We extend the auxiliary-mass-flow (AMF) method originally developed for Feynman loop integration to calculate integrals which also involve phase-space integration. The flow of the auxiliary mass from the boundary ( ) to the physical point ( ) is obtained by numerically solving differential equations with respective to the auxiliary mass. For problems with two or more kinematical invariants, the AMF method can be combined with the traditional differential-equation method, providing systematic boundary conditions and a highly nontrivial self-consistency check. The method is described in detail using a pedagogical example of at NNLO. We show that the AMF method can systematically and efficiently calculate integrals to high precision.

The plasmodial slime molds is the largest group in the phylum Amoebozoa. Its life cycle includes the plasmodial trophic stage and the spore-bearing fruiting bodies. However, only a few species have their complete life cycle known in details so far. This study is the first reporting the morphogenesis of Didymium laxifilum and Physarum album. Spores, from field-collected sporangia, were incubated into hanging drop cultures for viewing germination and axenic oat agar plates for viewing plasmodial development and sporulation. The spores of D. laxifilum and P. album germinated by method of V-shape split and minute pore, respectively. The amoeboflagellates, released from spores, were observed in water film. The phaneroplasmodia of two species developed into a number of sporangia by subhypothallic type on oat agar culture. The main interspecific difference of morphogenesis was also discussed.

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We compute the matching coefficient between the quantum chromodynamics (QCD) and the non-relativistic QCD (NRQCD) for the flavor-changing scalar current involving the heavy charm and bottom quark, up to the three-loop order within the NRQCD factorization. For the first time, we obtain the analytical expressions for the three-loop renormalization constant $$\tilde{Z}_s(x,R_f)$$ Z ~ s ( x , R f ) and the corresponding anomalous dimension $$\tilde{\gamma }_s(x,R_f)$$ γ ~ s ( x , R f ) for the NRQCD scalar current with the two heavy bottom and charm quark. We present the precise numerical results for those relevant coefficients $$(C_{FF}(x_0), \ldots , C_{FBB}(x_0))$$ ( C FF ( x 0 ) , … , C FBB ( x 0 ) ) with an accuracy of about thirty digits. The three-loop QCD correction turns out to be significantly large. The obtained matching coefficient $$C_s(\mu _f,\mu ,m_b,m_c)$$ C s ( μ f , μ , m b , m c ) is helpful to analyze the threshold behaviours when two different heavy quarks are close to each other and form the double heavy $$B_c$$ B c meson family.