Revisiting representations of quark mixing matrix

Using unitarity, unlike the approaches available in the literature, we have constructed 9 independent representations of CKM matrix starting with each of the 9 elements of the matrix. The relationship of these independently constructed representations with the already available ones in the literature has been compared and discussed. Further, the implications of these representations have been explored for some of the CKM parameters such as \delta, J and \epsilon_k. Interestingly, we find that the PDG representation which is equivalent to our first representation seems to be most appropriate to incorporate the hierarchy of the elements of the CKM matrix as well as to describe the related phenomenology.


Introduction
Over the last few decades, Cabibbo-Kobayashi-Maskawa (CKM) [1,2] phenomenology has registered remarkable progress on the experimental as well as theoretical front.On the experimental front, significant developments have been made in generating large amount of data for the measurement of various CKM parameters.Several groups like Particle Data Group (PDG) [3], CKMfitter [4], HFLAV [5], UTfit [6], etc., have been actively engaged in continously updating their analyses to arrive at more and more refined conclusions.At present, we have several CKM parameters which are determined with good deal of accuracy, e.g., the matrix elements |V us | = 0.2243 ± 0.0008, |V cb | = (40.8± 1.4)10 −3 are determined within an accuracy of a few percent [3].Similarly, the angles of the unitarity triangle are also known within an error of few percent.In particular, the parameter sin 2β, representing angle β of the unitarity triangle, is well measured with its world average being (22.2±0.7) • [3,5].Similarly, the angle α of the unitarity triangle is also known within a few percent level, e.g., the world average is (85.2 +4.8 −4.3 ) • [3].
On the theoretical front, CKM paradigm has played a crucial role in understanding several important features of flavor physics.The CKM matrix, characterised by 3 mixing angles and a CP violating phase, can have only 9 independent representations or parametrizations.In the literature [7]- [11] several representations of the CKM matrix have been discussed.In particular, Refs.[7] and [8] adopt the methodology of arriving at the representations by writing the CKM matrix as a product of three rotation matrices.In Ref. [9] attempt has been made to construct the possible representation using the unitarity constraints of the CKM matrix.Somewhat recently [10,11], attempt has been made to incorporate one of the angles of unitarity triangle as CP violating phase of the CKM matrix, resulting into several possible representations of CKM matrix involving 4 measurable parameters.
A closer look at the above attempts reveals that none of these emphasise clearly the fact that the given representations are the only 9 possible independent ones, nor do these explore the relation between these different representations.Also, keeping in mind the present level of measurement of the CKM parameters, it is to be noted that these attempts do not explore explicitly the usefulness of a particular representation.It is also not clear that the recent attempt [10,11] involving 4 directly measurable CKM parameters, including one of the angle of unitarity triangle in a particular representation, would lead to any advantage in carrying out the phenomenological analyses.It, therefore, becomes interesting to find 9 independent representations of the CKM matrix starting from the basic constraints of unitarity and also to check the co-relation of these with the already existing representations given by different authors.It would also be interesting to check whether any particular representation can be preferred for carrying out particular phenomenological analysis.
Keeping the above issues in mind, the purpose of the present paper is to construct all possible independent parametrizations of CKM matrix in rigorous and ab-initio manner.The relationship of these independently constructed representations with the already available representations in the literature would also be explored.Further, the implications of these representations, incorporating unitarity constraints, on some of the CKM parameters would be explored using the latest data.

