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Cabibbo-Kobayashi-Maskawa matrix: parameterizations and rephasing invariants

The paper is devoted to a discussion of general properties of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. First we propose a general method of a recursive construction of the CKM matrix for any number of generations. This allows to set up a parameterization with desired properties. As an application we generalize the Wolfenstein parameterization to the case of 4 generations and obtain restrictions on the CKM suppression of the fourth generation. Motivated by the rephasing invariance of the CKM observables we next consider the general phase invariant monomials built out of the CKM matrix elements and their conjugates. We show, that there exist 30 fundamental phase invariant monomials and 18 of them are a product of 4 CKM matrix elements and 12 are a product of 6 CKM matrix elements. In the Main Theorem we show that all rephasing invariant monomials can be expressed as a product of at most 5 factors: 4 of them are fundamental phase invariant monomials and the fifth factor consists of powers of squares of absolute values of the CKM matrix elements.


I. INTRODUCTION
The Standard Model has 18 adjustable parameters and 13 of these parameters have their origin in the Yukawa couplings of the Higgs boson with fermions. These parameters are: 3 lepton masses, 6 quark masses and 4 parameters of the CKM matrix 1 . The quark masses are the running masses and they are obtained from the eigenvalues of the Yukawa couplings in the process of diagonalization by the bi-unitary transformation. The CKM matrix is obtained from the left diagonalizing matrices of the up-and down quark mass matrices.
The CKM matrix is a 3 × 3 unitary matrix, which has additional properties stemming from the rephasing freedom of the quark fields. This is the reason why the CKM matrix has only 4 real parameters and one of them is a phase. The non vanishing phase is the source of the CP violation in the Standard Model.
The CKM matrix has 9 complex matrix elements, which are parameterized by 4 real parameters. The choice of the parameters is not unique. There exist various equivalent parameterizations, which were chosen to fulfill various needs. Let us start with the standard parameterization of the PDG 2 . This parameterization is exact and uses 3 angles and 1 phase and can be represented as the product of 3 real rotation matrices and the diagonal matrices with phase terms.
Another widely used parameterization is the one proposed by Wolfenstein 3 . Initially it was considered to be an approximate representation of the CKM matrix, because it was chosen in such a way as to reproduce the suppression for the weak transitions of quarks between the generations. Later, it was improved 4-6 and also it was made exact 7 .
The particular choice of parameters in principle is not important, because all exact parameterizations are mathematically equivalent. But an appropriate choice of parameterization for a given physical situation can lead to natural relationship between the parameters and physical observables. Fo example, in the parameterization of Wolfenstein the Jarlskog's invariant 8 J = Aηλ 6 + O(λ 8 ) has simple form, while in the standard parameterization of the CKM matrix 2 it is more complicated.
In Section II we discuss the recursive construction of the CKM matrix for a general case of n generations, assuming that the CKM matrix for (n−1) quark generations is known. There exist various approaches to the problem of parameterization of the CKM matrix, which fulfill various requirements. The parameterization of Harari and Leuler 9 stresses the simplicity and a possibility of generalization to an arbitrary number of quark generations. Jarlskog proposed the recursive parameterization also for an arbitrary number of quark generations that represents the CKM matrix as a product of special unitary matrices 10,11 . Bjorken and Dunietz 12 construct the CKM matrix out of rephasing invariants: the absolute values of the CKM matrix elements |V iα |, i < α and the phase of "plaquette" arg(V iα V jβ V * iβ V * jα ). There are many more parameterizations [13][14][15][16][17][18][19][20][21][22][23][24] which help to illustrate various features of the CKM matrix, but their discussion is beyond the scope of our paper.
Our algorithm for the construction of the parameterization of the CKM matrix has the flexibility that allows to adjust the parameterization to the required physical situation. It can be used to generalize the standard and Wolfenstein parameterizations to the case of 4 quark generations. From the Wolfenstein-type parameterization for 4 generations we then obtain the restrictions on the suppression factors for the 4-th generation, which are compatible with the present data for the CKM matrix.
The next topic, discussed in Section III, is the quark fields rephasing properties of the CKM matrix. The Yukawa couplings (mass matrices) of the standard model are not invariant under the rephasing of the quark fields. From this it follows that the CKM matrix is not invariant either. Since observable effects cannot depend on the choice of the phase of the quark fields it means that all observables including those containing the CKM matrix must not depend on arbitrary phases of the quark fields. This is the reason of the reduction of the number of the physically significant parameters of the unitary CKM matrix from 9 to 4. The general observables of CKM matrix are usually monomials built out of the CKM matrix elements V ij and its conjugates V * ij . The most important observables obtained from the rephasing invariant monomials of the CKM matrix 25 are the Jarlskog invariant 8 26 J and the unitarity triangle angles: In Section III we derive the conditions for the rephasing invariance of an arbitrary monomial built from the CKM matrix elements and then demonstrate in the Main Theorem that an arbitrary rephasing monomial can be expressed as the product of five factors: powers of absolute values of the CKM matrix elements multiplied by powers of 4 fundamental rephasing monomials, which are built out of 4 and 6 CKM matrix elements.
The last section of the paper contains a summary and conclusions.

