1 Introduction

July 14, 2019

DESY 09–218

Analytic result for the one-loop scalar pentagon

integral with massless propagators

Bernd A. Kniehl, Oleg V. Tarasov111On leave of absence from Joint Institute for Nuclear Research, 141980 Dubna (Moscow Region) Russia.

II. Institut für Theoretische Physik, Universität Hamburg,

Luruper Chaussee 149, 22761 Hamburg, Germany

The method of dimensional recurrences proposed by one of the authors [1, 2] is applied to the evaluation of the pentagon-type scalar integral with on-shell external legs and massless internal lines. For the first time, an analytic result valid for arbitrary space-time dimension and five arbitrary kinematic variables is presented. An explicit expression in terms of the Appell hypergeometric function and the Gauss hypergeometric function , both admitting one-fold integral representations, is given. In the case when one kinematic variable vanishes, the integral reduces to a combination of Gauss hypergeometric functions . For the case when one scalar invariant is large compared to the others, the asymptotic values of the integral in terms of Gauss hypergeometric functions are presented in , , and dimensions. For multi-Regge kinematics, the asymptotic value of the integral in dimensions is given in terms of the Appell function and the Gauss hypergeometric function .

PACS numbers: 02.30.Gp, 02.30.Ks, 12.38.Bx, 12.40.Nn
Keywords: Feynman integrals, Appell hypergeometric functions, multi-Regge kinematics

1 Introduction

Theoretical predictions for ongoing and future experiments at the CERN Large Hadron Collider (LHC) and an International Linear Collider (ILC) must include high-precision radiative corrections. The complexity of the evaluation of such radiative corrections is related, in particular, to the difficulties in calculating integrals corresponding to Feynman diagrams with many external legs depending on many kinematic variables. Purely numerical evaluation of such integrals cannot provide sufficiently high precision within reasonable computer time. The evaluation of one-loop integrals corresponding to diagrams with two, three, and four external legs was studied in numerous publications.

As for integrals associated with diagrams with five and more external legs, the situation is quite different. Not so many results for such integrals are available in the literature. Various authors [3] discussed the reduction of pentagon integrals to box integrals in space-time dimension . They were able to express these integrals as linear combinations of five different loop integrals with four external legs. Infrared divergences if any were supposed to be regulated by introducing a small fictitious mass. The representation of the dimensionally regularized pentagon integral in terms of box-type integrals was considered in Refs. [4, 5]. However, in the calculation of multi-loop radiative corrections, higher orders in are needed, and, therefore, one should extend such an expansion beyond the “box approximation.” The first step in this direction was recently taken in Ref. [6], where an analytic result for the one-loop massless pentagon integral with on-shell external legs as well as several terms of its expansion were presented in the limit of multi-Regge kinematics. However, no analytic results for arbitrary kinematics are available until now. Practically nothing is known about the analytic structure of on-shell pentagon integrals with unconstrained kinematics in arbitrary space-time dimension . For very simplified kinematics, a pentagon-type integral for arbitrary value of was given in Ref. [7] in terms of Euler gamma functions. A representation of the pentagon integral in terms of a four-fold Mellin-Barnes integral may be found in Ref. [8].

Important applications that require the evaluation of Feynman integrals with massless propagators include the study of jet production in QCD [9], which allows for a high-precision extraction of the strong-coupling constant , the investigation of the iterative structure of supersymmetric Yang-Mills (SYM) amplitudes [10], and tests of the scattering-amplitude/Wilson-loop duality [11].

In this paper, we perform a first analytic study of the on-shell pentagon integral with massless internal lines and arbitrary kinematic invariants. We use the method of dimensional recurrences proposed in Refs. [1, 2], which was already applied to the calculation of one- and two-loop integrals in Refs. [2, 12, 13] and, quite recently, also to the calculation of three- and four-loop integrals in Ref. [14].

This paper is organized as follows. In Section 2, we introduce our notations, explain the recurrence relation for the pentagon-type integral with respect to the space-time dimension , and we present a detailed derivation of its solution in Section 3. In Section 4, we consider a particular case of the on-shell pentagon integral with one vanishing kinematic variable and present an analytic expression in terms of the Gauss hypergeometric function . In Section 5, we specify the asymptotic values of the pentagon integral in , , and space-time dimensions when one of the scalar invariants is much larger than the others. In Section 6, we present an analytic expression for the pentagon integral in the limit of multi-Regge kinematics in terms of the Appell function and the Gauss hypergeomteric function . In the Conclusions, we summarize the accomplishments of the present paper and offer some perspectives for the application of the method of dimensional recurrences to six-point integrals. In Appendix A, we collect useful formulae for hypergeometric functions used in this paper. Appendix B contains intermediate results from Section 3.

2 Definitions and dimensional recurrences

We consider the following integral with five massless propagators:




The pentagon diagram associated with the integral is depicted in Fig. 1. The labeling of the momenta in Fig. 1. corresponds to Eq. (2.1). The Lorentz invariants are defined as:


In the present paper, we take the squares of the external momenta to be vanishing,


and work in the Euclidean region, . In what follows, we only keep non-vanishing variables as arguments of and use the notation


for the on-shell case of Eq. (2.4).

