ON ONE PARAMETRIZATION OF KOBAYASHI-MASKAWA MATRIX

Transcription

1 ELEMENTARY PARTICLE PHYSICS ON ONE PARAMETRIZATION OF KOBAYASHI-MASKAWA MATRIX P. DITA Horia Hulubei National Institute for Nuclear Physics and Engineering, P.O.Box MG-6, RO Bucharest-Magurele, Romania, Received May 0, 2011 An analysis of Wolfenstein approximation to the Kobayashi-Maskawa unitary matrix shows that it has a serious flaw: it depends on three independent parameters instead of four as it should be. Because this approximation is currently used in phenomenological analyses from the quark sector, the reliability of many phenomenological results is called in question. Such an example is the PDG fit, [1], page 150, for the Kobayashi-Maskawa matrix moduli. The parametrization cannot be fixed since even when it is brought to an exact form it has the same flaw and its use leads to many inconsistencies. At the beginning of 70s Kobayashi and Maskawa (KM), [2], have introduced the six quark model by making the observation that in such a model the unitary matrix representing charged weak currents has one phase parameter in addition to the real mixing angles. This extra phase introduces CP violation in a natural way as a result of weak mixing between quarks, and experiments at the mid of 70s have shown that in addition to the light quarks that make up ordinary hadrons, there is a charmed heavy quark. It was easily observed that the anomaly cancellation, true and delicate in the four-quark model, can be restored in the Weinberg-Salam model if there are two more quarks, b and t. In this way the first pillar of the future Standard Model was built. A decade latter Wolfenstein used an other form of the KM matrix, [], the so called Murnagham form, [4], that is still used today by the flavour community, see paper [1], whose form is U = c 12 c 1 c 1 s 12 s 1 e iδ c 2 s 12 c 12 s 2 s 1 e iδ c 12 c 2 s 12 s 2 s 1 e iδ s 2 c 1 s 12 s 2 c 12 c 2 s 1 e iδ c 12 s 2 s 12 c 2 s 1 e iδ c 2 c 1 (1) Even if Maiani, [5], and other theorists have shown that all possible phases from the first row and last column can be eliminated because all the quark fields are not measurable quantities the nowadays form contains a (unobservable) phase of the b quark field, which coincides with the CP -violation phase, see Eq.(1). For a better understanding of what Wolfenstein did we start with his footnote (c) Rom. RJP Journ. 56(9-10) Phys., Vol , Nos. 9-10, 2011 P , Bucharest, 2011

2 1088 P. Dita 2 no where he says: My notation is more closely related to that of L. Maiani [5]. The KM matrix mixing parameters have been written by Wolfenstein as s θ = λ, s γ = λ 2 A, and s β e iδ = λ A(ρ iη) (2) The present day notation for the above mixing angles is s 12 = s θ, s 2 = s γ and s 1 = s β () The above quoted statement is not true because s β in Maiani form of KM matrix is not multiplied by the exponential factor e iδ, see [5]. Wolfenstein gave an approximate form of KM matrix written in the form (1) by an expansion in parameter λ that was considered small enough. The first remark is that λ enters the definition of all three mixing angles, s ij, see relation (2), which implies that there is a close relationship between them, even if they are considered independent parameters in matrix (1). This relation can be obtained from paper [6], whose authors had the idea to use the exact form of KM matrix which follows by using relations (2) in the form (1). Thus from equations (2) they obtained the following relations ρ = s 1 s 12 s 2 cosδ, η = s 1 s 12 s 2 sinδ (4) see the formulae (2a)-(2b) in their paper [6] which are our starting point. The preceding two equations are equivalent to the following two ( ) 2 ρ 2 + η 2 s1 =, tanδ = η (5) s 12 s 2 ρ Our second remark is that the first relation (5) shows that ρ and η are not independent parameters because the ratio of the independent mixing angles s ij is a another independent quantity. Without much ado ρ and η are the cathetas of a rectangular triangle, and s 1 /s 12 s 2 is the hypotenuse. Pythagoras theorem says that the three leg lengths of such a triangle are not independent. That means that the matrix obtained by the substitution of formulae (2) into KM matrix form (1) leads to a matrix parametrized by three independent parameters λ, A, and either ρ, or η, instead of four, as it should be. In other words the relations (2) do not give a one-to-one transformation between the parameters s ij and δ entering (1), and λ, A, ρ, and η entering (2). By consequence, even if the resulting matrix could be unitary, the above fact implies that there is an entire class of matrices which cannot be recovered from experimental data when using the last group of parameters, i.e. those that explicitely depend on four independent parameters, and nobody estimated the systematic error implied by such a parametrization. On the other hand the ρ and η parameters are not rephasing invariant, see e.g. [7], and thus they are not acceptable from a phenomenological point of view. (c) RJP 56(9-10)

3 On one parametrization of Kobayashi-Maskawa matrix 1089 To see that the above parametrization is flawed we adhere to Jarlskog s point of view which consists in determination of quark mixing matrix in terms of measurable invariants. Two of them are the KM matrix moduli and the celebrated J invariant, see paper [8]. It is well known that KM moduli enter in leptonic and semileptonic decays through products of the form U pp f P and U pp f + (q 2 ), the first in leptonic, and the second in semileptonic decays, where f P and f + (q 2 ) denote the corresponding decay constants, and respectively, form factors. Thus the physics reality suggests the use of KM moduli in any phenomenological analysis from flavour physics sector [9], because the mixing angles are not measurable quantities. On the other hand by using KM moduli as parameters, instead of mixing angles, we lost the unitarity property of KM matrix (1). Hence an important problem that has to be solved is the consistency problem with unitarity of the KM moduli which amounts to find the necessary and sufficient conditions on the set of numbers V ij = U ij for this set to represent the moduli of an exact unitary matrix. This problem was solved in [10]. The new form of unitarity constraints says that the four independent parameters s ij and δ should take physical values, i.e. s ij (0,1), and cosδ ( 1,1), when they are computed via equations set: Vud 2 = c 2 12c 2 1, Vus 2 = s 2 12c 2 1, Vub 2 = s2 1, Vcb 2 2c 2 1, Vtb 2 1c 2 2, Vcd 2 = s 2 12c s 2 1s 2 2c s 12 s 1 s 2 c 12 c 2 cosδ, Vcs 2 = c 2 12c s 2 12s 2 1s 2 2 2s 12 s 1 s 2 c 12 c 2 cosδ, (6) Vtd 2 = s 2 1c 2 12c s 2 12s 2 2 2s 12 s 1 s 2 c 12 c 2 cosδ, Vts 2 = s 2 12s 2 1c c 2 12s s 12 s 1 s 2 c 12 c 2 cosδ relations easily obtained from (1). Now we choose as independent parameters the moduli V us = a, V ub = b, V cb = c, directly related to λ, A, ρ, η, and V cd. With the above notation we find from relations (6) V us s 12 = = 1 Vub 2 relations which are used in (5) to get a 1 b 2, s 2 = c 1 b 2, s 1 = b (7) ( ρ 2 + η 2 Vub (1 Vub 2 = ) ) 2 ( b(1 b 2 ) 2 ) = (8) V us V cb ac relation that shows again that ρ and η are not independent parameters. The above relations, (7) and (8), show that all the parameters λ, A, ρ, η depend on three independent moduli. To compare the two approaches we consider first and academic problem, the reconstruction of a unitary matrix when we know exactly all U matrix moduli, (c) RJP 56(9-10)

4 1090 P. Dita 4 and the simplest case is that of equal moduli Vij 2 = 1/, i, j = 1,2,, which are the moduli of the unitary Fourier matrix, such that all the computations can be done by hand. In this case the standard unitarity triangle, that is used in almost all phenomenological analyses, is an equilateral triangle, such that its legs length could be taken equal to unity. Then and from second Eq. (5) we get ρ = 1 2, η = 2 (9) From relations (7) we find tanδ = η ρ =, δ = π = 60 (10) s 1 = 1, s 12 = s 2 = 1 2 (11) Because all parameters entering (1) are determined, see relations (9)-(11), we can use them for the determination of all KM moduli of this academic case. Using them in Eqs. (6) we found Vud 2 = V us 2 = Vub 2 = V cb 2 = V tb 2 = 1 Vcd 2 = V ts 2 = , V cs 2 = Vtd 2 = 1 12 From the above numerical results on can see that by using Wolfenstein parametrization one cannot recover from data the simplest unitary matrix even when the parametrization is an exact one, and the above example represents the simplest test proving its inconsistency. In our approach the mixing angles take the same values, (11), and from the sixth relation (6) we find (12) cosδ = 0, δ = ± π 2 (1) The above relation shows that in the case of Fourier matrix we have an ambiguity in choosing δ, and to resolve it our choice is ImU 21 > 0, that implies ImU 22 < 0, ImU 1 < 0, and ImU 22 > 0. However we must take into account that U 1 in (1) is a complex number. By consequence we have to multiply (1) at right by the diagonal matrix d 1 = (e iδ,1,1), followed by a second diagonal matrix d 2 = (1,e iδ,e iδ ) to bring it at its rephaised form, such that, for example, U 21 has the form (c) RJP 56(9-10) U 21 = c 2 s 12 e iδ c 12 s 2 s 1 (14)

5 5 On one parametrization of Kobayashi-Maskawa matrix 1091 In our case ImU 21 is positive when δ = π 2, and we get U 21 = i 2 = e2πi/, (15) U 22 = 1 2 i 2 = e4πi/, etc recovering in this way the known form, up to equivalence, of the -dimensional Fourier matrix. The big difference between δ = π, obtained from the second Eq. (5), and the true value δ = π 2, shows that Wolfenstein parametrization is senseless, and the flavour community has to give it up. Up to now we discussed an exact academic example. In the following we show that the central values of KM moduli matrix given in paper [1], p. 150, are not compatible with unitarity constraints. For proving that we need to use another independent modulus. If this is V cd, from the sixth equation (6) one gets cosδ = (1 b2 )(V 2 cd (1 b2 ) a 2 ) + c 2 (a 2 + b 2 (a 2 + b 2 1)) 2abc 1 a 2 b 2 1 b 2 c 2 (16) and three similar formulae from the last three equations. The above relation shows that cosδ is an other invariant in the Jarlskog sense that depends on four independent moduli, and CP -violation phase can be measured via relations such as (16). If one makes use of the last four relations (6) we get only one solution for the mixing angles and cosδ. Because there are 57 such groups of four independent moduli one get 165 different formulae for cosδ. They take the same numerical value if and only if all the six relations similar to V 2 ud + V 2 us + V 2 ub = 1 (17) are exactly satisfied. If the moduli matrix generated by four independent moduli is compatible with unitarity then cosδ ( 1,1), and outside this interval when the corresponding matrix is not compatible. In the following we make use of data from numerical matrix (11.27) [1], p. 150, which is ± ± ± U = ± ± ± (18) The central values entering (18) do not come from an exact unitary matrix, which means that the six relations similar to (17) take different values, as the follow- (c) RJP 56(9-10)

6 1092 P. Dita 6 ing computation shows Vud 2 + V us 2 + Vub Vcd 2 + V cs 2 + Vcb Vtd 2 + V ts 2 + Vtb Vud 2 + V cd 2 + V td V 2 us + V 2 cs + V 2 ts Vub 2 + V cb 2 + V td results that from a phenomenological point of view could be acceptable. If we make use of all 165 different formulae for cosδ and compute them with the central values from (18) one get cosδ = i (19) σ cosδ = i (20) where i = 1 is the imaginary unit. The above result shows that the central values of matrix (18) are not compatible with unitarity. The complex values in (20) come from the matrix determined by the following four independent central values, written now as rational numbers: V ud = , V cd = , V cb = , V ts = (21) If we use relations similar to (17) to find all the entries of the corresponding KM moduli matrix, we find, e.g., Vub 2 = 1 V ud 2 V 2 ts, etc. If we compute the previous expression with the numerical values (21) we get Vub 2 = 72761/1010, i.e. V ub = i. If we compare with the corresponding value from (18) we see that both of them are of the same order of magnitude, but the second one is an imaginary quantity. Thus the V ub error entering (18) is meaningless, and the central values matrix (18) do not satisfy all unitarity constraints. As a matter of fact this is a general phenomenon. For example if we use the independent moduli V us, V ub, V cd, V cb then from relation (16) we got cos δ 0.641, showing that the corresponding moduli matrix comes from an exact unitary matrix. If we modify the previous numerical V us value by adding to it the small quantity 10 4, smaller than the uncertainty entering (18), the mixing parameters are still physical, s 12 is modified by a very small quantity, and also V ud, V td,andv ts moduli, and these modifications lead to cos δ This result shows that the errors attached to the moduli of the numerical matrix have no physical significance. Thus the central values matrix (18) is not unitary because not all cosδ take physical values, i.e. cos δ ( 1, 1), and they do not satisfy the physical conditions cosδ (i) cosδ (j), i j. (c) RJP 56(9-10) cd V cb 2 +V 2

7 7 On one parametrization of Kobayashi-Maskawa matrix 109 In conclusion the above few numerical computations obtained by using the central values from the last PDG fit in paper [1], p.150, values obtained by using Wolfenstein parametrization, show that this parametrization is wrong even when it is made exact, and must be abandoned if we want to obtain physical reliable numbers for flavour physics parameters. How such a phenomenological analysis can be done see our paper [11]. Acknowledgements. We acknowledge a partial support from ANCS contract no 15EU/ REFERENCES 1. C. Amsler et al. [Particle Data Group], Phys.Lett. B 667, 1 (2008). 2. M.Kobayashi and T.Maskawa, Progr.Theor.Phys. 49, 652 (197).. L. Wolfenstein, Phys.Rev.Lett. 51, 1945 (198). 4. F.D. Murnagham, The unitary and rotation groups, Spartan Books, Washington D.C. (1962). 5. L. Maiani in Proc. Intern.Symp. on Lepton and Photon Interactions at High Energy 1977, p. 867, 1977 (DESY, Hamburg, 1977). 6. M. Schmidtler and K.R. Schubert, Z.Phys. C 5, 47 (1992). 7. A. Höcker and L. Ligeti, Annu.Rev.Nucl.Part.Sci. 56, 501 (2006); see p.507 bottom 8. C. Jarlskog, Phys.Rev.Lett. 55, 109 (1985). 9. C. Jarlskog and R. Stora, Phys. Lett. B 208, 268 (1998). 10. P. Diţă, J.Math.Phys. 47, (2006). 11. P. Diţă, arxiv: (2010). (c) RJP 56(9-10)