16 Parametrizations of the CKM Matrix

The CKM matrix is usually parametrized in some specific way. The purpose of the parametrizations is to incorporate the constraints of $3×3$ unitarity. Some parametrizations also incorporate experimental information, in particular the pattern of moduli in eqn (13.47); this is the case of the Wolfenstein parametrization and of related parametrizations, which in practice are the ones most often used.

Rephasing-invariance is the possibility of changing the overall phase of any row, or of any column, of the CKM matrix, without changing the physics contained in that matrix. We may use this freedom to constrain five matrix elements to be real, or else to fix their phase in any other desirable way. It follows that the $3×3$ unitary CKM matrix should in principle be parametrized by three rotation angles and one phase.³⁵

Even if five matrix elements may have their phases fixed, it is important to notice that those five matrix elements cannot be chosen at will. This is because the quartets are rephasing-invariant. One must be careful not to implicitly fix the phase of any quartet when choosing a phase convention for the CKM matrix.

All phase conventions used in practice have one thing in common: the matrix elements $Vud$ and $Vus$ are chosen real and positive. The reason for this choice is the central role played by $λu≡VudVus∗$ in the physics of the neutral-kaon system. If $λu$ is real and spurious phases are neglected, then $Γ12≈A0∗A¯0$ is real at tree level, as will be shown in Chapter 17. This is an advantageous simplification. Still, this phase convention is in no way necessary, because physics is rephasing-invariant.³⁶

It seems natural to parametrize V by means of three Euler angles—the angles of three successive rotations about different axes—and one phase. The phase must be introduced in such a way that it cannot be eliminated by means of a rephasing of the quark fields.

The first parametrization of the CKM matrix was put forward by Kobayashi and Maskawa (1973). They wrote

Here, $ci$ and $si$ are shorthands for $cosθi$ and $sinθi,$ respectively, where $θ1,θ2,$ and $θ3$ are Euler angles. One of the rotations is on the xy plane, and the other two rotations are on the yz plane. The phase $δ$ appears as a rephasing of the third generation; as the rephasing occurs in between two rotations involving that generation, it is impossible to identify $δ$ with a rephasing of the quark fields.

The first row and the first column of V have implicitly been chosen to be real, by use of the rephasing freedom of the CKM matrix. Without loss of generality $θ1,θ2,$ and $θ3$ may be constrained to lie in the first quadrant, provided one allows $δ$ to be free, $0≤δ<2π.$ Indeed, putting one of the Euler angles in any other quadrant is equivalent to a physically meaningless rephasing of V, sometimes coupled with the transformation $δ→δ+π.$ For instance, if $θ1$ was chosen to lie in the second quadrant, then we might bring it into the first quadrant by means of the transformation $c1→−c1.$ This transformation is equivalent to $eiδ→−eiδ$ together with

which is a change of the sign of the fields u, s, and b.

In the Kobayashi–Maskawa parametrization

From this it is easy to derive that the maximum possible value of J is $1/(63),$ which is obtained when $δ=π/2,θ2=θ3=π/4,$ and $c1=1/3,$ i.e., when all matrix elements of V have modulus $3−1/2,$ cf. eqn (13.49).

Chau and Keung (1984) have introduced a different parametrization, the use of which has been advocated by the Particle Data Group (1996):

Here, $cij$ and $sij$ are shorthands for $cosθij$ and $sinθij,$ respectively. The three rotation angles $θ12,θ13,$ and $θ23$ may be restricted to lie in the first quadrant provided one allows the phase $δ13$ to be free. Only four matrix elements are chosen to be real; still, only one physical phase appears in the parametrization.

The $sij$ are simply related to directly measurable quantities:

because experimentally $∣Vub∣$ is very small. On the other hand,

and therefore

is related in a complicated way to rephasing-invariant quantities.

We call a parametrization ‘rephasing-invariant’ when its parameters are defined to be rephasing-invariant quantities, for instance the moduli of some matrix elements, or the phases of some quartets. In contrast, the rotation angles in the previous section can be related to measurable quantities—see for instance eqns (16.4)—but they are not directly defined to be rephasing-invariant quantities.

We shall next present three rephasing-invariant parametrizations. As they are not used very often, some readers may prefer to skip this section.

