A Feynman Integral and its Recurrences and Associators
Abstract
We determine closed and compact expressions for the –expansion of certain Gaussian hypergeometric functions expanded around half–integer values by explicitly solving for their recurrence relations. This –expansion is identified with the normalized solution of the underlying Fuchs system of four regular singular points. We compute its regularized zeta series (giving rise to two independent associators) whose ratio gives the –expansion at a specific value. Furthermore, we use the well known one–loop massive bubble integral as an example to demonstrate how to obtain all–order –expansions for Feynman integrals and how to construct representations for Feynman integrals in terms of generalized hypergeometric functions. We use the method of differential equations in combination with the recently established general solution for recurrence relations with non–commutative coefficients.
Max–Planck–Institut für Physik
Werner–Heisenberg–Institut
80805 München, Germany
MPP–2015–259
Contents:
 1 Introduction

2 Hypergeometric Function with half–integer parameters
 2.1 Harmonic polylogarithms, integral operators and the generalized operator product
 2.2 Differential equations for hypergeometric functions
 2.3 Recurrence relations for generalized hypergeometric functions
 2.4 Fuchsian system and its associators
 2.5 Hypergeometric function expansion from associators
 3 A simple Feynman integral
 4 Concluding remarks
1 Introduction
Scattering amplitudes describe the interactions of physical states and play an important role to determine physical observables measurable at colliders. In perturbation theory at each order in the expansion scattering amplitudes are comprised by a sum over Feynman diagrams with a fixed number of loops. Each individual Feynman diagram is represented by integrals over loop momenta and integrates to functions, which typically depend on the Lorentz–invariant quantities of the external particles like their momenta, masses and scales. These functions are generically neither rational nor algebraic but give rise to a branch cut structure following from unitarity and the fact that virtual particles may go onshell.
The class of functions describing Feynman integrals are iterated integrals, elliptic functions and perhaps generalizations thereof. To obtain physical results one is interested in their Laurent series expansion (–expansion) about the integer value of the space–time dimension (typically ). In the parameter space of the underlying higher transcendental functions this gives rise to an expansion w.r.t. small parameter around some fixed numbers, which may be integer or rational numbers. Expansions around integer values is in general sufficient for the evaluation of loop integrals arising in massless quantum field theories. However, the inclusion of particle masses in loop integrals or the evaluation of phase space integrals may give rise to half–integer values, cf. e.g. [1].
The module of hypergeometric functions [2] is ubiquitous both in computing tree–level string amplitudes [3] and in the evaluation of Feynman diagrams with loops cf. e.g. [4]. Therefore, finding an efficient procedure to determine power series expansion of these functions is an important problem. Their underlying higher order differential equations lead to recurrence relations, which can be solved explicitly by the methods recently proposed in [5]. This procedure gives an explicit solution to the recurrence relation providing for each order in a closed, compact and analytic expression. This way we get hands on the –expansions of this large family of functions [5]. For a subclass of the latter the coefficients of their expansions generically represent multiple polylogarithms (MPLs), which give rise to periods of mixed Tate motives in algebraic geometry. Then, expansions around rational numbers naturally yield –th roots of unity in the arguments of the MPLs [6, 7].
Amplitudes in field–theory very often can be described by certain differential equations or systems thereof with a given initial value problem subject to physical conditions [8]. For generic parameters the corresponding differential equations for generalized hypergeometric functions are Fuchsian differential equations with the regular singular points at and . For a specific subclass (of at least hypergeometric functions to be specified later) at rational values of parameters (real parameters shifted by ) after a suitable coordinate transformation the underlying first order differential equations become a Fuchs system with regular singular points at and with (Schlesinger system). By properly assigning the Lie algebra and monodromy representations of this linear system of differential equations their underlying fundamental solutions can be matched with the –expansion of specific generalized hypergeometric functions. As a consequence, each order in is given by some combinations of MPLs and group–like matrix products carrying the information on the parameter. Furthermore, at special values the latter can be given in terms of their underlying regularized zeta series. The latter give rise to independent associators, which are defined as ratio of two solutions of the specific differential equation. This way one obtains a very elegant way of casting the full –expansion of certain generalized hypergeometric functions into the form dictated by the underlying Lie algebra structure and the analytic structure of MPLs. Computing higher orders in the –expansion is then reduced to simple matrix multiplications. In this work we explicitly work out the case , which is relevant to a Feynman integral to be discussed in this work and comment on the generic case .
