# A numerical test of differential equations for one- and two-loop sunrise diagrams using configuration space techniques

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## Abstract

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.

## Keywords

Recurrence Relation Momentum Space Configuration Space Modify Bessel Function Mass Case## 1 Introduction

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 [3]. This work supplements the work of Kotikov [4] 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

*n*-loop

*n*-particle irreducible correlation function

*x*is given by the product of the propagators,

*m*in

*D*-dimensional (Euclidean) space-time is given by

*D*=2

*λ*+2).

*K*

_{ λ }(

*z*) is the McDonald function (modified Bessel function of the second kind). Note that

*p*and

*x*in the integral expression of Eq. (3) are

*D*-dimensional Lorentz vectors, i.e.

*p*⋅

*x*=

*p*

_{ μ }

*x*

^{ μ }, while the quantity

*x*in the rightmost expression of Eq. (3) (and, therefore, also in the argument of the propagator) denotes the absolute value \(x=\sqrt{x_{\mu}x^{\mu}}\). In the limit

*mx*→0 at fixed

*x*, the propagator simplifies to

*λ*) is Euler’s Gamma function.

*n*-particle irreducible correlator function in (Euclidean) momentum space. The momentum space

*n*-particle irreducible correlator function is given by the Fourier transform of the

*n*-particle irreducible correlator function

*Π*(

*x*) in configuration space,

*Π*(

*x*) in (5) depends only on the absolute value \(x=\sqrt{x_{\mu}x^{\mu}}\). Therefore, one proceeds by first integrating the exponential factor over the

*D*−1 dimensional hypersphere. We write \(d^{D}x=x^{D-1}\,d^{D}\hat{x}\,dx\) where \(d^{D}\hat{x}\) denotes the

*D*−1 dimensional integration measure over the

*D*−1 dimensional hypersphere. The integration of the exponential factor over the unit sphere gives

*J*

_{ λ }(

*z*) is the Bessel function of the first kind. As before,

*p*and

*x*on the right hand side of Eq. (6) stand for the absolute values \(p=\sqrt{p_{\mu}p^{\mu}}\) and \(x=\sqrt{x_{\mu}x^{\mu}}\). Therefore, the correlator in momentum space depends only on the absolute value of the momentum,

## 3 The one-loop case

In Ref. [5], 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.

*v*=

*k*,

*p*(or a linear combination of both) leads to a surface term which can be assumed to vanish (up to Stokes’ contributions). The integration-by-parts technique consists in calculating the integral explicitly and one then expresses the result in terms of scalar integrals

*α*

_{1}=

*α*

_{2}=1 which is relevant for the differential equation, and for

*D*=2 space-time dimensions. As mentioned in Ref. [3], even though other dimensions are feasible, this choice avoids singular contributions and serves for the simplest integrand. The equation to be checked is Starting from the integrals

*S*(

*α*

_{1},

*α*

_{2}) with higher (integer) values of

*α*

_{ i }can be obtained as partial derivatives with respect to the masses, and accordingly

*p*as function of

*m*

_{1}and

*m*

_{2}. The 3D-plot in MATHEMATICA shows stochastic fluctuations around the expected value of 1 of the order of 10

^{−9}.

^{−13}.

^{−8}.

*p*,

*m*

_{1}and

*m*

_{2}. In order to check whether one can derive further relations by using integral identities in configuration space, we have used the general ansatz

*i*=0,1,2). By choosing random values for

*p*,

*m*

_{1}and

*m*

_{2}, and solving the resulting system of equations, one obtains

*a*is an arbitrary parameter. This leads to Eq. (19) and which is the difference of Eq. (19) and the same equation with

*m*

_{1}and

*m*

_{2}interchanged. We conclude that no more recurrence relations can be found that go beyond Eq. (19).

## 4 The two-loop case with equal masses

*D*=2 space-time dimensions. We write

*S*(1,1,1) in a form which is easily adapted to the non-degenerate mass case to be discussed later on. One has

*λ*=0 to compactify the result, These results have been checked with a precision of the order of 10

^{−8}.

## 5 The two-loop case with arbitrary masses

*p*

_{ i }(

*t*) (

*i*=0,1,2,3) are given by and where

*p*and

*m*

_{3}, in terms of

*m*

_{1}and

*m*

_{2}we obtain a 3D-plot with MATHEMATICA which shows again stochastic fluctuations of the order of 10

^{−4}. In the course of our numerical checks we were able to identify two typos in the coefficients of

*c*(

*t*,

*m*

_{1},

*m*

_{2},

*m*

_{3}) in the preprint version of Ref. [3] which we have corrected.

## 6 Conclusions

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.

## Notes

### Acknowledgements

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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