The product of the gluon dressing function and the square of the ghost dressing function in the Landau gauge can be regarded to represent, apart from the inverse power corrections 1/Q 2n , a nonperturbative generalization A(Q 2 ) of the perturbative QCD running coupling a(Q 2 ) (≡ αs(Q 2 )/π). Recent large volume lattice calculations for these dressing functions indicate that the coupling defined in such a way goes to zero as A(Q 2 ) ∼ Q 2 when the squared momenta Q 2 go to zero (Q 2 1 GeV 2 ). In this work we construct such a QCD coupling A(Q 2 ) which fulfills also various other physically motivated conditions. At high momenta it becomes the underlying perturbative coupling a(Q 2 ) to a very high precision. And at intermediate low squared momenta Q 2 ∼ 1 GeV 2 it gives results consistent with the data of the semihadronic τ lepton decays as measured by OPAL and ALEPH. The coupling is constructed in a dispersive way, resulting as a byproduct in the holomorphic behavior of A(Q 2 ) in the complex Q 2 -plane which reflects the holomorphic behavior of the spacelike QCD observables. Application of the Borel sum rules to τ -decay V + A spectral functions allows us to obtain values for the gluon (dimension-4) condensate and the dimension-6 condensate, which reproduce the measured OPAL and ALEPH data to a significantly better precision than the perturbative MS coupling approach.3 It is possible to show that pQCD renormalization schemes exist in which pQCD coupling a(Q 2 ) is holomorphic for Q 2 ∈ C\(−∞, −M 2 thr ] and at the same time reproduces the high-energy QCD phenomenology as well as the semihadronic τ -lepton decay physics [23][24][25]. 4 MiniMOM scheme is known at present to four loops [18][19][20]. 5 In this scheme, however, we rescale Q 2 from the Λ MM to the usual Λ MS convention. 6 In Ref. [29], the matching of A(Q 2 ) and dA(Q 2 )/d ln Q 2 at an IR/UV transition scale Q 2 0 ∼ 1 GeV 2 is imposed, fixing the values of A(0) > 0 and Q 2 0 . On the other hand, our coupling A(Q 2 ) will be holomorphic, no explicit IR/UV matching scale will exist. Instead of the matching, we will impose various physically motivated conditions which will affect simultaneously the behavior of A(Q 2 ) in the UV and IR regimes.10 In principle, we could construct A in any other scheme, e.g., in MS scheme, but then it would not be clear how such a coupling compares with A latt of Ref.[32] in the deep IR regime. For an application and discussion of the MiniMOM scheme in pQCD, see Ref.[56].

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In a scenario with two almost mass degenerate heavy sterile Majorana neutrinos with a mass ∼ 1 GeV, we present the semileptonic decay widths of heavy charged pseudoscalars such as B mesons, either lepton-number-violating (B ± → µ ± e ± π ∓ ), or lepton-number-conserving (B ± → µ ± e ∓ π ± ), mediated by such on-shell neutrinos. It had been explained in the literature that such decays may be eventually detected, and that they can present even detectable CP violation effects. In this work we point out that, in addition, such decays may present detectable effects of heavy neutrino oscillation, allowing us to extract the oscillation length and thus the heavy neutrino mass difference ∆MN , as well as a CP-violating Majorana phase.