A. Introductory remarks
The CKM matrix is unitary, but the rephasing freedom for the quark fields reduces the number of the physically significant parameters. The (n × n) unitary matrix is described by n 2 parameters. The up-and down-quarks rephasing freedom reduces the number of the parameters by (2n − 1), so the CKM matrix for n quark generations is described by (n − 1) 2 parameters. These parameters are divided into two classes: angle-like and phases. Angle-like parameters are derived from the (n×n) real unitary matrix 27 (rotation or orthogonal matrix) and there are n(n−1) 2 such parameters. The remaining (n−1)(n−2) 2 parameters are phases. One can observe that if the number of quark generations is incremented from (n−1) to n then the number of angle-like parameters increases by (n − 1) and the number of phases by (n − 2).

B. Recursive construction of the CKM matrix
In this section we outline how to construct and parameterize the (n × n) CKM matrix V (n) , if we know the CKM matrix V (n−1) . The presented algorithm does not impose any conditions on the parameterization of the matrix V (n−1) , so this method allows to introduce such properties of the CKM matrix that are required by the physical conditions for each generation separately.
The CKM matrix V (n) is built from V (n−1) in two steps: a. We construct n real column vectors with n rows e 1 , . . . , e n that depend on (n − 1) angle-like independent parameters θ 1,n , . . . , θ n−1,n , are normalized to 1 and orthogonal: b. The columns of the CKM matrix V (n) are obtained from the vectors e k and the elements of the matrix V

Example 1. Standard parameterization for 3 quark generations
We will show here how one can obtain the CKM matrix V (3) in the standard parameterization using the procedure outlined above.
The matrix V (2) depends on one angle θ 1,2 (it is the 2 × 2 rotation matrix) where we use the notation s i,j = sin θ i,j and c i,j = cos θ i,j . We now choose the vectors e 1 , e 2 and e 3 in the following way The vectors in Eq. (13) fulfill the condition (8).
Now, according to Eq. (9) we construct the columns of the matrix The form of the matrix (15) is not exactly the same as that of the standard parameterization 2 , but both forms are equivalent, because after rephasing of the matrix V (3) in Eq. (15) by multiplying the first and second column by e iδ 1,3 and the first row by e −iδ 1,3 one obtains exactly the standard parameterization.

