On Cabibbo angle from theory

We find an expression for the Cabibbo angle from quark flavour generators of the first two generations. The flavour generators operate on the toroidal components in an intrinsic dynamics for colour degrees of freedom. The generators have led to parton distributions for u and d valence quarks of the proton that compare well with those derived from experiment. The present result for the cosine of the Cabibbo angle compares rather well with the experimentally established value for the up-down quark mixing element 0.97420+/−0.00021 of the Cabibbo-Kobayashi-Maskawa matrix.

Introduction. -In 1963 Nicola Cabibbo introduced a connection between strangeness-conserving and strangeness-changing processes expressed in cosine and sine factors of a certain angle [1]. Cabibbo remarked as a consequence that the vector coupling constant in beta decay is not the full Fermi constant G Fμ but contains a factor cos θ C where θ C has come to be known as the Cabibbo angle. This was a first sign of slight differences in the description of purely leptonic decay and leptonic decays of hadrons. Cabibbo's description was taken up by Glashow, Iliopoulos and Maiani in 1970 with the introduction of the charm quark [2]. A few years later Kobayashi and Maskawa introduced a three-generation quark mixing [3] whose elements have come to be understood as mutual quark coupling strengths. For instance the original cosine factor is now understood to express a coupling between a u and a d quark flavour. We shall introduce in the present work, quark flavour generators from which it is possible to derive an expression for the Cabibbo angle.
In the Standard Model of elementary particles [4], the strong interactions are described by quantum chromodynamics where quarks in strong interaction mass eigenstates can change colour by emission or absorption of gluon interaction quanta. The weak interactions are described by the electroweak theory (quantum flavourdynamics), where quarks in weak interaction eigenstates can change flavour by emission or absorption of intermediate vector bosons, W ± . Figure 1 shows the situation in a schematic form. Note that the colour transformations do not change the flavours. Note also that the flavours come in three generations. The generations can be mixed by flavour transformations but not by colour transformations 1 .
The mixing between generations is a consequence of the mixing between flavour mass eigenstates and flavour interaction states. The mass eigenstates u, c, t, d, s, b constitute the base for the strong interactions, the (weak) interaction eigenstates u , c , t , d , s , b constitute the base for the weak interactions. The connection between the two base sets is usually described by the Cabibbo-Kobayashi-Maskawa matrix such that the u-type flavours are chosen to be the same for both interactions, whereas the d-type flavours are trans- . The Cabibbo-Kobayashi-Maskawa matrix VCKM transforms mass eigenstates (unprimed) to weak interaction states (primed) and mix all three generations. The Cabibbo mixing angle θC describes the mixing between the first two generations. It was originally introduced by Cabibbo [1] before charm, beauty and truth (topness) were discovered. We introduce quark flavour generators Tu, T d , Ts (5) that act on colour components (21) and find | sin θC| = |TrT † u Ts| (24). The result 2 9 = 0.222 · · · is quite close to the experimentally extracted value [4] for the Wolfenstein parameter λ = 0.22453 ± 0.00044 from CKMfitter and λ = 0.22465 ± 0.00039 from UTfit.
For an updated Standard Model presentation of the common origin of quark masses and mixing from the Yukawa couplings to the Higgs field, see Ceccucci, Ligeti and Sakai in [4].
The Standard Model has no clue on how to fix the elements of the mixing matrix V CKM except that the unitarity requirement-corresponding to assuming transformations to stay within exactly three generationsreduces the number of independent parameters to three angles, θ 12 , θ 23 , θ 13 where c 12 = cos θ 12 , s 12 = sin θ 12 and so on. The two angles θ 23 and θ 13 are considerably smaller 2 than θ 12 and setting them equal to zero corresponds to having the Cabibbo 2 From the Wolfenstein parameters λ = 0.22453 ± 0.00044, A = angle θ C = θ 12 whereby the third generation is decoupled and we have the Cabibbo mixing matrix for the first two generations [5], So far the parameters in V CKM -and thus in the approximate V C -have had to be settled by experiment. We shall use the same flavour generators here and shall include also a strangeness generator, T s to have all the three flavours below the charm threshold, Flavour in colour. -The physical origin of the connection between flavour and colour that we use in the present work and used in [6] to generate the parton distribution functions in fig. 2 is deeply rooted in an underlying model for baryon structure.