Revisiting representations of the CKM matrix
Before proceeding further, a brief discussion of the presently known representations is perhaps desirable.To begin with, let us define the CKM matrix, e.g., The mixing matrix V CKM being a 3 × 3 unitary matrix can have only 9 independent representations.We first discuss the approach given by C. Jarlskog [7], for the sake of readability as well as to facilitate discussion, we reproduce their methodology here.According to Ref. [7], the CKM matrix can be written as a product of three rotation matrices, e.g., R 12 , R 23 and R 13 , given by where s 12 , s 23 and s 13 denote the sines of the three mixing angles.The author mentions 12 different ways to arrange product of these rotation matrices, yielding where To obtain a possible unitary representation of CKM matrix, it was suggested that phase factor δ could be added in 3 different ways leading to 36 representations of the CKM matrix, obviously all of these cannot be independent.Considering the possibility R = R 23 (θ 23 )R 13 (θ 13 )R 12 (θ 12 ), the phase factor δ can be added in 3 possible ways as R 23 (θ 23 , δ) R 13 (θ 13 , 0) R 12 (θ 12 , 0) or R 23 (θ 23 , 0) R 13 (θ 13 , δ) R 12 (θ 12 , 0) or R 23 (θ 23 , 0) R 13 (θ 13 , 0) R 12 (θ 12 , δ).For example, R 23 (θ 23 , δ) = H.Fritzsch and Z. Z. Xing [8], after an analysis of the 12 combinations given by C. Jarlskog, mentioned in equation (3), found that only 9 out of these are 'structurally' different.They also noted that the phase factor δ can be associated in 3 different manners with any of the rotation matrix, however, it can be shown that these are all equivalent due to the facility of rephasing invariance.The 9 possible representations of CKM matrix given by them are shown in Table 1.
A. Rasin [9] had also attempted to construct possible representations of the CKM matrix using the unitarity constraints of the CKM matrix, the different possibilities, without changing their notations, have been presented in Table 2.We have closely examined these representations and find that only 6 of these are independent.For example, the representation 9 of Table 2  (4) Using the facility of rephasing invariance, multiplying the above matrix from the left side by the matrix    one obtains representation 9. Similarly, representations 5 and 6 are related to 7 and 8 respectively.In an another approach [10,11], attempt has been made to use experimentally measurable quantities, i.e., magnitudes of the CKM matrix elements and angles α, β or γ of the unitarity triangle as the CP violating phase of the CKM matrix resulting into several possible representations of the CKM matrix involving 4 measurable parameters.Again to facilitate discussion, we reproduce some essentials of these attempts here.For example, considering angle γ as the phase of the CKM matrix, 4 parametrizations have been obtained, referred to as the γ angle parametrizations.This angle can be expressed in terms of the elements of the CKM matrix as The phase γ can be allocated along with either of the 4 CKM matrix elements appearing in the definition of γ, i.e., V ud,ub,cd,cb , leading to only one of these being complex and all others being real and positive, e.g., The above defines 4 parametrizations(γ 1 , γ 2 , γ 3 and γ 4 ) of the CKM matrix in which γ is explicitly the CP violating phase, all 4 of these being equivalent.To obtain parametrization γ 3 , one can use γ, |V cd |, |V cs |, |V td | as independent variables and express others as functions of them, resulting into where The other 3 γ parametrizations can be obtained in a similar manner.Considering angles α and β as the CP violating phase of the CKM matrix, the corresponding α and β parametrizations can also be obtained.Interestingly, the authors have shown that the 4 parametrizations for α or β or γ are all equivalent and also these 12 parametrizations can be transformed from each other and again are all equivalent.

Cartesian derivation of independent representations of the CKM matrix
To understand the issue of construction of 9 independent representations of the V CKM , we have attempted to carry out this task in a rigorous ab-initio manner, without involving the rotation matrices, henceforth these would be referred to as Cartesian representations.To this end, we follow an approach wherein the 9 independent representations are constructed using any individual element of a 3 × 3 complex unitary matrix V given by It may be noted that the CKM matrix is sandwiched between quark fields which allows 5 out of 6 phases of above 3 × 3 unitary matrix to be removed using rephasing invariance, leaving the matrix having 3 independent angles and 1 non removable phase.Further, the elements of the CKM matrix should obey the following unitarity constraints where α, β ≡ (1, 2, 3) and i, j ≡ (1, 2, 3).Taking into consideration the physical structure of CKM matrix, while constructing its representations one needs to consider the diagonal elements of matrix V, given in equation (8), to be nearly equal to unity whereas the off diagonal elements should be much smaller than unity.
To illustrate our procedure, we consider an example wherein we begin with a complex element a 21 of the matrix V, defined as Following the unitarity constraints, one may introduce two more angles θ 2 and θ 3 such that where c i = cos θ i and s i = sin θ i , with i = 1, 2, 3.It is interesting to note that this is a unique way to define the above elements in terms of the mixing angles and any other way would disturb the unitarity relations.Using these, the matrix V given in equation ( 8) becomes The above equation can also be written as (13) Further, using unitarity conditions mentioned in equation ( 9), one gets As a next step, we now derive the remaining 4 complex elements of the CKM matrix, i.e., a 12 , a 13 , a 32 and a 33 .Using the already defined 5 elements of the CKM matrix, mentioned in equations (10) and (11), the following unitarity constraints a 11 a * 21 + a 12 a * 22 + a 13 a * 23 = 0, can be re-written as By splitting the complex elements a 12 , a 13 into real and imaginary parts, we get From equation (18), for non-zero c 1 , Im(a 12 ) and Im(a 13 ) can be expressed with a real coefficient α as Im(a 12 ) = −αs 3 and Im(a 13 ) = αc 3 .
Using these relations, equation ( 19) becomes Therefore, the elements a 12 and a 13 can be written as Similarly, using the unitarity relations Finally, using equations ( 23) and(25), matrix V can be written as (26) The factored out phases can be removed by using the facility of rephasing invariance, yielding the following representation of the CKM matrix in terms of 3 mixing angles and 1 non removable phase Similarly, the other independent representations can also be constructed by using a different starting element of the matrix V.In Table 3, we have summarized the 9 independent Cartesian parametrizations of the CKM matrix along with the corresponding starting element of unitary matrix V.
As a next step, it is desirable to explore their relationship with the representations available in literature.For example, considering the representation 1 of the CKM matrix given in Table 3, one can obtain the commonly used parametrization [12] adopted by PDG [3], e.g., where c ij = cos θ ij , s ij = sin θ ij for i, j=1, 2, 3, with θ 12 , θ 23 , θ 13 and δ being the 3 mixing angles and the CP violating phase respectively.To do so, we need to re-designate s 1 → s 13 , s 2 → s 12 and s 3 → s 23 as well as carry out rephasing of the quark fields using the multiplication of matrices respectively on the left and right side of Cartesian representation 1.The Kobayashi-Maskawa (KM) representation [2,13] can be shown to be related to the Cartesian representation 7.

Unitarity based numerical evaluation of the Cartesian representations
After having discussed the relationship of 9 Cartesian representations with other similar representations [7]- [11], we have carried out a unitarity based analysis, using the latest data, to understand the significance of these representations.To begin with, we have calculated mixing angles and CP violating phase for each parametrization for numerical evaluation of the corresponding CKM matrix.As a first step, we have considered the representation 1 of Table 3, this being equivalent to the PDG representation.In order to find θ 1 , we have evaluated |V ub | using a unitarity based analysis involving the 'db' triangle [14], shown in Figure 1.From this triangle, using the relationship between its angles and sides, one gets Using the PDG values [3] of |V cd | = 0.221 ± 0.004, |V ud | = 0.97373 ± 0.00031 and the value of |V cb | = (40.6 ± 0.9) × 10 −3 as advocated by Belle collaboration [15] as well as the values of α and β as given by PDG [3], we get This is a rigorous unitarity based value of V ub , which is in agreement with values given in Refs.[14,16].This value of V ub implies the ratio V ub V cb = 0.08506 ± 0.00373, in agreement with measurements from Λ b → ρµν and B s → Kµν decays [5].Further, considering the latest PDG value [3] of |V us |, one gets Considering the value of |V cb | as mentioned above, we obtain As a next step, to evaluate the phase δ corresponding to this representation, we first discuss the relationship of phase δ with the angle γ of the unitarity triangle.To do so, one can express γ in terms of the elements of the mixing matrix, mentioned in equation ( 6).This can be further expressed as The above equation can be simplified as This can be solved further to obtain For the present unitarity based analysis, the angle γ can be found using the closure property of the angles of the unitarity triangle yielding γ = (72.6 ± 4.6) • .Using the above expression one gets implying that for representation 1 of Table 3, the CP violating phase δ be considered to be equal to the angle γ of the unitarity triangle.After having found the three mixing angles and the phase δ, we obtain the corresponding CKM matrix for the representation, i.e., 0.97451 ± 0.00018 0.2243 ± 0.0008 0.00349 ± 0.00015 0.2242 ± 0.0008 0.97371 ± 0.00019 0.0406 ± 0.0009 0.00872 ± 0.00020 0.03981 ± 0.00088 0.99917 ± 0.00004 A look at this matrix reveals that this shows an excellent overlap with the one obtained by PDG [3]  .This, as well as unitarity, leads to hierarchy amongst the 9 elements of the CKM matrix, as defined in Refs.[17]- [19].
After numerically constructing this representation of the CKM matrix, as a next step we have found the angles and phases of the remaining parametrizations in order to arrive at the numerical values of the corresponding matrix elements.The numerical evaluation of other representations is not straight forward as in these cases the CP violating phase cannot be considered to be nearly equal to the angle γ of the unitarity triangle.Therefore, for these representations, along with |V us |, |V cb | and |V ub | as inputs, instead of phase δ, we consider the numerical value of the element |V td | from the CKM matrix given in equation (40).It may be noted that the element |V td | captures the effects of CP violating phase δ as is emphasized in the literature [20].Using these inputs, for each Cartesian parametrization, we can then find the values of the 3 mixing angles, θ 1 , θ 2 , θ 3 and the CP violating phase δ, these have been presented in column 2 of Table 4.It may be noted that for different parametrizations, magnitudes of the corresponding CKM matrix elements have not been given here as these are rephasing invariant quantities.
Corresponding to the different representations, we have also found the CP violating rephasing invariant Jarlskog's parameter J [7] defined as For all the Cartesian representations of the CKM matrix, in column 3 of Table 4, we have presented the corresponding expressions of J. On numerical evaluation, as expected, its value comes out to be same for all the representations, also being in agreement with the PDG value [3], i.e., (3.08 +0.15 −0.13 )×10 −5 .
θ 1 = 0.0087 ± 0.00020 θ 2 = 0.2261 ± 0.0008 θ 3 = 0.0398 ± 0.0009 δ = (22.9± 1.1) Further, for different representations, we have evaluated ϵ k , the CP violation defining parameter in the K − K system.Following Ref. [20] and using the Cartesian representation 1, this being equivalent to PDG representation, expressing V cs , V cd , V ts and V td in terms of the corresponding mixing angles and δ as well as using the numerical values of these inputs, we get this being largely in agreement with the one given by PDG, i.e., (2.228 ± 0.011) × 10 −3 .The same exercise has been carried out for the remaining parametrizations.Intriguingly, out of the 9 representations, we find that the representations 1, 2, 3, 4, 7 and 8 are able to provide an appropriate fit to the parameter ϵ k .The other 3 representations, i.e., 5, 6 and 9 are very much off the mark.This is not a surprising conclusion keeping in mind the hierarchical nature of the elements of the CKM matrix as well as the accuracy with which these are measured.This has also been discussed in a different context in a recent paper by Xing et.al. [19] while evaluating parameter J in terms of the magnitudes of the CKM matrix elements.This also brings to fore whether there is a preferred representation of the CKM matrix for carrying out phenomenological analyses.To this end, we find PDG representation, perhaps, provides a viable answer to this question.Interestingly, in the PDG representation, within the level of fraction of a percent, the mixing angles capture the hierarchy of the well measured 3 CKM matrix elements.Further, as discussed earlier, the CP violating phase δ is almost equal to one of the angles of the unitarity triangle.Interestingly, if we compare the PDG representation with the γ representation mentioned in equation (7), one finds that the best measured CKM matrix element V us is expressed in terms of other lesser known elements, unlike the PDG representation.One may also like to compare the PDG representation with the Wolfenstein representation, interestingly, the well known CKM matrix elements as well as the CP violating phase enter indirectly in the Wolfenstein representation unlike the PDG representation.

Summary and Conclusions
In the literature, several representations of the CKM matrix have been discussed, however, none of these attempts emphasize clearly the fact that the given representations are the only 9 possible independent ones, nor do these explore the relation between these different representations.Further, keeping in mind the present level of measurement of CKM parameters, these attempts do not explore explicitly the usefulness of a particular representation.In the present work, we have attempted to construct 9 possible independent parametrizations of CKM matrix in rigorous and ab-initio manner, starting with each of the 9 elements of the matrix.The relationship of these independently constructed representations with the already available ones in the literature has been discussed.Further, incorporating unitarity constrains, the implications of these representations have been explored for some of the CKM parameters such as δ, J and ϵ k .It has been observed that the PDG representation, perhaps, provides the best option for carrying out CKM phenomenology at the present level of measurements.

Table 3 :
Cartesian representations of the CKM matrix.
It needs to be emphasized that the matrix given in equation (40) has been obtained using well measured CKM parameters and the unitarity based constraints.It is interesting to mention that in the representation considered by us, the hierarchy of the CKM matrix elements is very well captured by the hierarchy of the mixing angles, i.e., |V us | ∼ s 2 ≫ |V cb | ∼ s 3 ≫ |V ub | ∼ s 1

Table 4 :
Calculated parameters using different representations