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Analysis of a Zero of a Beta Function Using All-Orders Summation of Diagrams

Conventionally, one calculates a zero in a beta function by computing this function to a given loop order and solving for the zero. Here we discuss a different method which is applicable in theories where one can perform a partial diagrammatic summation to infinite-loop order. We show that this method, compared with the conventional method, yields much better agreement with exact results in the case of an asymptotically free gauge theory with ${\cal N}=1$ supersymmetry. Applications to other field theories are also discussed.