amplitude T and let us focus our attention on its flavor structure. Since the baryon number is conserved (remember that we are in the hadronic sector o f the standard model) any ingoing quark A must emerge as an outgoing quark B. Thus the amplitude T factorizes into products o f the type

T B A , A = ( a or a ) , B=(/3 or b ) , (16)

where the Greek (Latin) letters denote the up- (down-) typequarks. Thusa ,p=u ,c , . . . and a,b =d,s,. . . .

In general, we are interested in the transition rates and must compute TBA / 2. As shown in the last section, TBA is o f the form

where the U matrices appearing here are the appropriate hooks, which as before keep track of the identity o f the external quarks A and B. Thus U 1 equals U ( U ' ) i f A is up (down) type, etc. Furthermore, f ( S , S ' ) is an n x n matrix which we learned how to write down in the last section. Taking the square o f the modulus o f the above expression gives

Here the E A and EB are so-called elementary matrices.' The elementary matrix EA is defined to have only one nonvanishing element which is a 1 at the position A along the diagonal. The elementary matrices above are not just mathematical constructions but have a precise physical meaning because they are related to the quark mass matrices. For A = a , the E A is calculable from the diagonal matrix D2=diag(x l , x2 , . . . ) and higher (up to the nth) powers of it. W e have

and thus

r = 1 -n. From this system o f n equations we may solve for the n "unknowns" E,. A simple computation gives

Here x = m 2 , a denotes the flavor ( a= 1 denotes the up

quark, etc.), v is a Vandermonde-type determinant7 given by

This quantity is well known to be given by

where a,/3= 1 -n with the restriction f l> a. Thus

etc. The quantity u , in Eq. (21) is defined by

Thus this quantity is a n X n matrix. For example, for n = 3 ,

L ~ ~ ( D ~ ) = ( x ~ - x 2 ) ( x 3 - D 2 ) ( x 2 - D 2 ) . As before, for the down quarks the bookkeeping requires that we use latin indices and primes. In other words, the A =a, a-a, D-+Dl, v-+u', and x-x ' , where

with x i = m d 2 , etc. Remembering that the matrix U 1 ( U2 ) is the one which diagonalizes the mass matrix o f the A- ( B - ) type quarks [see Eq. (2)] we may transform the elementary matrices to the weak basis, viz.,

U ~ E , uI = E A ( S , S 1 ) , (26)

where

E a ( S , S ' ) = S v , ( S ) / ( x a v )

and

Substituting Eq. (26) into Eq. (18) yields

This result clearly demonstrates that the measurable on the LHS is indeed, as expected, an invariant function o f mass matrices.

Another class o f measurables is the "forward flavor amplitude" T A A . It arises when an ingoing quark, perhaps after having undergone several flavor changes, end up with its initial flavor. T A A is an invariant function o f mass matrices, as it is given by

A measurable o f the type in Eq. (29) is the electric dipole moment o f the quark (which in the impulse approxima- tion leads, e.g., to the electric dipole moment o f the neu- tron). This quantity is o f great interest8 in connection with CP and T nonconservation. When computing the

35 - MATRIX REPRESENTATION OF SYMMETRIES IN FLAVOR . . . 1689

electric dipole moment of the quark A, we are interested in the imaginary part of the quantity T A A EA being Hermitian, what then matters is whether or not f ( S , S 1 ) is Hermitian. The Hermitian pieces in f give real contribu- tions to the trace in (29) and thus cannot contribute to the electric dipole moment. Of course, in general, there will be non-Hermitian pieces in f, because S and S' do not commute. Writing out f ( S , S f ) in terms of links, etc., we can quickly examine which diagrams contribute and which ones do not. The computation of the contributing ones may also become considerably simplified due to the fact that Eq. (29) exhibits the symmetry properties of the individual diagrams whereby the Hermitian pieces may be left out.

As an example consider the diagram in Fig. 2, where a photon is attached to the outer virtual W. The relevant flavor factor is then given by

where Q and Q' denote the four-momenta of the virtual up-type and down-type quarks, respectively. Note that the quarks a and P have the same four-momenta. Furth- ermore, the external quarks have the same flavor and we are only interested in the imaginary part of the diagonal element Fii, where i is not summed. Since the links are Hermitian, we see immediately that F is Hermitian whereby its diagonal elements are real and there is, therefore, no contribution to the electric dipole moment from the above diagram. The advantage of the above formalism is that it allows us to obtain this well-known result very quickly and to understand its origin (Hermiticity) in a trivial fashion.

