# Box ladders in non-integer dimension

Gonzalez I, Kondrashuk I (2013)

Theoretical and Mathematical Physics 177(2): 1515-1539.

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We construct a class of triangle-ladder diagrams which can be calculated bymaking a use of Belokurov-Usyukina loop reduction technique in d = 4 -2edimensions. The main idea of the approach consists in generalization of thisloop reduction technique existing in d = 4 dimensions. The recursive formularelating L-loop result for any triangle ladder of that class and (L-1)-loopresult for another triangle ladder of the same class is derived. Since themethod proposed in the present paper combines analytic and dimensionalregularizations, at the end of the calculation we have to remove the analyticregularization by taking the limit in which the parameters of the analyticregularization are vanishing. In this limit on the left hand side of therecursive relations we obtain in the position space the diagram in which theindices of the rungs are 1 and all the other indices are 1-e. Fouriertransforms of this type of diagrams are the momentum space diagrams which haveindices of the rungs equal to 1-e and all the other indices 1. Via conformaltransformation of the dual space image of this momentum space representation werelate such a class of triangle ladder momentum diagrams to a class of the boxladder momentum diagrams in which the indices of the rungs are equal to 1-e andall the other indices are 1. Thus, the proposed generalization of theBelokurov-Usyukina loop reduction technique to non-integer number of dimensionsallows us to calculate this class of box-ladder diagrams in the momentum spaceexplicitly in terms of Appell's hypergeometric function F_4 without expandingin powers of parameter e in an arbitrary kinematic region in the momentumspace, since any diagram from this class can be reduced to one-loop diagram,including the important on-shell case.

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### Cite this

Gonzalez I, Kondrashuk I. Box ladders in non-integer dimension.

*Theoretical and Mathematical Physics*. 2013;177(2):1515-1539.Gonzalez, I., & Kondrashuk, I. (2013). Box ladders in non-integer dimension.

*Theoretical and Mathematical Physics*,*177*(2), 1515-1539. doi:10.1007/s11232-013-0120-3Gonzalez, I., and Kondrashuk, I. (2013). Box ladders in non-integer dimension.

*Theoretical and Mathematical Physics*177, 1515-1539.Gonzalez, I., & Kondrashuk, I., 2013. Box ladders in non-integer dimension.

*Theoretical and Mathematical Physics*, 177(2), p 1515-1539.I. Gonzalez and I. Kondrashuk, “Box ladders in non-integer dimension”,

*Theoretical and Mathematical Physics*, vol. 177, 2013, pp. 1515-1539.Gonzalez, I., Kondrashuk, I.: Box ladders in non-integer dimension. Theoretical and Mathematical Physics. 177, 1515-1539 (2013).

Gonzalez, Ivan, and Kondrashuk, Igor. “Box ladders in non-integer dimension”.

*Theoretical and Mathematical Physics*177.2 (2013): 1515-1539.
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