Figure 1: Feynman diagram corresponding the integral .

Due to the symmetry with respect to permutations of all propagators, the integral considered as a function of the kinematic variables must fulfill the following relations:


As was shown in Refs. [1, 5], the integrals and fulfill the following relation:




Here, are integrals corresponding to Feynman diagrams with four external legs defined as


where are defined in Eq. (2.2). An analytic expression for this integral with arbitrary kinematics in arbitrary space-time dimension was recently obtained in Ref. [15].

3 Solution of the dimensional recurrence relation

Redefining the integral as


we obtain the relation


which has the following solution:


where is an arbitrary periodic function depending on the scalar invariants and satisfying the condition


Another solution of Eq. (3.14) reads:




The correctness of both solutions, Eqs. (3.15) and (3.17), may be easily verified by direct substitution into Eq. (3.14). Thus, for example, using Eqs. (3.17) and (3.18), we have


in agreement with Eq. (3.14). Furthermore, the solution in the form of Eq. (3.17) may be easily obtained from Eq. (3.15) by adding to and subtracting from the expression on the right-hand side of Eq. (3.15) the sum


In fact, the combination


is invariant with respect to the change , where is integer, so that this sum may be absorbed into the periodic constant that we denoted by . Changing the summation index in the remaining sum as , we obtain Eq. (3.17). To obtain the solution of the difference equation (3.14) in terms of convergent series, one may choose either Eq. (3.15) or Eq. (3.17) depending on the kinematics.

The dependence of the arbitrary periodic functions and on the scalar invariants may be constructed from a system of differential equations which follows from the one for the integral . For the integral , we derive a system consisting of 5 differential equations of the form


where is not summed over, are the box integrals defined in Eq. (2.10), and and are polynomials in . To derive this system of equations, we use the method proposed in Ref. [16]. Some details of the derivation are presented in Appendix B. The derivation of such equations is done with the help of the computer program package Maple. Explicit expressions for and are given in Eqs. (10.76) and (10.77) in Appendix B, respectively. Using Eq. (3.13), one may obtain from Eq. (3.22) a system of equations for the integral . Substituting Eq. (3.15) into this system, we obtain the following system of equations for the periodic function after a rather tedious calculation:


These differential equations do not depend explicitly on and are, thus, much simpler than those for the integral itself. The solution of this system of differential equation with respect to for obtained with the help of the computer program package Maple reads:


where is an arbitrary periodic constant,


which is independent of the scalar invariants . The system of differential equations for looks similar to Eq. (3.23). An arbitrary periodic constant may be determined from calculated for some particular kinematics. Usually, setting some of the scalar invariants to zero greatly simplifies the computation of the integral. But in our case, as may be seen from Eq. (3.24), cannot be determined from calculated for such kinematics because the term proportional to drops out. For the same reason, one cannot use calculated for kinematics with . Instead, we determine the periodic function by comparing the limiting value of our analytic result for with the analogous value obtained from the integral representation of Eq. (9.63) by exploiting the steepest-descent method described in details in Ref. [12].

Without loss of generality, we henceforth assume the following hierarchy between the scalar invariants:


For the case when the scalar invariants satisfy the conditions of Eq. (3.26) and additionally


the integrals in Eq. (2.10) may be written as




and is the one-loop massless propagator-type integral


An explicit derivation of these results using the method of dimensional recurrences may be found in Ref. [15]. Results for these integrals were also obtained in Ref. [4] using a different method.

To evaluate , we use the solution in the form of Eq. (3.15). This solution may be used if


As we shall see later, the quantities emerge as expansion parameters in the resulting hypergeometric series. If condition (3.31) is fulfilled, then we may obtain a convergent series using Eq. (3.15). If this condition is not fulfilled, then we may use Eq. (3.17). In this case, the inverse quantities, , will be the expansion parameters in the resulting hypergeometric series.

In the following, we obtain an analytic result assuming that Eq. (3.31) is fulfilled for all scalar invariants . If this condition is not fulfilled for a particular term in , then an analytic result may be obtained by performing analytic continuations of the hypergeometric functions in the final result. Another possibility to obtain the analytic result in this case is to repeat the calculation using the solution of the form of Eq. (3.17). It should be noted that, if Eq. (3.27) is not satisfied, then the arguments of all hypergeometric functions generated by the integral exceed unity. Analytic continuation of the result for this integral given in Eq. (3.28) yields


Adopting Eq. (3.31), exploiting Eq. (3.28) and, if , also Eq. (3.32) for , we obtain a result in which each term is real.

The infinite sums resulting from Eq. (3.15) may be written in terms of known hypergeometric functions. As is evident from explicit expressions for the integrals, we must compute three different types of sums. The first one is related to the function ,




The result of the summation of a term including the function reads:


The simplest type of infinite series originates from the two terms in Eq. (3.28) without functions. The contribution arising from the term proportional to reads: