A numerical test of differential equations for one- and two-loop sunrise diagrams using configuration space techniques
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We use configuration space methods to write down one-dimensional integral representations for one- and two-loop sunrise diagrams (also called Bessel moments) which we use to numerically check on the correctness of the second order differential equations for one- and two-loop sunrise diagrams that have recently been discussed in the literature.
KeywordsRecurrence Relation Momentum Space Configuration Space Modify Bessel Function Mass Case
Sunrise-type diagrams have been under investigation since many years. Exact analytical results can be obtained only for special mass or kinematic configurations such as for the equal or zero mass cases or for the threshold region. For example, threshold expansions of the non-degenerate massive two-loop sunrise diagram have been studied in Refs. [1, 2]. The construction of differential equations for the corresponding correlator function provides some hope that by solving these differential equations, a general analytical solution can be obtained. Recently, mathematical methods were used to construct the coefficients of such a differential equation in a systematic way . This work supplements the work of Kotikov  and Remiddi et al. [5, 6, 7] on the same subject. While traditionally the correlator is calculated in momentum space, configuration space techniques allow for a surprisingly simple solution for sunrise-type diagrams: The correlator in configuration space is just a product of single propagators which in turn can be expressed by modified Bessel functions of the second kind. Transforming back to momentum space, one ends up with a one-dimensional integral over Bessel functions, known as Bessel moments [8, 9]. As outlined in a series of papers [2, 10, 11, 12, 13, 14, 15, 16], the corresponding one-dimensional integral can be easily integrated numerically for an arbitrary number of propagators with different masses in any space-time dimension. Therefore, configuration space techniques can be used to numerically check the differential equations for the correlator function obtained by other means. This will be detailed in this note.
The paper is organized as follows: In Sect. 2 we introduce the configuration space techniques which will be used in Sect. 3 to check the differential equations for one-loop sunrise-type diagrams. In Sect. 4 we check the differential equations for the two-loop sunrise diagrams for the equal mass case, while in Sect. 5 we will deal with nondegenerate cases. Our conclusions can be found in Sect. 6. Even though the configuration space techniques are well suited to treat general D≠4 space-time dimensions, we will mainly deal with the case of D=2 space-time dimensions in this paper. For reasons of simplicity, throughout this paper we work in the Euclidean domain. The transition to the Minkowskian domain can be obtained as usual by a Wick rotation (or, equivalently, by replacing p 2→−p 2).
2 Configuration space techniques
3 The one-loop case
In Ref. , Remiddi explains how to obtain the differential equation for the one-loop sunrise-type diagram with arbitrary masses and dimensions. By applying the integration-by-parts technique to the correlator in momentum space, recurrence relations can be obtained. Finally, Euler’s theorem for homogeneous functions connects the loose ends of the iterative steps involving partial derivative with respect to p 2. We have numerically checked all these steps and have found numerical consistency—up to Stokes’ contributions due to surface terms in integer space-times dimensions.
4 The two-loop case with equal masses
5 The two-loop case with arbitrary masses
Using configuration space techniques, we were able to check numerically the differential equations for sunrise-type diagrams found in the literature. The precision of our numerical test is still quite moderate, but gives sufficient confidence in the validity of the differential equations derived by other means. For example, the introduction of artificial “typos” in the coefficients of the differential equations are easily discovered. More rigorous tests would require the use of more stable integration routines than those provided by MATHEMATICA. For the future we hope to find independent routes to discover further relations between Bessel moments which may lead to generalizations of the present findings to cases involving three-loop or even higher order sunrise-type diagrams.
We want to thank Stefan Weinzierl and Anatoly Kotikov for helpful discussions and Volodya Smirnov for encouragement. S.G. acknowledges the support by the Estonian target financed Project No. 0180056s09, by the Estonian Science Foundation under grant No. 8769, and by the Deutsche Forschungsgemeinschaft (DFG) under No. KO 1069/14-1. A.A.P. acknowledges partial support by the RFFI grant No. 11-01-00182-a.
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