In context with quark models we frequently come across CKM(Cabibbo–Kobayashi–Maskawa matrix) matrix. I need a clear idea on how CKM matrix helps.
he Cabibbo–Kobayashi–Maskawa matrix (CKM matrix, quark mixing matrix, sometimes also called KM matrix) is a unitary matrix which contains information on the strength of flavour-changing weak decays. Technically, it specifies the mismatch of quantum states of quarks when they propagate freely and when they take part in the weak interactions. It is important in the understanding of CP violation.
it is necessary to count the number of parameters in this matrix, V which appear in experiments, and therefore are physically important. If there are N generations of quarks (2N flavours) then
An N × N unitary matrix (that is, a matrix V such that VV† = I, where V† is the conjugate transpose of V and I is the identity matrix) requires N2 real parameters to be specified.
2N − 1 of these parameters are not physically significant, because one phase can be absorbed into each quark field (both of the mass eigenstates, and of the weak eigenstates), but an overall common phase is unobservable. Hence, the total number of free variables independent of the choice of the phases of basis vectors is N2 − (2N − 1) = (N − 1)2.Of these, N(N − 1)/2 are rotation angles called quark mixing angles.
The remaining (N − 1)(N − 2)/2 are complex phases, which cause CP violation.
For the case N = 2, there is only one parameter which is a mixing angle between two generations of quarks. Historically, this was the first version of CKM matrix when only two generations were known. It is called the Cabibbo angle after its inventor Nicola Cabibbo.
For the Standard Model case (N = 3), there are three mixing angles and one CP-violating complex phase.
Cabibbo's idea originated from a need to explain two observed phenomena:
the transitions u ↔ d, e ↔ νe, and μ ↔ νμ had similar amplitudes.
the transitions with change in strangeness ΔS = 1 had amplitudes equal to 1/4 of those with ΔS = 0.
Cabibbo's solution consisted of postulating weak universality to resolve the first issue, along with a mixing angle θc, now called the Cabibbo angle, between the d and squarks to resolve the second.
Thank you so much for the answer. It would be really nice if you could attach some articles on CKM matrix.
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