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Conformal window and Landau singularities

A physical characterization of Landau singularities is emphasized, which should trace the lower boundary N_f^* of the conformal window in QCD and supersymmetric QCD. A natural way to disentangle ``perturbative&#39;&#39; from ``non-perturbative&#39;&#39; contributions below N_f^* is suggested. Assuming an infrared fixed point is present in the perturbative part of the QCD coupling even in some range below N_f^* leads to the condition gamma(N_f^*)=1, where gamma is the critical exponent. This result is incompatible with the existence of an analogue of Seiberg free dual magnetic phase in QCD. Using the Banks-Zaks expansion, one gets 4<N_f^*<6. The low value of N_f^* gives some justification to the infrared finite coupling approach to power corrections, and suggests a way to compute their normalization from perturbative input. If the perturbative series are still asymptotic in the negative coupling region, the presence of a negative ultraviolet fixed point is required both in QCD and in supersymmetric QCD to preserve causality within the conformal window. Some evidence for such a fixed point in QCD is provided through a modified Banks-Zaks expansion. Conformal window amplitudes, which contain power contributions, are shown to remain generically finite in the N_f=-\infty one-loop limit in simple models with infrared finite perturbative coupling.