Branco and Lavoura (1988a) have suggested parametrizing the CKM matrix by means of four linearly independent $Uαi.$ This is a convenient choice, because the moduli constitute the most reliable information on the CKM matrix. A convenient set of four squared moduli is

Indeed, $∣Vus∣,∣Vub∣,$ and $∣Vcb∣$ are small and relatively well measured, while $∣Vtd∣$ is crucial in $Bd0−Bd0¯$ mixing. In order to reconstruct the full CKM matrix from the parameters in eqn (16.7), one first uses the normalization of the rows and columns of V, which allows one to compute the values of the remaining five moduli. One then uses the relation

which is derived in an analogous way as for eqn (13.27). One can thus find the real part of each quartet. The imaginary part of the quartets, J, is given, as in eqn (13.29), by

together with eqn (16.8). The sign of J cannot be found from the moduli alone. This is because the transformation $V→V∗$ leaves the moduli invariant but changes the sign of J. As a consequence, the parametrization of Branco and Lavoura requires that sign J be given together with the parameters in eqn (16.7).

Bjorken and Dunietz (1987) were the first authors to put forward a rephasing-invariant parametrization. They chose the following phase convention:

They used as parameters the rephasing-invariant quantities

It follows from the definition of $ϕ$ and from the phase convention in eqn (16.10) that $Vub=Uubexp(−iϕ),$ while $Vus=Uus$ and $Vcb=Ucb.$

The full CKM matrix may be reconstructed in the following way. Firstly, as $Vud$ and $Vtb$ are real and positive by convention,

Then,

constitutes a quadratic equation for $∣Vcs∣,$ which gives (remember eqn 16.10)

The orthogonality conditions yield the remaining three matrix elements of V:

In the SM with $ng$ generations, one may eliminate $2ng−1$ phases from the initial $na2$ phases of the matrix elements of V through a rephasing of the quark fields. The number of rephasing-invariant phases is thus

At this stage we are not yet imposing unitarity. It is remarkable that $nphases$ equals the number of parameters necessary to parametrize the $ng×ng$ unitary matrix V: $nphases=Nparam$⁠.

The idea of Aleksan et al. (1994) was to parametrize V by four $ωαiβj.$ We already know that $ωαiβj=ωβjαi=−ωαjβi=−ωβiαj.$ Therefore, for three generations one needs to consider only nine phases: $ωtbud,ωtbcd,ωcbud,ωtbus,ωtbcs,ωcbus,ωtsud,ωtscd,$ and $ωcsud.$ From these nine phases only four are linearly independent. We may choose as parameters

The first two phases are related to $β$ and $γ$ in eqns (13.31) by

where we have taken the argument of a complex number to lie between $−π$ and $+π.$ Similarly, $ωtbcs$ and $ωcsud$ can be related to the two phases

through³⁷

In the SM, these four angles obey a strong hierarchy (Aleksan et al. 1994): although $β$ and $γ$ may be large, $ϵ$ and $ϵ′$ must be small: $ϵ≲0.05$ and $ϵ′≲0.0025.$

The four phases $β,γ,ϵ,$ and $ϵ′$—or, equivalently, $ωtbcd,ωcbud,ωtbcs,$ and $ωcsud$—can be used to parametrize the $3×3$ unitary CKM matrix. The other five $ωαiβj$ can be readily obtained from

Equations (16.22) follow from the algebra of complex numbers. Unitarity is only needed in order to compute the moduli of the matrix elements from the phases of the quartets. It follows from the normalization of the $ith$ column of V that

We therefore need to know the ratios $Uti/Uui$ and $Uci/Uui.$ They are found by applying the law of sines to the unitarity triangles. Let (i, j, k) be a permutation of the indices (d, s, b) and consider the unitarity triangles arising from the orthogonality of the columns of the CKM matrix. Then,

By using eqns (16.23) and (16.24) one obtains the moduli of all matrix elements as functions of the sines of linear combinations of the parameters in eqn (16.17). Of course, since the elements of the CKM matrix are not rephasing-invariant, one must choose a specific phase convention before the matrix elements themselves can be written in terms of the manifestly rephasing-invariant moduli and $ωαiβj.$ Clearly, once one knows the moduli of all matrix elements and the phases of all quartets, we are in possession of all the physical information in the CKM matrix.

In 1983 it was realized that the bottom quark decays predominantly to the charm quark: $∣Vcb∣≫∣Vub∣.$  Wolfenstein (1983) then noticed that $∣Vcb∣∼∣Vus∣2$ and introduced an approximate parametrization of V—a parametrization in which unitarity only holds approximately—which has since become very popular. He wrote

The parameter $λ≈0.22$ is small and serves as an expansion parameter. On the other hand, $A∼1$ because $∣Vcb∣∼∣Vus∣2.$ Finally, $∣Vub∣/∣Vcb∣∼λ/2$ and therefore $ρ$ and $η$ should be smaller than one. Thus, one may estimate the order of magnitude of any function of the matrix elements of V by considering the leading term of its expansion in $λ.$³⁸

One easily checks that the unitarity relations—normalization of each row and column of V, and orthogonality of each pair of different rows or columns—are satisfied up to order $λ3$ by the matrix in eqn (16.25). An expansion of V up to a higher power of $λ$ must be made if one wants to obtain a better approximation to unitarity.

