By Carroll B.W., Ostlie D.A.

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Q M. 1/a ) is related to the position of the Landau pole, cf. Eq. 78). The derivation * Q of Eqs. 84) requires some care and can be found in Beneke and Braun (1995a) and Ball et al. ective coupling is continuous at "! and * approaches a "nite value as P0.

51) Table 2 Comparison of contributions of various diagrams to the leading UV and IR renormalon behaviour. , for the IR renormalon (! /2)Ln!. In the case of Fig. 9b we refer to the diagram with a chain inserted into a chain analogous to Fig. 4b Diagram Fig. 1 Fig. 9a Fig. 9b UV IR n 1 n 1/n n ln n ln n Ignoring O(1) "xed by gauge invariance. 44 M. 52) s " ! # # ! c 2 2 2 2 follows. erent signs of the anomalous dimension terms compared to Eq. 48). ) The global normalization K is not determined.

1982), Palanques-Mestre and Pascual (1984), Beneke and Braun (1994) and Ball et al. (1995a). If R needs infrared subtractions and receives only virtual corrections, the procedure is essentially identical. The case when R requires IR subtractions and receives real and virtual corrections has not been worked out in detail so far. 2. The dispersive method The bubble diagrams can also be calculated by using the dispersion relation 1 Im ( ) 1 1 1 * d d " # ( ! ) * k! "1# ( )" k!