We assume baryons to be stationary states of a Hamiltonian on the configuration space U (3), where the length scale a is determined from the classical electron radius r e = e 2 /(4π 0 m e c 2 ) [4] as πa = r e [6] and the configuration variable u ∈ U (3) contains nine dynamical angular variables θ j , α j , β j u = e iχ = e i(θj Tj +αj Sj /h+βj Mj /h) , j = 1, 2, 3.
0.836 ± 0.015,ρ = 0.122 +0.018 −0.017 ,η = 0.355 +0.012 −0.011 in [4] we find a clear hierarchical structure θ 12 = 12.98 • , θ 23 = 2.42 • , θ 13 = 0.21 • , δ = 71.03 • with only a minute angular admixture (but large phase) of the third generation into the first. 3 The "flavour in colour" relation is not an algebraic relation between quantum numbers. Instead it implies flavour degrees of freedom to be considered as specifically generated vector fields on an intrinsic three-dimensional torus.  [6] to compare with experimental determinations [4,7] (insert). The theoretical distributions are from an approximate state and are generated by Tu and T d (5), respectively acting on toroidal degrees of freedom in the intrinsic protonic state. The generators Ts (5) and Tc (31) give zero strangeness and charm content in the protonic state at what would correspond to q 2 → 0 GeV 2 . For finite q 2 , sea quarks (and antiquarks) from other generations will evolve via the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi convolutions because of gluon emission leading to pair creation into different generations already in next-to-leading-order processes, see, e.g., pp. 407 in [8]. Bottom: proton spin structure function (solid) [9] based on flavour generators Tu and T d (5) and overlaid on data from several experimental groups [4,7].
The generators iT j = ∂ ∂θj are diagonal and we identify them as three colour generators. In topological language, they generate the maximal torus -U (3) is a compact space. The generators S j commute as intrinsic angular momentum operators. The mixing operators M j -in combination with the other two sets-generate flavour degrees of freedom and commute like Laplace-Runge-Lenz operators, pp. 236 in [10]. The total set of nine degrees of freedom are excitable kinematically from momentum, angular momentum and Laplace-Runge-Lenz operators in laboratory space as seen in a coordinate representation with projection x j = aθ j . For instance S 3 = aθ 1 p 2 − aθ 2 p 1 and The potential 1 2 Trχ 2 is periodic and only depends on the eigenangles θ j (see footnote 4 ). With a suitable parametrization of the Laplacian [12] the off-diagonal degrees of freedom can therefore be integrated out to get where R is antisymmetric in the three angles. The total potential W (θ 1 , θ 2 , θ 3 ) [6] contains the centrifugal term from the Laplacian (8) and depends on hypercharge and isospin through the value of S 2 k + M 2 k . From the measure-scaled toroidal wave function R, we can generate colour fields c j by use of the exterior derivative expanded on the torus forms dθ j that are conjugate to the left-invariant coordinate fields ∂ j = uiT j generated by T j , i.e., dθ i (∂ j ) = δ ij . Thus, we have dR = c j dθ j (10) and can read off colour components along tracks on the torus, generated by any combination T = k 1 T 1 + k 2 T 2 + k 3 T 3 of generators. In particular up and down flavour tracks leading to the parton distribution functions in fig. 2 for the proton correspond to apply- (5) on an approximate expression for R. The fractional coefficients -which correspond to the quark electrical charges-enter because U (3) and SU (3) only share off-diagonal generators whereas the diagonal generators are different. Generators X i j of U (3) may be defined from creation and annihilation operators a † i and a j , i, j = 1, 2, 3 for the 3-dimensional harmonic oscillator, see pp. 71 and 221 in [13], Here E ij is a 3 × 3 matrix representation used on (21) with the ij th element equal to 1 and all other elements 0. This set is equivalent to our set of T j , S j /h, M j /h, e.g., 4 This follows from the fact that the eigenvalues of u are not changed by conjugation and that the trace is invariant under conjugation: Any u = e iχ with eigenvalues e iθ 1 , e iθ 2 , e iθ 3 can be diagonalized via conjugation with a particularly chosen v ∈ U (3) From the cyclic property of the trace, TrA −1 BA = TrB, it follows that Trχ 2 = Tr ξ 2 = P 3 j=1 θ 2 j , with −π ≤ θ j ≤ π for the shortest geodesic. See also [11]. 5 Measure-scaling "Jacobian" J = Q 3 i<j 2 sin 1 2 (θ i − θ j ).
where λ 1 , λ 2 are two of the six off-diagonal Gell-Mann matrices (see p. 209 in [10]). Generators for SU (3) may be defined as It is the factor 1/3 that makes the fractional charges appear in the generators (5) if one assumes the common Gell-Mann-Nakano-Nishijima formula and for U (3) define charge and hypercharge operators as Thus, eq. (5) mixes quark charges e q = 2 3 , − 1 3 with baryonic hypercharges y = 1, 0.
To sum up the interpretation: The colour degrees of freedom are contained in the torus of the U (3) configuration space and the flavour structure follows from the Laplacian on this configuration space. The interconnection between the two originates from the assumed Hamiltonian in (6) (see footnote 6 ). The model thus contains both the exact colour su c (3) symmetry and the broken flavour su f (3); the first reflecting the U (3) Lie group configuration space, the second reflecting the u(3) algebra of the generators from which the Laplacian (8) is constructed [12]. It can be shown [9,16] that the exterior derivative of the wave function generates colour quark fields, the c j 's in (11), and gluon fields that transform under su(3) as the fundamental and adjoint representations, respectively. Thus, left invariance of the coordinate fields used for the intrinsic configuration space leads to local gauge invariance for the fields generated in the space-time frame of the laboratory space.
Cabibbo angle from strange decay. -Let us consider a strangeness-changing, non-leptonic decay At tree level, this decay is described by the Feynman diagram in fig. 3. We follow the notation conventions of [5].
In the standard description for a two-generation model one inserts a factor cos θ C in the upper vertex and sin θ C in the lower vertex to have the decay amplitude 6 Historically it grew out of work on lattice gauge theory with Manton's action [14] in a Kogut-Susskind Hamiltonian [15], which is here completely reinterpreted [6]. Leaving the scene, one observes a baryon, the proton p and a meson, the negative pion, π − . We discuss the factor sin θC in the lower vertex (17).
Here the integration over the W propagator has been carried out under the condition that the fourmomentum exchange q = p 1 − p 3 fulfils q 2 (M W c) 2 , where M W is the W mass. Further, g W is the electroweak coupling constant g 2 W = e 2 / sin 2 θ W = 4πα/ sin 2 θ W with θ W the electroweak mixing angle and u and v are particle and anti-particle spinors, respectively. Unitarity of the Cabibbo matrix (4) is automatically fulfilled by the parametrization via a single angle θ C , We want to derive sin θ C . The lower vertex in fig. 3 involves quarks of strange and up flavours, respectively. We can generate these flavours according to (11) from acting on colour components in the following way: Here the colour states are [5] 7 Thus, we include colour in the incoming and outgoing quark states and sum over colour with c j (1) = T s c j and c j (3) = T u c j . The Dirac matrices γ μ and γ 5 operate on the spinors independently of the operation of the flavour generators on the colour components. We can therefore factorize the colour algebra from the spinor algebra to get Note that there are no gluon propagators here, so this is not a strong interaction effect as such. Comparing the last expression to the standard expression in (17) we find the Cabibbo angle determined by This yields cos θ C = 1 9 √ 77 = 0.974996 · · · ≈ |V ud | = 0.97420 ± 0.00021 [4].
The comparison seems quite promising. An exact agreement should not be expected considering that V ud = cos θ 12 cos θ 13 contains a mixing into the third generation. This mixing, however, is rather small and therefore cos θ 13 ≈ 1 and |V ud | ≈ cos θ C . Let us now consider the (leptonic) neutron decay depicted in fig. 4. At the lower vertex one would traditionally include the Cabibbo factor cos θ C , whereas our model would read This substitutes cos θ C by Tr T † u T d which yields 7

9
-somewhat off the value of V ud . A radical solution to this discrepancy is to absorb Tr T † u T d in a redefined coupling strength with g 2 0 = e 2 /(sin θ W cos θ W ) 2 as an a priori coupling strength for weak interactions and then to keep only cos θ C in the spinor brackets such that V C remains unitary by virtue of (19). From (17) we would then conclude which -as it should-compares rather well with [4]  Charm and GIM-mechanism. -Let us try to enlarge the model to charm quarks. We take the following charm flavour generator to supplement the set in (5) Just like T s , neither will T c generate any partons in the protonic state we referred to in fig. 2. We consider the d → c vertex analogous to the s → u vertex in fig. 3, (32) Using the same idea as in (23) we would conclude a Cabibbo factor It is the same value of the trace as in (23) but this time should compare to − sin θ C which is opposite in sign to the relation sin θ C = −2/9 in (24). Can this immediate inconsistency be remedied? Yes, we can exploit the freedom of choice of an overall phase factor in V CKM . Instead of the standard V CKM we may use As (35) With U CKM in stead of V CKM we get the following vertex Cabibbo factors for the lower vertex in fig. 3 to compare with a similar generation-changing charm decay: In the second decay we used the vertex factor rule in the right part of fig. 5. Thus, the immediate problem of opposite signs on sin θ C in (23) and (33) is eliminated, albeit at the price of a common non-zero phase, i = e iπ/2 -a phase which we would like to understand better. The reader may remember in the back of his or her mind that the opposite signs on the sin θ C in the Cabibbo matrix (4) are the basis of the standard exposition of the Glashow-Iliopoulos-Maiani mechanism [2] that supported the charm quark hypothesis before charm degrees of freedom were observed, namely by explaining the suppression of the leptonic kaon decay Will the GIM-mechanism survive our phase change in the CKM-matrix? Figure 6 shows the Feynman diagrams to explain the mechanism. We will check that they still cancel using U CKM from (34) instead of V CKM in the standard parametrization (3) which is approximated by the Cabibbo matrix (4). Using fig. 5, we list the Cabibbo factors in the four W -quark-vertex processes needed We see that the opposite signs in the GIM-mechanism have been shifted to the cosines. In other words the GIM-mechanism still works: the product of the two upper Cabibbo factors in the list (38) refers to the left diagram in fig. 6 and cancels the product of the two lower Cabibbo factors referring to the right part of fig. 6. This is as it should be because U CKM is related to V CKM by a simple phase factor.
Discussion. -To see clearly the hierarchy, we expose the underlying factorization of V CKM in three pairwise rotations among the three generations [4], (39) For θ 23 = θ 13 = 0 this expression reduces to the Cabibbo matrix (4) which has been our concern here. Further development of the present model would require a suggestion on the origin of the mixing into the third generation, i.e., Fig. 6: GIM-mechanism in leptonic kaon decay, V * cs V cd = −V * us V ud for any phase convention in the Cabibbo-Kobayashi-Maskawa matrix VCKM (1) at θ13 = 0 and θ23 = 0. The two diagrams therefore cancel (except for effects due to mc = mu). As a consequence, the decay K 0 → μ + + μ − is heavily suppressed. θ 23 = 0 and θ 13 = 0. Attempts in the literature tend to combine the quark and lepton sectors [17][18][19][20][21] in search for a common origin of the hierarchical mass and mixing structure observed in both sectors. de Medeiros Varzielas, Rasmussen and Talbert [21] suggest that corrections to fermionic mixings should be sought via other mechanisms than finite group strategies. We note that Kobayashi and Maskawa [3] actually consider a coupling via the Higgs field to strong interactions to explain CP -violation before they suggest their 6-plet model which has now developed into the standard description used with bottom and top quarks in the third generation. Our strategy is to relate electroweak and strong interaction structure where quark degrees of freedom are at play. We see quarks as degrees of freedom excited in hadrons both by electroweak and by strong interactions -but askew with respect to each other. In this sense the Cabibbo angle expresses a mixing between interactions.
Conclusion. -We have relaxed on the independent treatment of the electroweak and strong interactions. We used a relation between flavour generators and colour components to find a possible physical origin of the Cabibbo angle. The relation is supported by previously derived parton distribution functions from the same set of flavour generators that are here related to the Cabibbo angle. The model needs improvement to include mixing into the third generation. * * * I appreciated the anonymous referee queries into the physical background of flavour in colour. I thank Mogens Stibius Jensen and Henrik Georg Bohr for helpful discussions. I thank the Technical University of Denmark for an inspiring working environment.