Example 2. Standard parameterization for 4 quark generations
In this example we will construct an analogue of the standard parameterization for 4 gener- Then, using Eqs. (9) we construct the vectors v  (17)

C. Wolfenstein parameterization for 4 quark generations
The other widely used parameterization of the CKM matrix is the one proposed by Wolfenstein 3 in which the matrix elements are expressed as powers of the parameter λ For 3 generations the CKM matrix in the Wolfenstein parameterization has the form by assuming a one to one correspondence between the Wolfenstein parameters A, λ, ρ and η and the parameters of the standard parameterization s 12 ≡ λ, s 23 ≡ Aλ 2 and s 13 e iδ 13 ≡ Aλ 3 (ρ+iη). 4 We shall generalize the parameterization (19) to the case of 4 quark generations using the method outlined earlier.
First we have to construct the vectors e i , which are expressed in the spirit of the Wolfenstein parameterization in terms of the powers of λ. The vectors e i are real and are chosen in the following way Here The powers k i are considered to be constants (suppression factors) and A and the CKM matrix is equal The matrix V (4) in Eq. (22) is described by 9 parameters: λ, A, ρ, η of the matrix (19), introduced by Wolfenstein, 3 , δ 1,4 , δ 2,4 of the vectors in (21).
Not all these parameters can be determined from the experimental data. On the other hand we can derive some restrictions on the powers k i of the supression factors for the 4-th generation from the experimental information for the CKM matrix for 3 generations. We have the following information 28 Now, using the information in Eq. (23) with the explicit representation of the 4 × 4 CKM matrix in Eqs. (21) and (22) we obtain the following restrictions on the powers k i which can be resolved and give The vector v 4 for the minimal values of k i in Eq. (25) has the following form This result is rather surprising, because in the case of 3 generations the suppression has totally different structure. The full analysis of the 4 × 4 CKM matrix based on Eq. (22) will be published elsewhere. However, we would like to note that the real suppression may be different, because we have only obtained the lower limits of the suppression powers.
To conclude this section we will compare the values of two following rephasing invariants For 4 generations Eq. (28) does not hold and we have Experimentally J A is of the order λ 6 and J B is rather poorly known, because it contains the CKM matrix elements that are known with large errors. The sum J In this section we will present the systematic study of the most general rephasing invariant monomials that can be built out of the CKM matrix elements and its conjugates. These monomials can be considered as building blocks of general observables of the CKM matrix.
Next we will show that such invariant monomials can be expressed as powers of a finite number of elementary rephasing monomials.
The discussion of the rephasing invariant monomials depends on the number of generations and we will discuss here in detail the case of 3 generations. The generalization to 4 or more generations is in most cases simple, but may be tedious.
Let us denote by P (m, n) the most general monomial constructed from the CKM matrix elements and its conjugates: Here m and n are 3×3 matrices with integer 35 , non negative matrix elements and [m] ij = m ij and [n] ij = n ij . The mapping between the monomial P (m, n) and the matrices m and n is one to one 36 .
The monomial P (m, n) in general is not rephasing invariant. Suppose that we make the following phase transformation of the CKM matrix then the monomial P (m, n) is transformed in the following way P (m, n) → e iφ 1 (m 11 +m 12 +m 13 −n 11 −n 12 −n 13 ) P (m, n), i.e., if the sum of the elements of the first row of the matrices m and n are equal. From this one obtains

Example 3. Rephasing invariant and pure rephasing invariant (PRI)
The rephasing invariant is not a pure rephasing invariant, because it is equal to i.e., one can factor out the square |V 11 | 2 out of it. On the other hand the remaining part The PRIs can be represented by two matrices m and n, as in Eq. (32), but they can also be represented by one 3 × 3 matrix p with the following properties: 1. The matrix elements of p are integers (positive, negative or 0).
2. The sum of the elements of p in each row and column is equal to 0.
3. A permutation of the rows and columns of the p matrix is reversible and the resulting matrix is also the p matrix of pure rephasing invariant.
The PRI, which we call B(p), corresponding to the matrix p is constructed in the following way: It is easy to show that B(p) constructed in such a way is rephasing invariant and that it cannot be factored out into the squares of the CKM matrix elements and other rephasing invariant, so it is indeed the PRI.
The one to one mapping between the p matrix and PRI B(p) has the following additional (39)