A further example of the application of the above formalism is provided by the measurables of the quark mixing matrix. These may all be expressed as invariant func- t i o n ~ . ~ For example, the measurable / V,, 1 which enters in the rate of W + a +Z is given by

Here the derivation of this result uses Eqs. (21)-(24) and the definition of V. This formula is rather compact and simple. The point is that, given S and S' , the computation of the eigenvalues (masses) is in general a difficult task because it involves solving nth-degree equations. However, once the masses are computated (perturbatively, or otherwise) the determination of the measurables of the quark mixing matrix is trivial, in the sense that it only in-

volves elementary operations. These operations may be tedious and the formula (30) may be left to the computer to do. The advantage of the above method is that we do not have to keep track of the matrices U and U' which diagonalize S and S' , respectively, and multiply them with each other and worry about which phases are physical and which ones are removable by redefinition of the quark fields. The relation (30) does all the necessary thinking for us and automatically supplies us with a basis and phase convention independent answer. Note also that the quantity in the square brackets in Eq. (30) seems to be a polynomial of order n both in S and St. However, by using the secular equation, one may reduce the order to n - 1, in both S and S'. Thereby, the measurable / V,, 1 may be expressed as a function of the variables I,,, see Eq. ( 5 ) .

As a specific example of Eq. (30), consider the case of three families. Given S and S' , we find

for a = 1, j = 1. The remaining 8 quantities Vaj ( are obtained from this equation by cyclic permutation. Note that the secular equation has been used to reduce the order of the polynomial which appears as the argument of the trace. Evidently, the RHS in the above expression may be written as a function of the variables introduced in Eq. ( 5 ) .

The expression (30) also exhibits the scaling properties of the I V,, 1 . These quantities are invariant under the transformation

and (31)

where N and N' are nonzero constants. Under this transformation the eigenvalues are rescaled, viz.,

and

This invariance implies that the quantities Vaa / depend on the ratio of masses x a / x S and x i / x L but not on the individual masses. This scaling property also follows from the definition of V with some additional reasoning. Relation (30) exhibits the scaling property manifestly. Other measurables of the quark mixing matrix may also be expressed as invariant functions (see Ref. 6).

VI. CONNECTIONS WITH THE COMMUTATOR FORMALISM A N D CP NONCONSERVATION

FIG. 2. A skeleton diagram for computation of the electric dipole moment of the down-type quark qi . A photon is to be attached to this diagram in all possible ways.

In this section, I shall confine myself to the case n = 3 (i.e., the standard model with three families) and discuss the relationship of this work with some earlier results on CP nonconser~ation.~

1690 C . JARLSKOG 3 5 -

The present understandingi0 of the observed phenom- enon'' of CP violation in the context of the electroweak model is based on the very important observation that the elements of the quark mixing matrix need not be relative- ly real, if there are at least three f ami~ ies . '~ There is the possibility of having a phase S in the quark mixing matrix. Introducing the three mixing angles and a phase in the quark mixing matrix, 14 conditions have to be satis- fied in order to get CP violation. These are based on the following principles.

(i) No two quarks with the same charge are allowed to be degenerate in mass. This gives six conditions, m u f m , , m d f m s , etc.

(ii) None of the angles in the quark mixing matrix is permitted to assume its maximally or minimally allowed value [e.g., Bi+0;rr/2, i =I-3 and S#O,?i, in the ~ o b a ~ a s h i - ~ a s k a w a " (KM) parametrization].

These 14 conditions are unified in a single relation as follow^.^ We introduce the commutator of the "square" of the mass matrices S and S' [see Eq. (I)] :

[S, S ' ] - iC , (32)

where by definition C is Hermitian and traceless. Al- though the elements of C are not measurables its eigenvalues are and in particular their product, i.e., the determinant of C is very interesting as it unifies the 14 above conditions in a single relation. Indeed CP is violated if and only if

