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].
The supergraph technique for calculations in supersymmetric gauge theories where supersymmetry is broken in a "soft" way (without introducing quadratic divergencies) is reviewed. By introducing an external spurion field the set of Feynman rules is formulated and explicit connections between the UV counterterms of a softly broken and rigid SUSY theories are found. It is shown that the renormalization constants of softly broken SUSY gauge theory also become external superfields depending on the spurion field. Their explicit form repeats that of the constants of a rigid theory with the redefinition of the couplings. The method allows us to reproduce all known results on the renormalization of soft couplings and masses in a softly broken theory. As an example the renormalization group functions for soft couplings and masses in the Minimal Supersymmetric Standard Model up to the three-loop level are calculated.
We are interested in the structure of the Lcc vertex in the Yang-Mills theory, where c is the ghost field and L the corresponding BRST auxiliary field. This vertex can give us information on other vertices, and the possible conformal structure of the theory should be reflected in the structure of this vertex. There are five two-loop contributions to the Lcc vertex in the Yang-Mills theory. We present here calculation of the first of the five contributions. The calculation has been performed in the position space. One main feature of the result is that it does not depend on any scale, ultraviolet or infrared. The result is expressed in terms of logarithms and Davydychev integral J(1, 1, 1) that are functions of the ratios of the intervals between points of effective fields in the position space. To perform the calculation we apply Gegenbauer polynomial technique and uniqueness method.
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