Determination of f+π(0) or extraction of |Vcd| from semileptonic D decays

By globally analyzing all existing measured branching fractions for D→πe+νeD→πe+νe decays, partial decay rates in different four-momentum transfer-squared q2q2 bins, as well as products of the decay form factor f+π(q2) and the Cabibbo–Kobayashi–Maskawa (CKM) quark-mixing matrix element |Vcd||Vcd|, we obtain f+π(0)|Vcd|=0.1428±0.0019−0.0011+0.0019. This product, in conjunction with the |Vus||Vus| determined from (semi-)leptonic K   decays and the relation of |Vcd|=|Vus|=λ|Vcd|=|Vus|=λ from the unitarity of the CKM matrix, implies a value for the D→πD→π semileptonic form factor f+π(0)=0.634−0.010+0.012±0.002, which is consistent within error with those calculated in theory based on quantum chromodynamics (QCD). Alternately, using this product together with the most accurate form factor calculated in Lattice QCD (LQCD), we find |Vcd|D→πe+νe=0.2144−0.0033+0.0040±0.0093. Combining this |Vcd|D→πe+νe|Vcd|D→πe+νe with |Vcd|D+→μ+νμ=0.2160±0.0049±0.0014|Vcd|D+→μ+νμ=0.2160±0.0049±0.0014 extracted from both the BESIII and CLEO-c measurements of D+→μ+νμD+→μ+νμ decays, we find the most precisely extracted |Vcd||Vcd| to be |Vcd|=0.2157±0.0045|Vcd|=0.2157±0.0045 up to date. From these determined quantities we find [mD+f+π(0)/fD+]exp=5.81±0.17, which is in excellent agreement with [mD+f+π(0)/fD+]LQCD=5.85±0.26 calculated in LQCD, indicating that the LQCD approach to the charm quark sector is excellent. Using this |Vcd||Vcd| together with the PDG'2014 |Vud||Vud| and |Vtd||Vtd|, we check for the first column unitarity and find |Vud|2+|Vcd|2+|Vtd|2−1=−0.004±0.002|Vud|2+|Vcd|2+|Vtd|2−1=−0.004±0.002, which deviates from unitarity by 2σ.