Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion

Fleischer J, Smirnov VA, Tarasov OV (1997)
ZEITSCHRIFT FUR PHYSIK C-PARTICLES AND FIELDS 74(2): 379-386.

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A method of calculating Feynman diagrams from their small momentum expansion [1] is extended to diagrams with zero mass thresholds. We start from the asymptotic expansion in large masses [2] (applied to the case when all M-i(2) are large compared to all momenta squared). Using dimensional regularization, a finite result is obtained in terms of powers of logarithms (describing the zero-threshold singularity) times power series in the momentum squared. Surprisingly, these latter ones represent functions, which not only have the expected physical ''second threshold'' but have a branchcut singularity as well below threshold at a mirror position. These can be understood as pseudothresholds corresponding to solutions of the Landau equations. In the spacelike region the imaginary parts from the various contributions cancel. For the two-loop examples with one mass M, in the timelike region for q(2) approximate to M-2 we obtain approximations of high precision. This will be of relevance in particular for the calculation of the decay Z --> in the m(b) = 0 approximation.
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Fleischer J, Smirnov VA, Tarasov OV. Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion. ZEITSCHRIFT FUR PHYSIK C-PARTICLES AND FIELDS. 1997;74(2):379-386.
Fleischer, J., Smirnov, V. A., & Tarasov, O. V. (1997). Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion. ZEITSCHRIFT FUR PHYSIK C-PARTICLES AND FIELDS, 74(2), 379-386.
Fleischer, J., Smirnov, V. A., and Tarasov, O. V. (1997). Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion. ZEITSCHRIFT FUR PHYSIK C-PARTICLES AND FIELDS 74, 379-386.
Fleischer, J., Smirnov, V.A., & Tarasov, O.V., 1997. Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion. ZEITSCHRIFT FUR PHYSIK C-PARTICLES AND FIELDS, 74(2), p 379-386.
J. Fleischer, V.A. Smirnov, and O.V. Tarasov, “Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion”, ZEITSCHRIFT FUR PHYSIK C-PARTICLES AND FIELDS, vol. 74, 1997, pp. 379-386.
Fleischer, J., Smirnov, V.A., Tarasov, O.V.: Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion. ZEITSCHRIFT FUR PHYSIK C-PARTICLES AND FIELDS. 74, 379-386 (1997).
Fleischer, Jochem, Smirnov, VA, and Tarasov, OV. “Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion”. ZEITSCHRIFT FUR PHYSIK C-PARTICLES AND FIELDS 74.2 (1997): 379-386.
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