Astroparticle Physics

2.1 Examples of Interaction Processes 31It appears as if the s quark can sometimes behave likethe d quark. It is, in fact, the d ′ and s ′ quarks that couple toweak interactions, rather than the d and s quarks. The d ′ ands ′ quarks can be described as a rotation with respect to the dand s quarks. This rotation is expressed byd ′ = d cos θ C + s sin θ C ,s ′ =−d sin θ C + s cos θ C ,(2.7)where θ C is the mixing angle (Cabibbo angle).The reason that angles are used for weighting is based onthe fact that the sum of the squares of the weighting factors,cos 2 θ + sin 2 θ = 1, automatically guarantees the correctnormalization. θ C has been experimentally obtained to beapproximately 13 degrees (sin θ C ≈ 0.2235). Since cos θ C ≈0.9747, the d ′ quark predominantly behaves like the d quark,albeit with a small admixture of the s quark.The quark mixing originally introduced by Cabibbo wasextended by Kobayashi and Maskawa to all three quark families,such that d ′ , s ′ ,andb ′ are obtained from d, s, b by a rotationmatrix. This matrix is called the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix),⎛ ⎞ ⎛⎝ d′s ′ ⎠ = U ⎝ d ⎞s ⎠ . (2.8)b ′ bThe elements on the main diagonal of the (3 × 3) matrixU are very close to unity. The off-diagonal elements indicatethe strength of the quark-flavour violation. A similarcomplication in the neutrino sector will be discussed later,where the eigenstates of the mass are not identical with theeigenstates of the weak interaction (see Sect. 6.2.1).The Standard Model of electroweak and strong interactionscannot be the final theory. The model contains toomany free parameters, which have to be adjusted by hand. Inaddition, the masses of all fundamental fermions are initiallyzero. They only get their masses by a mechanism of spontaneoussymmetry breaking (Higgs mechanism). Another veryimportant point to note is that gravitation is not consideredin this model at all, whereas it is the dominant force in theuniverse as a whole. There have been many attempts to formulatea Theory of Everything (TOE) that unites all interactions.A very promising candidate for such a global descriptionis the string theory. String theory is based on theassumption that elementary particles are not point-like, butCabibbo angleCKM matrixlimits of the Standard ModelHiggs mechanismTheory of Everythingstring theory