Example 4. Analytic expression of PRI for a given matrix p
Suppose that the matrix p is equal then the rephasing invariant defined by p is equal and it fulfills all the properties of a PRI.
Let us introduce now the notion of the fundamental rephasing invariant (FRI). (40) 6-th order fundamental rephasing invariants (I 1 , I 2 , . . . , I 12 ) For each FRI in Eqs. (40) and (41) there corresponds a p matrix, e.g., The important condition in Theorem 2 is that the powers of the invariants are positive.
The next important point is that in the decomposition there may be no more than 1 fundamental rephasing invariant of the 6-th order. Without these conditions the decomposition of a pure rephasing invariant into the fundamental invariants is not unique. The proof of this theorem is given in the Appendix.
The inverse theorem is not true, the product of two or more FRIs is rephasing invariant, but it does not have to be the PRI.
From Theorem 2 follows and there are analogous formulas for the remaining I i 's.
To conclude this section let us briefly discuss some properties of the rephasing invariants for 4 generations of quarks.
The notion of the fundamental rephasing invariant is generalized to contain no more Let us also briefly discuss the unitarity properties of the FRIs with 4 CKM matrix elements. From simple considerations one can find out that there are 36 such invariants (and its conjugates), e.g., Unitarity gives 48 relations between them, e.g., If we take the imaginary parts of all these unitarity relations, then we obtain 48 linear homogeneous equations for 36 variables. Not all these equations are linearly independent and eventually it turns out that only 9 of these equations are linearly independent. In Ref. 8 it has been shown that from unitarity of the CKM matrix one can obtain further linear relations between these 9 imaginary parts and only 3 imaginary parts are sufficient to express the remaining ones. The coefficients of the relations in the latter step depend on the real parts of the invariants and the squares of the CKM matrix elements, so the final expressions are complicated. We will address this problem and discuss the consequences of the unitarity of the CKM matrix for 4 generations elsewhere.

IV. CONCLUSIONS
We have discussed here two important topics concerning the CKM matrix: parameterizations and the rephasing invariants of the CKM matrix. The elements of a general p matrix in (38) are positive or negative integers and zeros.
In Table I we list all possible distributions of the number of the elements of the p matrix which are positive, negative or zero. The Type 3, 4, 6 and 7 contain two subtypes of distributions, which are related by complex conjugation and thus do not require separate proofs. As we know one can make permutations of rows and columns of the p matrix and this operation has an inverse. In such a way we can simplify the proof by organizing the elements of the p matrix in the standard form without losing the generality.
We will consider now each type of the p matrix listed in Table I.
After a suitable permutation the p matrix of the Type 1 has the following standard form Then from Eqs. (39), (40) and (42) one sees, that p 1 matrix corresponds to the J 1 fundamental invariant taken to the power n, that is Type 2. Here we have two types of nonequivalent p matrices. The first one in the standard form reads and B(p 2 A ) has the representation The second nonequivalent p matrix of the Type 2 has the form and B(p 2 B ) has the representation Type 3. The p 3 matrix in the standard form is equal and B(p 3 ) has the representation Type 4. The p 4 matrix after a suitable permutation of rows and columns takes the following standard form , n 1 , n 2 , n 3 > 0 (A. 9) and B(p 4 ) has the representation p 4 → B(p 4 ) = (I 4 ) n 1 · (J 3 ) n 2 · (J 7 ) n 3 . (A.10) Type 5. The standard form of the matrix p 5 is equal , n 1 , n 2 , n 3 > 0 (A.11) and B(p 5 ) has the representation and B(p 7 ) has the representation p 7 → B(p 7 ) = (J 5 ) n 1 · (J 9 ) n 2 · (J 6 ) n 3 · (I 1 ) n 4 . (A. 16) We have considered here all possible types of pure monomial rephasing invariants and thus Theorem 2 follows from explicit calculation. This completes the proof. 26 Also the following formula holds Im(V αi V βj V * αj V * βi ) = J γ,k ε αβγ ε ijk = J(3(δ αi δ βj − δ αj δ βi ) + δ αi + δ βj − δ αj − δ βi ).
. 27 For example the parameter λ for the 2 × 2 rotation matrix will be called the angle-like variable.