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We present a rigorous discussion of the analyticityproperties of the beta function and of the effective potential for thetheory of the ground state of a one dimensional system of many spinlessfermions.  We show that their analyticity domain as a function of therunning couplings is a polydisk with positive radius bounded below,uniformly in all the cut offs (infrared and ultraviolet) necessary togive a meaning to the formal Schwinger functions. We also prove thevanishing of the scale independent part of the beta function showingthat this implies the analyticity of the effective potential and of theSchwinger functions in terms of the bare coupling.  Finally we show thatthe pair Schwinger function has an anomalous long distance behaviour

Beta function and Schwinger functions for a many fermions system in one dimension. Anomaly of the fermi surface