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Register a new userHypergeometric functions, their epsilon expansions and Feynman diagrams
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Authors
M. Yu. Kalmykov
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We review the hypergeometric function approach to Feynman diagrams. Special consideration is given to the construction of the Laurent expansion. As an illustration, we describe a collection of physically important one-loop vertex diagrams for which this approach is useful.
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We will present some (formal) arguments that any Feynman diagram can be understood as a particular case of a Horn-type multivariable hypergeometric function. The advantages and disadvantages of this type of approach to the evaluation of Feynman diagrams is discussed.
We evaluate three typical four-loop non-planar massless propagator diagrams in a Taylor expansion in dimensional regularization parameter $epsilon=(4-d)/2$ up to transcendentality weight twelve, using a recently developed method of one of the present coauthors (R.L.). We observe only multiple zeta values in our results.
We describe the application of differential reduction algorithms for Feynman Diagram calculation. We illustrate the procedure in the context of generalized hypergeometric functions, and give an example for a type of q-loop bubble diagram.
The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of such functions with parameters whose values differ from the original ones by integers, is discussed in the context of evaluating Feynman diagrams. Where this is possible, we compare our results with those obtained using standard techniques. It is shown that the criterion of reducibility of multiloop Feynman integrals can be reformulated in terms of the criterion of reducibility of hypergeometric functions. The relation between the numbers of master integrals obtained by differential reduction and integration by parts is discussed.
We briefly sketch a proof concerning the structure of the all-order epsilon-expansions of generalized hypergeometric functions with special sets of parameters.
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