Feynman integrals are classified according to their topologies. A topology includes all integrals, that consist of the same set of propagators but have different powers thereof. There are integration–by–parts (IBP) identities [9], which allow to reduce all integrals of a topology to a set of master integrals (MIs). One way to compute MIs is to apply the method of differential equations [8] (for reviews, see [10]). The idea of this method is to take derivatives of MIs w.r.t. kinematic invariants and masses. The results are combinations of integrals of the same topology, which can again be written in terms of MIs using the IBP reduction. This way one obtains differential equations for the MIs. In practice the goal is to solve these equations in a Laurent expansion around . If the differential equations take a suitable form the coefficient functions of the –expansion can be given in a straightforward iterative form [11]. By replacing integrations with integral operators the iterative solutions can be written as recurrence relations with non–commutative coefficients. Solving these recurrences yields the all–order –expansions for Feynman integrals. This approach allows to give the Laurent expansions as infinite series explicitly in terms of iterated integrals, thereby providing solutions, which are exact in . This outcome is equivalent to the representation generic to hypergeometric functions and their generalization, with the advantage that the behaviour of the Feynman integral at can be extracted directly.
The present work is organized as follows. In section 2 we show how to systematically and efficiently derive –expansions of hypergeometric function with one half–integer parameter. In section 2.1 we introduce harmonic polylogarithms (HPLs), integral operators and related objects. In section 2.2 we discuss the differential equation satisfied by the hypergeometric function . A given order of its –expansion can be derived from this differential equation based on computing successively all lower orders. Next, in section 2.3 by transforming the differential equation to recurrence relations we derive the terms in the power series solution for any given order in . This strategy yields an all–order expansion, i.e. an infinite Laurent series in explicitly in terms of iterated integrals. In section 2.4 we investigate the underlying Fuchs system involving four regular singular points describing the hypergeometric function with one half–integer parameter. For the latter we derive analytic solutions in terms of hyperlogarithms and group–like matrix elements encoding the information on the parameters and . One of these solutions will be matched with the relevant –expansion of the hypergeometric function . As a consequence the coefficients of the –order in the power series expansion are given by a set of the MPLs of degree entering the fundamental solution supplemented by matrix products of matrices. Furthermore, we construct the two regularized zeta series generic to the underlying Fuchs system. The latter gives rise to two independent associators whose ratio will be related to the hypergeometric function at the special point . In section 3 as an example we discuss the –expansion of Feynman integrals using a massive one–loop integral. The all–order expansion is derived via the method of differential equations in section 3.1 and this result is used to construct a representation in terms of a hypergeometric function in section 3.2. Benefiting from the fact that the all–order result is exact in , we can give the hypergeometric representation of the Feynman integral not only in dimensions but also in general dimensions . Finally, in section 4 we present for the hypergeometric function the underlying Fuchs system involving regular singular points. We comment on its generic solutions and the underlying associators.
2 Hypergeometric Function with half–integer parameters
The Gaussian hypergeometric function is given by the power series [2]
(2.1) 
with parameters and with the Pochhammer (rising factorial) symbol:
The series (2.1) converges absolutely at the unit circle if the parameters meet the following condition:
(2.2) 
In the sequel we want to investigate hypergeometric functions (2.1) with some of their parameters shifted by . This gives rise to the following five possibilities
(2.3) 
and:
(2.4) 
It has been shown in [12] that the class of four hypergeometric functions (2.3) can algebraically be written in terms of only one, e.g.:
If the functions (2.3) (with the same argument ) have in their parameters additional integer shifts, they can be expressed in terms of (2.3) and first derivatives thereof and some polynomials in the parameters and . Therefore, in this section we shall apply our new technique [5] to solve for recurrences to compute the following –expansion:
(2.5) 
The coefficient functions are expressible in terms of harmonic polylogarithms (HPLs) with rational coefficients [13]. On the other hand, the type (2.4) involves elliptic functions and will not be discussed here. E.g. we have: with the elliptic function of first kind.
2.1 Harmonic polylogarithms, integral operators and the generalized operator product
HPLs of weight are defined recursively as [14]
(2.6) 
with the multiple index and
(2.7) 
Weight functions are simple logarithms:
(2.8) 
In the next subsection we will encounter differential equations satisfied by functions for which we know their boundary conditions at . Therefore, it is advantageous to introduce the following integral operators
(2.9) 
with their integrations starting at the point 1. We shall use a shorter notation for products of these operators:
(2.10) 
Products of integral operators (2.9) acting on functions independent of , like the constant function
(2.11) 
with , e.g.:
(2.12) 
Since HPLs with argument 1 can have divergences, the second terms on the r.h.s. of eqs. (2.11) reveal that some products of integral operators are illdefined. This happens when is the rightmost operator of a product acting on a constant. It can be seen in the following sections that these cases never appear in our results. Divergent HPLs also arise for other operator products, however, they can be removed using the underlying product algebra. Consider for example and its representation in terms HPLs following from eqs. (2.11):
(2.13) 
According to the identity
(2.14) 
the divergences in the last two terms of (2.13) cancel each other.
The essential difference between the integral operators (2.9) and the integrations in the definition (2.6) of HPLs is the lower bound. See also eq. (2.37) or Ref. [15] for polylogarithms with lower integration limits not chosen to be zero.
Products of integral operators (2.9) are closely related to the ‘symbol’ of the underlying function, together with the prescription that the integration path goes from 1 to z. Indeed, a sequence of integral operators immediately determines a ‘symbol’, which when integrated against this path turns it into the Chen iterated integral representation of the function (see e.g. Ref. [16] for an introduction for physicists).
Furthermore we introduce the generalized operator product
(2.15) 
as the sum of all the
(2.16) 
possible distinct permutations of non–commutative factors , each one appearing times (). We will call the non–negative integers indices and the factors arguments of the generalized operator product (2.15). Some examples are:
(2.17) 
A recursive definition and basic properties of the generalized operator product can be found in [5]. The object (2.15) is useful to handle non–commutative quantities, such as the integral operators (2.9).
2.2 Differential equations for hypergeometric functions
In this section we describe some of the achievements originally developed in [13] for the calculation of expansions (2.5). Some of the formulas derived here are the foundation of the method we use in the next section to obtain the all–order expansion.
Applying the differential operator on the series (2.1), it is easy to show, that hypergeometric functions satisfy:
(2.18) 
Combining these relations yields the differential equation:
(2.19) 
For the hypergeometric function
(2.20) 
the second order differential equation (2.19) in can be written as a system of two first order differential equations in the variable
(2.21) 
for and :
(2.22) 
Inserting the expansion (2.5) and using, that the resulting differential equations are valid at any order in , yields iterative differential equations for the coefficient functions and :
(2.23) 
Eq. (2.1) can be used to determine boundary conditions at . This point transforms to the point under (2.21), so that the boundary conditions are for and for . The system (2.23) can then be solved iteratively for via:
(2.24) 
From eqs. (2.1) and (2.2) follow the lowest orders and for . Starting with those we can use eqs. (2.24) to straightforwardly calculate the –expansion (2.5) order by order, e.g.:
(2.25) 
We used eqs. (2.11) to give and in terms of HPLs. This can be achieved for and higher order coefficient functions as well.
The iterative computation of Laurent expansions of hypergeometric functions described in this section has first been presented in [13]. It has been applied to similar function, e.g. to generalized hypergeometric functions with integer parameters in [17]. Eqs. (2.24) are the main results of this subsection. They allow to straightforwardly calculate the expansion (2.5) up to any order in . However, is is not possible to obtain a given order without knowing its lower orders. This problem will be solved in the next section.
2.3 Recurrence relations for generalized hypergeometric functions
In this section we present the all–order expansions for the hypergeometric functions (2.5). The idea is to write the differential equations for the coefficient functions as recurrence relations
More importantly, the general solution for this type of recurrence relations has been presented in [5]. Hence the all–order expansion of (2.5) can now systematically be constructed and straightforwardly be given in a closed form. By all–order we mean a representations for the coefficient functions , which include as a variable and therefore hold for all orders in . In contrast to that, the method of the previous section allows to compute –expansions order by order starting with , , and so on. In other words the formula for , no matter if in the form of a iterative solution to differential equations as discussed in the previous section or as a recurrence relation as presented in the following, is not given in terms of HPLs or similar functions. Instead, coefficient functions of lower orders are included. On the other hand our all–order result obtained from the solution of the recurrence relation gives coefficient functions for all orders explicitly in terms of iterated integrals.
Combining eqs. (2.24) yields the second order recurrence relation
(2.26) 
with the non–commutative coefficients
(2.27) 
With the initial values and the solution of eq. (2.26) reads
(2.28) 
Let us demonstrate that, in contrast to the findings in [13], this result allows to obtain any order of the expansions (2.5) directly without using lower orders. For instance in (2.28) yields the condition for the sum over non–negative integers and . This equation has three solutions: , and . It is easy to check, that they give
respectively. This is in accordance with the expression for given in (2.25). Higher orders can be evaluated the same way, e.g. for there are 13 terms with the following summation indices:
Every summand can be calculated straightforwardly, e.g. for we get .
According to the discussion at the beginning of this section a variety of other hypergeometric functions with parameters, which differ from those in (2.5) by integer or half–integer shifts, can algebraically be expressed as linear combinations of the function (2.5) and derivatives thereof. Thus our result (2.28) allow to construct the all–order expansions for these functions as well. One of these functions appears in section 3 in a representation of a Feynman integral.
2.4 Fuchsian system and its associators
The differential equation (2.19)
(2.29) 
for the function
(2.30) 
can be written as a system of two first order differential equations of Fuchsian class in the variable defined in (2.21)
(2.31) 
with the singularities , the vector
and (cf. also (2.22)):
(2.32) 
In (2.31) the matrices are given by:
(2.33) 
The system (2.31) has four regular singular points at and and is known as Schlesinger system. Recently, in Ref. [5] we have thoroughly discussed the latter and its generic solution given in terms of hyperlogarithms [20].
Hyperlogarithms are defined recursively from words built from an alphabet (with ) with letters:
(2.34) 
Generically, we have
(2.35) 
with the MPLs:
(2.36) 
More generally one considers the iterated integrals [21]
(2.37) 
with some base point , which can be entirely expressed in terms of the objects (2.34). Of course, we have . In the case under consideration with the three singular points and the alphabet consists of three letters and is directly related to the differential forms
appearing in (2.31). The corresponding hyperlogarithms (2.34) boil down to the HPLs discussed in section 2.1.
A grouplike solution to (2.31) taking values in with the alphabet can be given as formal weighted sum over iterated integrals (with the weight given by the number of iterated integrations)
(2.38) 
with the MPLs (2.34). The solution (2.38) can be constructed recursively and built by
Picard’s iterative methods
(2.39) 
respectively. Generalizing (2.38) to
(2.40) 
we arrive at the following expansions
(2.41) 
(2.42) 
and: