ANOTHER METHOD FOR A GLOBAL FIT OF THE CABIBBO-KOBAYASHI-MASKAWA MATRIX

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1 ANOTHER METHOD FOR A GLOBAL FIT OF THE CABIBBO-KOBAYASHI-MASKAWA MATRIX PETRE DIÞÃ Institute of Physics and Nuclear Engineering, P.O. Box MG6, Bucharest, Romania Received February 1, 005 Recently we proposed a novel method for doing global fits on the entries of the Cabibbo-Kobayashi-Maskawa matrix [1]. The new used ingredients were a clear relationship between the entries of the CKM matrix and the experimetal data, as well as the use of the necessary and sufficient condition the data have to satisfy in order to find a unitary matrix compatible with them. This condition writes as 1 cosδ 1 where δ is the phase that accounts for CP violation. Numerical results are provided for the CKM matrix entries, the mixing angles between generations and all the angles of the standard unitarity triangle. 1. INTRODUCTION The Cabibbo-Kobayashi-Maskawa matrix parameterizes the weak charged current interactions of quarks and is a lively subject in particle physics. The four independent parameters of this matrix govern all flavor changing transitions of quarks in the Standard Model, and their determination is an important task for both experimenters and theorists. The large interest in the subject is also reflected by the workshops organized in the last years whose main subject was the CKM matrix [, 3, 4]. At the level of experts there exists a consensus concerning the method of extraction from experimental data information about the CKM matrix entries [ 7], and the usual method is to consider the orthogonality of the first and third columns. Relying on an approximation that comes from Wolfenstein [8] this method is not reliable as we shall show in the paper. In our opinion the two main shortcomings of the current approach are the following: a) one uses one unitarity triangle instead of six; b) one works with an approximation of the standard parameterization which may lead to inconsistencies. We construct a theoretical model and we use it to test the unitarity property of the data as they are provided by the Particle Data Group (PDG) [9] and the published fits [7]. We find that one can reconstruct from PDG data a unitary matrix and the phase δ is close to π/. dita@zeus.theory.nipne.ro Rom. Journ. Phys., ol. 50, Nos. 3 4, P , Bucharest, 005

2 80 P. Diþã. THEORETICAL MODEL Our theoretical framework is as follows. We use the standard parameterization advocated by PDG that we write it in a completely rephasing invariant form as c c c s s U = c s c s s c c s s s s c s s c c s c s s c s c c eiδ eiδ eiδ ei δ with c ij = cosθ ij and s ij = sinθ ij for the generation labels ij = 1, 13, 3, and δ is the phase that encodes the breaking of the CP-invariance. The rephasing invariance is equivalent to choosing the phases of five arbitrary CKM matrix entries equal to zero, and the above form is the simplest one. We remind that phase invariance of the U-transformed quark wave functions is a requirement for physically meaningful quantities. From experiments one measures a matrix whose entries are positive ud us ub = cd cs cb td ts tb As the notation suggests we make a clear distinction between the unitary CKM matrix U and the positive matrix provided by the data. The main theoretical problem is to see if from a matrix as () one can reconstruct a unitary matrix as (1). Thus we have to provide a relationship between the two objects defined above, and it is = U that leads to the following relations ud = c1c13, us = s1c13, ub = s13 cb = s3c13, tb = c13c3, cd = s1c3 + s13s3c1 + s1s13s3 c1c3 cosδ cs = c1c3 + s1s13s3 s1s13s3 c1c3 cosδ td = s 13c1c3 + s1s3 s1s13 s3c1c3 cosδ = s s c + c s + s s s c c cosδ ts It is easily seen that CP-violation requires θ 0, ij = 1, 13, 3, θ π/, and θ π/ ij From the relation (19) one gets 1 3 (1) () (3)

3 3 Global fit of the Cabibbo-Kobayashi-Maskawa matrix 81 i= d, s, b i= u, c, t ji 1= 0, j= u, c, t ij 1= 0, j= d, s, b (4) We stress that the above relations does not test the unitarity, as it is usually stated in many papers; they are necessary but not sufficient conditions. The class of positive matrices satisfying Eqs. (0) is considerable larger than the class of positive matrices coming from unitary matrices. The set (0) is known in the mathematical literature as doubly stochastic matrices, and the subset coming from unitary matrices ij = Uij is known as unistochastic ones [10]. The double stochastic matrices have an important property, they are a convex set, i.e., if 1 and are doubly stochastic so is their convex combination α (1 α ), α [0, 1] as it is easily checked. The first problem to solve is to find a necessary and sufficient criterion for discrimination between the two sets. For that we need to choose four independent parameters; these can be four measurable quantities as ij [11], four independent invariant phases [1], or λ, A, ρ, η as in [8]. We will always choose these parameters as four experimentally measurable quantities, i.e., ij, that means that we can get values for cos δ. We will focus on cosδ because it is the quantity that allows us to separate the unistochastic matrices from the double stochastic ones. For example by using the numerical values from PDG data one can get values for cosδ outside the physical range [ 1, 1], or even complex as we will show in the following. The above relations provides the necessary and sufficient condition the data have to satisfy in order the matrix () comes from a unitary matrix, and this condition is 1 cosδ 1 (5) Since in relations (3) δ enters only through the cosine function we can take δ [0,π] without loss of generality. The last four relations (3) provide us formulae for cosδ and their explicit form depends on the independent four parameters we choose to parameterize the data. To see that we choose the following four independent parameters us, ub, cb, ij, where ij take one value from the set (1,, 31, 3). The first three parameters are those employed in the standard unitarity approach. It is easily seen that we get us cb 1 =, 13 = ub, and, 3 = 1 1 ub ub s s s

4 8 P. Diþã 4 and from the last four relations (3) we obtain four independent formulae for cos δ, one of them being cosδ= 4 4 cd cbub cd ub + cbub + cdub us + cbus + ubus + cbub us (6) = 1 1 us ub cb ub us ub cb The above relations provides us the necessary and sufficient conditions the data have to satisfy in order that the matrix () comes from a unitary one, and they are 0 s ij 1, and 1 cosδ 1 (7) s ij The first conditions 0 1 are also satisfied by the double stochastic matrices and only the last one, 1 cosδ 1, discriminates between the two sets. Thus if the data are compatible to the existence of a unitary matrix these mixing angles have to be equal, and also the different values for cos δ, no matter how we choose the four independent parameters. As a warning what we said before can be summarized as follows: the unitarity property is a property of all the CKM matrix elements and not the property of a row and/or a column, as it is considered by many people working in the field. Now we define a test function that should take into account the double stochasticity property expressed by the conditions (4), the unitarity (6) and the numerical values of data. Our proposal is χ ( ) ( ) 1 = (cos i cos j δ δ ) + ji 1 i< j j= u, c, t i= d, s, b + 1, 1 cos ( i) ij δ 1 j= d, s, b i= u, c, t term that expresses the full content of unitarity, and a second one of the form ij ij χ = σ i= u, c j= d, s, b ij (8) (9) where ij is a numerical matrix that describes the experimental data, and σ is the matrix of errors associated to ij. In the last sum we use only the data coming from the first two rows because the entries of the third row are not yet measured. The above expressions will be used to test globally the unitarity property of the experimental data.

5 5 Global fit of the Cabibbo-Kobayashi-Maskawa matrix RESULTS In the following we test our method on the published data, i.e. we want to see if our necessary and sufficient criterion, 1 cosδ 1, could constrain enough the data. For that we use the PDG [9] data, the fit [6] and its recent up-to-date results [7], and we consider only the central values from the above cited papers. These values are given in relation (10) where we used the notation to denote the numerical CKM data matrix as it is usually provided. All matrices (10) satisfy quite well the stochasticity property (4). PDG CKMfG CKMfG (6) = (7) (9) [5] = (30) (3) [6] = (33) (34) Here CKMfG denotes the CKM fitter Group. As we said before cosδ depends on the four independent parameters we use to obtain it and for comparison we took six groups of independent parameters that lead to 17 independent cos δ ; see Table. It is easily seen that the central values from all PDG and CKMfG are not compatible with unitarity, although they are compatible with the double stochasticity property to an acceptable accuracy. More, the data show that cos δ can take even imaginary values and these values can not be properly processed in the usual approach of the unitarity triangle and Wolfenstein approximation. In all the cases it is s 13 that causes the trouble. For the PDG data s 3 13 = cs + ts ud = i and, respectively, s 13 = cd + cs + td + ts 1= 10 i; for the CKM fitter Group results [6] it has the form s 3 13 = 1 ud us = i. On the other hand such an incompatibility cannot, in principle, be detected by using the Wolfenstein parameterization, because quantities as s 13 = 1 ud us, that may become imaginary in some cases, are usually approximated by s 13 = ub and never appear in the usual approach. The conclusion is that unitarity requires (10)

6 84 P. Diþã 6 a very fine tuning between all the entries of the matrix (7) and our method could put strong constraints on the CKM matrix. For minimization of the χ we used the FindMinimum function provided by Mathematica. Special care was taken for properly treating the cases that lead to cosδ values outside the physical range. We started the fit by using the information provided by the first group of four parameters in Table since we wanted a comparison with the unitarity triangle approach that uses the same information. With no independent parameters we found a χ =χ 3 1 +χ = and we used the ij determined parameters to test all the seventeen cosδ () i. We found that nine values, for i = 1,, 4, 8, 13,, 17, are around 0. 05, their mean being < cosδ>= However we found also a few discrepancies: cosδ (5) = cosδ (6) = and cosδ (7) = 0. 0 ; even worse four values for i = 9,, 1 are outside the physical Table 1 All the central data (31) coming from [9], [6] and [7] are not compatible with unitarity, although the last fit [7] is considerably better than the published one. The left column contains the six groups of independent parameters used in our analysis. Para s cosδ PDG [9] CKMfG [6] CKMfG [7],, (1) cosδ cosδ ud ub cd, () cb ud us cd, (6) cs (3) cosδ (4) cosδ ,, (5) cosδ i cosδ i ,, cd cs td, (8) ts ud ub cs, (10) tb (7) cosδ i cosδ i ,, (9) cosδ ud ub cs, (14) td cosδ (11) cosδ (1) cosδ ,, (13) cosδ cosδ ud cd cs, (17) ts (15) cosδ ,, (16) cosδ i cosδ i

7 7 Global fit of the Cabibbo-Kobayashi-Maskawa matrix 85 region. We interpret this phenomenon as showing that the use of only one orthogonality constraint leads to non reliable results. Taking into account the cosδ () i provided by the second group of independent parameters the fit improves, all the cosδ () i being inside the physical region, and the maximum difference between cosines is of the order and δ is around 58, value compatible to that provided by the CKM fitter Group [7]. Including now in χ all the the first twelve cosδ () i we found that the results considerably improve, and the difference between cosδ () i is of the order The big surprise was that δ takes values close to 90! For comparison, with our method we obtained by using the up-to-date ij parameters provided by the CKM fitter Group [7] that cosδ() i , values that are listed in the last column of Table, with a meanvalue < cosδ>= and σ= which leads to a CP phase δ = , the interval of variation being δ , results that almost coincide with that given in [7] that are 50 δ γ 7. In our opinion these results show the limits for the prediction power of the unitarity triangle approach. In the following by using our method we found that χ takes values in the interval χ , while the same expression provides χ 1 = 95. when using the last ij values obtained by CKM fitter Group. We included step by step all the constraints implied by the six groups and obtained three different matrices; they were used to obtain another one by using the convexity property of the unistochastic matrices. Other matrices were obtained by relaxing the condition (9) by taking all combinations with five and respectively four ij that provided = new matrices. The set of these matrices was considered as 5 independent experiments on which the statistics was done. For cosδ these lead to 5 17 = 45 values that gave δ= ± The interval for the χ 1 values for all the 5 matrices was shown above and the final results are shown in the Table. In fact the obtained values can be improved, but this will be done elsewhere. Looking at the final results we see that ij values are not far from those provided by other fits. The striking feature concerns the values for the angles of the unitarity triangle, α, β and γ, angles that are not obtained from the fit, but from parameters provided by the fit. One sees that with a very good approximation the standard unitarity triangle is almost a rectangle one. We have δ>γ where their difference is about However this leads to the unexpected result, sinα sin β, relation that is not satisfied by the experimental data, which in particular are not very clean. We obtain that sinβ = , value that is not far

8 86 P. Diþã 8 from the world average of the BaBar [13] and Belle [14] experiments, that is sinβ= ± [15]. In fact both angles α and β are at the lower extremity of the typical ranges for them [15], 70 α 130, 0 β 30. We mention that a rectangle unitarity triangle was found in [16] but with α= 90. With our values we find Im ( U * ) ( ) 10 4 CKMts U CKMtd =. ±. which is compatible to that used in [17] for determination of the CP-violating ratio ε / ε. In conclusion we can say that a part of the unitarity triangle results are not reliable and that our approach could outperform by far all the other methods used to reconstruct a CKM unitary matrix from experimental data. Table Fit results and errors using the standard input from PDG data. The results show that there is a unitary matrix compatible with the data. The values for α, γ and δ strongly disagree with the previous determinations. Quantity Central value ± error ud ± us ± ub 3 ( ± 0.001) 10 cd ± cs ± cb ( ± )10 td 3 ( ± )10 ts ( ± )10 td ± sinθ 0.75 ± sinθ 3 13 ( ± )10 sinθ 3 ( ± )10 J 5 (3. 0 ± 0. 7)10 δ ± α ± 0. 0 β ± 0. 0 γ ± REFERENCES 1. P. Diþã, preprint hep-ph/ M. Battaglia, A. J. Buras, P. Gambino, and A. Stocchi (Eds.), Proceedings of the Workshop, The CKM Matrix and the Unitarity Triangle, February (00), CERN, Geneva, hep-ph/

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