Abstract
A preliminary determination of the Dirac phase in the PMNS matrix is $\dell\approx -\frac{\pi}{2}$. A rather accurately determined Jarlskog invariant $J$ in the CKM matrix is close to the maximum. Since the phases in the CKM and PMNS matrices will be accurately determined in the future, it is an interesting problem to relate these two phases. This can be achieved in a families-unified grand unification if the weak CP violation is introduced spontaneously {\it \`a la} Froggatt and Nielsen at a high energy scale, where only one meaningful Dirac CP phase appears.
Highlights
IntroductionMaskawa (CKM) matrix are rather accurately determined [1], which makes it possible to pin down the invariant phase δCKM into three possibilities α, β, and γ of the unitarity triangle [2]
At present, the real angles of the Cabibbo–Kobayashi–Maskawa (CKM) matrix are rather accurately determined [1], which makes it possible to pin down the invariant phase δCKM into three possibilities α, β, and γ of the unitarity triangle [2]
Even though the physical argument presented in the previous section is enough for relating δPMNS and δCKM, here we present a scheme in detail of how they are connected
Summary
Maskawa (CKM) matrix are rather accurately determined [1], which makes it possible to pin down the invariant phase δCKM into three possibilities α, β, and γ of the unitarity triangle [2]. Even that requirement is used only when we argue for the possibility of δCKM ±δPMNS based on the assumption that the Dirac phases in the CKM and PMNS matrices originates from the spontaneous CP violation mechanism [38] at a high energy scale à la Froggatt and Nielsen [20]. To have a relation without any other parameters, such as in the relation δPMNS = ±δCKM, only one phase must be introduced in the whole theory such as in the unification of GUT families.
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