What is the connection between the present work and the commutator formalism? The ~ o i n t is that detC is a verv good example of an invariant function of mass matrices, which has been the main topic of this paper. The detC is invariant under the transformation (4). In Ref. 9 I studied whether one could use this quantity to define a measure for CP violation. The work was stimulated by earlier pa- pers by several authors1* who had addressed the following question: What is the appropriate measure for CP violation in the standard model. After all, we talk of parity being maximally violated in charged-current interactions. "How much" is CP violated? In fact the question of the degree of CP violation has been of interest for a long time.13 The point is that many authors have had the feel- ing that the degree of CP violation must be related to some kind of a "phase" appearing in the quark mixing matrix and yet one could not simply take, say, the KM phase as the measure of the violation, such that sinS= 1 would correspond to having "maximal" CP violation. The reason is that there are many ways of introducing such a phase (for a few examples see Ref. 14). These phases are, in general, out of phase with each other in the sense that they are not maximal simultaneously. Since the measure of CP violation should be convention independent a priori, we cannot use any of them. The outcome of the study of Ref. 9 was that one could indeed define an invariant measure (convention independent) in terms of det(C), but the measure did not at all correspond to just having a maximal phase.'5 By using detC at least one is sure not to violate the fundamental principle which tells

us that equivalent classes of mass matrices [see Eq. (4)] are physically indistinguishable. This may seem obvious to the reader who has been following the concept of invariant functions of mass matrices but in fact in the literature there are examples of definitions of the so-called maximal CP violation where the definition given does not involve an invariant function and is, therefore, dependent on the choice of the basis. Below I shall give an example.

Consider the following supposition. We define CP to be maximally violated if, in the basis where the mass matrix of the up-type quarks is diagonal, the mass matrix of the down-type quarks is such that is has purely imaginary nondiagonal elements. The mass matrix is assumed to be Hermitian. Let us study whether or not the above definition is reasonable. We may immediately note that there is no problem with the assumption of Hermiticity, as one can prove'6 that in the standard model one may without loss of generality assume that the mass matrices are Her- mitian. Thus the assumption is that

Here m ' is the mass matrix for the down-type quarks and a, b, c, r, s, and r are assumed to be real numbers. It is obvious that the above definition as it stands cannot make sense because we may rotate both mass matrices with, e.g., the unitary matrix X=diag( i , l , l ) . From Eq. (41, the new and the old mass matrices are physically indistinguishable but the new m ' has no longer purely imaginary nondiagonal elements. Thus the property of having purely imaginary nondiagonal elements is not a statement about an invariant function of mass matrices. On the contrary, it is a basis-dependent statement and, therefore, can have no physical significance. We may now ask ourselves whether we nevertheless could keep the general idea of the above definition but somewhat modify it in order to turn it into a statement on an invariant function. Let us now examine this question. In order to do so we use the commutator formalism which is unique, in the sense that it provides an if and only if condition for CP violation, and in addition it is guaranteed to be invariant. The relevant quantity here is the det[m,ml]. In general this quantity is given by9

where

Returning to the specific model defined by Eqs. (34) and (35), we may easily compute the relevant determinant. It is given by

3 5 - MATRIX REPRESENTATION O F SYMMETRIES IN FLAVOR . . . 1691

(38) where

From Eqs. (36)-(391, we obtain

where cbij is the phase of the m;. Since we have a sine here, which necessarily must lie between + 1 and - 1, we may wish to employ it for defining the degree of CP violation, such that the values + 1 would be the signature of maximality. However, a priori we cannot do so because we have formulated the problem in a very particular basis, the one in which the mass matrix for the up-type quarks is diagonal. Obviously if we have taken the mass matrix of the down-type quarks to be diagonal we would have obtained a different answer. Thus we must first examine what happens if we choose a different basis. Indeed the commutator formalism provides the answer. From Eq. (36) it is evident that the only "angular looking object" which appears is the product of the "four V's":

Here the object in parentheses has a magnitude and a phase. In shorthand notation, we have

The LHS of this equation is a universal constant (up to a sign); i.e., it does not depend on the particular choice of the rows ( a , P ) and columns ( j , k ) a fact which has been realized by many authors." However, the RHS provides us not with a single angle but with nine angles, depending on which rows and which columns we decide to choose. Indeed within the context of the standard model there is no way to single out any of these angles and use it to give a measure of CP violation.

In conclusion, if one insists on defining a measure for CP violation one must make sure that the statement made involves an invariant function of mass matrices.

VII. CONCLUSIONS

In this paper I have discussed how the amplitudes of the electroweak model with any number of families may be written as matrices in flavor space. The advantage of

the formalism is that the symmetry properties become more transparent. Thus one may, e.g., quickly check the CP properties (Hermiticity) of the amplitudes. The expressions one obtains are concise and very easily visual- ized, i.e., they may immediately be written down. The concept of invariant functions of mass matrices is useful not only for avoiding pitfalls (as shown by the example in the previous section) but also for obtaining closed expressions for measurable quantities. Thus given a specific model for the quark mass matrices, the measurables are obtained from the formalism described in this paper without having to bother about questions such as which phases are physical and which ones are not, etc. Indeed, the formalism does much of the thinking for us. Finally, the commutator formalism provides a very powerful de- vice, for studying the CP structure of the standard model with three families. The determinant of the commutator is a very good example of the invariant functions discussed in this paper. As a final remark and for complete- ness, I will very briefly discuss some further possible applications of the commutator formalism.