The Wolfenstein parametrization is original for two main reasons. Firstly, it incorporates as ingredients not only unitarity, but also experimental information: $∣Vus∣≪1,∣Vcb∣∼∣Vus∣2,$ and $∣Vub∣≪∣Vcb∣.$ Secondly, it is only approximately unitary, with the approximation to exact unitarity being achieved in a series expansion.

In the Wolfenstein parametrization, to leading order,

This is the justification for the coordinates of the vertices of the unitarity triangle in Fig. 13.2.

While $λ=0.2205±0.0018$ and $A=0.824±0.075$ are relatively well known, the parameters $ρ$ and $η$—or, equivalently, the angles $α,β,$ and $γ$—are much more uncertain. The main goal of CP-violation experiments is to over-constrain these parameters and, possibly, to find inconsistencies suggesting the existence of physics beyond the SM.

Sometimes the expansion up to order $λ3$ in eqn (16.25) is not sufficient and one may want to use terms of higher order in $λ.$ One knows for instance that the imaginary parts of all quartets should be equal in absolute value. This is however not true when using eqn (16.25): $Qudcs$ and $Qcstb$ are real, while $ImQtdcb=A2λ6η$ and $ImQuscb=A2λ6η(1−λ2/2).$ Such imprecisions may become misleading and/or constitute a source of error when using eqn (16.25).

Expanding the Wolfenstein parametrization to a higher order in $λ$ is easier and more systematic when one is guided by an exact parametrization, i.e., by an exactly unitary matrix V. Indeed, one needs a definition of the way in which the series expansion in $λ$ is to be carried out to higher orders. A way to do this has been suggested by Branco and Lavoura (1988b). They have used as a guide the Bjorken–Dunietz parametrization. They have defined the parameters by means of the equations

together with the phase convention in eqn (16.10). In this way, $λ=∣Vus∣,A=∣Vcb/Vus2∣,μ=∣Vub/(VusVcb)∣,$ and $ϕ=ωuscb$ are directly related to measurable quantities. It is important to stress that eqns (16.27) are exact by definition: the expressions for $Vus,Vcb,$ and $Vub$ are not corrected by terms of higher order in $λ.$

We may reconstruct the full CKM matrix just as was done in the Bjorken–Dunietz parametrization. Thus,

Together with eqns (16.15) this fixes the CKM matrix. We may now perform the expansion as a series in $λ$ up to any desired order.³⁹ We present here the result of the expansion up to order $λ5.$ For ease of comparison with eqn (16.25), we substitute $μ$ and $ϕ$ by $ρ≡μcosϕ$ and $η≡μsinϕ.$ We obtain

Equations (16.27) and (16.29) coincide with eqn (16.25) up to order $λ3.$

Buras et al. (1994) have used the Chau–Keung parametrization as the basis for a different exact version of the Wolfenstein parametrization. They defined the parameters by means of the equations

Then,

Substituting these expressions in eqn (16.3) one obtains an exact parametrization of the CKM matrix, which one may then proceed to expand as a power series in $λ.$ In practice, the differences between the parametrizations of Buras et al. (1994) and of Branco and Lavoura (1988b) first arise only at order $λ6:$ eqns (16.29) are valid in both parametrizations.

In the parametrization of Branco and Lavoura (1988b)

The parameters $λα$ for the $K0−K0¯$ system, defined in eqn (13.50), are

For the $Bd0−Bd0¯$ system,

where

For the $Bs0−Bs0¯$ system,

It is useful to introduce a Wolfenstein-type parametrization of the CKM matrix with the following four parameters: $λ≡∣Vus∣,A≡∣Vcb∣/∣Vus∣2,Rt≡∣VtdVtb∣/∣VcdVcb∣,$ and $Rb≡∣VudVub∣/∣VcdVcb∣$⁠. As usual, we make the phase convention that $Vud$ and $Vus$ are real and positive; we also choose $Vcd$ negative and $Vcb$ positive, so that the product $VcdVcb∗$ is real and negative as in the unitarity triangle in Fig. 13.1. Finally, we choose $Vtb$ positive. In this phase convention, the phase of $Vub$ is $−γ$ and the phase of $Vtd$ is $−β$ (remember eqns 13.31).

Working out this parametrization, and making the usual series expansion in $λ,$ one obtains

If one uses for exp $(iβ)$ and for exp $(iγ)$ the expressions in eqns (13.34)–(13.36), one has a parametrization of the CKM matrix in terms of $λ,A,Rt,$ and $Rb.$ All matrix elements have been given up to order $λ5.$

Using this parametrization only up to order $λ3,$ one has the simple result (Buras and Fleischer 1998)

which will be extensively used in Part IV.