Niels Tuning (1) Exercise – Unitarity triangles 6) In the course you saw the unitarity triangle constructed from the orthogonality condition on the 1 st and 3 rd column –The angles , and the sides R u and R t all represent measurable observables from B 0 meson decays that are related to each other through the unitarity constraints. If all measurement obey these constraints – that is all measurements are consistent with CP violation exclusively originating from the CKM mechanism – then all components fit the geometric representation of a triangle (0,0)(1,0) (,)(,)   RuRu RtRt 

Niels Tuning (2) Exercise – Unitarity triangles (continued) 6)The unitarity triangle is not unique to the B 0 meson, there are in fact 3 of them. Construct the unitarity triangle for the K 0 system, i.e. the triangle that follows from the orthogonality condition on the 1 st and 2 nd column a)Write out the orthogonality condition of the 1 st and 2 nd column (ignore O( λ 7 ) terms) b)Represent the condition as a triangle in the complex plane c)Rescale and rotate the triangle such that the basis of the triangle is formed by the points (0,0) and (1,0) d)What are the coordinates of the apex of the triangle in terms of the Wolfenstein parameters - The nice feature of the ‘B 0 meson triangle’ is that all angles and sides are of the same magnitude, which makes it easy to check their compatibility e)Using the values of A,, and  (A=0.82, =0.22, =0.25 and =0.35), make a drawing of the K 0 triangle that accurately reflects the shape of the triangle (i.e. the height-to-width) ratio f)Calculate the surface area of the K 0 triangle in terms of the Wolfenstein parameters and compare it to the surface area of the B 0 triangle. (The surface is equal to 2*J, with J the Jarlskog invariant) Wolfenstein parametrization to O (λ 5 ):

Niels Tuning (3) Exercise – Adding amplitudes 7) To have observable CP violation in a process resulting from two interfering amplitudes one must have a phase difference between the amplitudes that changes under CP conjugation. The actual requirement is slightly more specific. The goal of this exercise is to formulate the more exact requirement. Given a decay process that can proceed through two amplitudes: amplitude A, with |A| = 1 and amplitude B with |B|=1 and phase difference f W =90 0 between A and B that is entirely due to phase factors in CKM matrix elements. –Draw the vector-addition diagram for the total amplitude A+B and calculate the magnitude |A+B| –Now draw the diagram for the CP-conjugate process. (What happens to f w under CP conjugation?) –Calculate the magnitude of |A+B|. Is it different from |A+B|? Is there observable CP violation in this process? –Redo the exercise with the following modification: the phase difference between A and B is now f=f s +f w, where f w =90 o is the phase difference due to CKM factors and f s =45 o, which is due to other other physics in amplitudes A and B that is invariant under CP conjugation.

Niels Tuning (4)