(i) Given a specific model for mass matrices one may use the commutator formalism to study the CP properties of the model, without having to diagonalize the mass matrices first. For an application see Ref. 18. One may then, e.g., check whether or not the model proposed gives CP violation as certain parameters in the model approach zero.

(ii) detC appears in physics'9 in the renormalization of the 6 parameter QCD. We know that QCD does not automatically respect CP, due to the presence of the so- called @ term2' in its Lagrangian. Electroweak interactions renormalize the 6 parameter:

Here the last term is expected due to CP violation in electroweak interactions. The general form of 6% is known2' since several years ago. In fact, by simple manipulations, one may easily show that l9

Finally, the commutator formalism has recently been used22 to compute the renormalization of the observed low-energy CP violation, when one goes to the energy scale of the grand unified theories.

ACKNOWLEDGMENT

This work has been supported by the Swedish National Research Council (NFR).

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2C. Mbller, Ann. Phys. 14, 531 (1932); S. Mandelstam, Phys. Rev. 115, 1741 (1959).

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4For a pedagogical introduction to the standard electroweak

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5S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2, 1285 (1970).

6C. Jarlskog and A. Kleppe, Nucl. Phys. B (to be published). 'See, for example, F. R. Gantmacher, Matrizenrechnung II

1692 C. JARLSKOG 35 -

(VEB Deutscher Verlag, Berlin, 1959). 8For a review and an extensive list of references see N. F. Ram-

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'OM. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).

"J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Phys. Rev. Lett. 13, 138 (1964).

12Many authors have been involved in this field. See, for example, B. Stech, Phys. Lett. 130B, 189 (1983); D. Hochberg and R. G. Sachs, Phys. Rev. D 27, 606 (1983); L. Wolfenstein, Phys. Lett. 144B, 425 (1984); M. Gronau and J. Schechter, Phys. Rev. Lett. 54, 385 (1985); M. Roos, Phys. Scr. 31, 315 (1985).

13The early works on CP violation are reviewed in, P. K. Kabir, The CP Puzzle (Academic, New York, 1968).

14For examples of such parametrization see L. Maiani, in Proceedings of the International Symposium on Lepton and Photon Interaction at High Energies, Hamburg, 1977, edited by F. Gutbrod (DESY, Hamburg, 19771, p. 867; L. Wolfen- stein, Phys. Rev. Lett. 51, 1945 (1983).

15For a review see, C. Jarlskog, in Physics in Collision V,

proceedings of the Fifth International Conference, Autun, France, 1985, edited by B. Aubert and L. Montanet (Editions Frontieres, Gif-sur-Yvette, France, 19851, p. 347; in Progress in Electroweak Interactions, proceedings of the XXIth Ren- contre de Moriond, Les Arcs, France, 1986. edited by J. Tran Thanh Van (Editions Frontieres, Gif-sur-Yvette, France, 19861, p. 389.

16P. H. Frampton and C. Jarlskog, Phys. Lett. 154B, 421 (1985). "This product has been used by many authors who have been

interested in CP computations. See, for example, E. P. Sha- balin, Yad. Fiz. 28, 151 (1978) [Sov. J. Nucl. Phys. 28, 75 (197811; L.-L. Chau and W.-K. Keung, Phys. Rev. Lett. 53, 1802 (1984); D. D. Wu, Phys. Rev. D 33, 860 (1986). See also Refs. 9 and 15.

I8L. Wolfenstein, Phys. Rev. D 34, 897 (1986). I9C. Jarlskog, Report No. USIP 85-22 (unpublished); C.

Jarlskog, in Physics in Collision V , Ref. 15. >OG. 't Hooft, Phys. Rev. Lett. 37, 8 (1976); Phys. Rev. D 14,

3432 (1976). >'J. Ellis and M. K. Gaillard, Nucl. Phys. 150, 141 (1979). 22G. G. Athanasiu, S. Dimopoulos, and F. J. Gilman, Phys.

Rev. Lett. 57, 1982 (1986).

matrix representation of symmetries in flavor space, invariant functions of mass matrices, and...

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