Probing CKM parameters through unitarity, εK and ∆mB d,s

A recently carried out unitarity based analysis [1, 2] involving the evaluation of Jarlskog’s rephasing invariant parameter J as well as the evaluation of the angles α, β and γ of the unitarity triangle is extended to include the effects of K − K̄ as well as B − B̄ mixings. The present analysis attempts to evaluate CP violating phase δ, |Vtd| and the angles α, β and γ by invoking the constraints imposed by unitarity, K − K̄ and B − B̄ mixings independently as well as collectively. By invoking the “full scanning” of input parameters, as used by Buras et al., our analysis yields the following results respectively for δ and |Vtd|, unitarity: 50 ± 20 (in I quadrant), 130 ± 20 (in II quadrant) and 5.1 × 10−3 ≤ |Vtd| ≤ 13.8 × 10−3, unitarity and εk: 33 o ≤ δ ≤ 70 (in I quadrant), 110 ≤ δ ≤ 150 (in II quadrant), 5.8 × 10−3 ≤ |Vtd| ≤ 13.6 × 10−3, unitarity and ∆md: 30 ≤ δ ≤ 70, 6.4×10−3 ≤ |Vtd| ≤ 8.9×10−3. Incorporating all the constraints together, we get 37 ≤ δ ≤ 70, 6.5× 10−3 ≤ |Vtd| ≤ 8.9× 10−3 , the corresponding ranges for α, β and γ are, 80 ≤ α ≤ 124, 15.1 ≤ β ≤ 31 and 37 ≤ γ ≤ 70, again in agreement with the recent analysis of Buras et al. as well as with other recent analyses. Our analysis yields 0.50 ≤ sin2β ≤ 0.88, in agreement with the the recently updated BABAR results. This range of sin2β is in agreement with the earlier result of BELLE, however has marginal overlap only with the lower limit of recent BELLE results. Interestingly, our results indicate that the lower limit of δ is mostly determined by unitarity, while the upper limit is governed by constraints from ∆md.