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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 906 (2016) 168–193

www.elsevier.com/locate/nuclphysb

A Feynman integral and its recurrences and associators

Georg Puhlfürst, Stephan Stieberger

∗

Max-Planck-Institut

für Physik, Werner-Heisenberg-Institut, 80805 München, