Suggest a numerical example if possible. How can I obtain the branching ratios for the tau decay from the CKM ignoring phase space and form factor and please suggest a numerical example if possible?
Hadronic decays of tau lepton (at rest ) cannot involve third generation quarks in the final decay product (mtau=1.777,mb=4.25,mt=175 in GeV units approx). Therefore, only first two rows and columns of the CKM matrix is required in this study. Next, because CKM matrix elements are different for different generations (the matrix is written in flavor basis), they are perhaps not expected to appear in expressions when only non-strange decay modes are studied.
The main hadronic decays are to πν (12%), ρν (25%) and a1ν (18%). Adding up the possibilities, the τ decays to 1 charged track about 85% of the time, and to 3 charged tracks about 15% of the time.
See: http://www.hep.man.ac.uk/u/roger/PHYS40521/lecture5.pdf
τ decays to a K in place of a π are restricted by Cabibbo suppression (that is a small off-diagonal element of the CKM matrix will enter into the calculation of this tree level diagram) . Thus the τ → K ν branching ratio is only 0.7%.
For further reading,
1. Determining the strange quark mass in Cabibbo suppressed tau lepton decays; K.G. Chetyrkin, Johann H. Kuhn (Karlsruhe U., TTP), A.A. Pivovarov (Mainz U., Inst. Phys.). Published in Nucl.Phys. B533 (1998) 473-493; e-Print: hep-ph/9805335
2. Strange quark mass determination from Cabibbo suppressed tau decays
Antonio Pich (Valencia U.), Joaquim Prades (Granada U., Theor. Phys. Astrophys.). Published in JHEP 9910 (1999) 004; e-Print: hep-ph/9909244
Of course third generation quarks (as does the charm quark, of the second generation, too) can and will contribute through loop corrections and that's where the CKM matrix elements, also, enter. Just draw the Feynman diagrams. The CKM matrix expresses the fact that the mass eigenstates of the quarks, defined by the strong interaction, are not identical to the flavor eigenstates, defined by the electroweak interaction, through which the tau lepton decays.
The quarks, that appear in the electroweak interaction vertices, include the appropriate CKM factors.
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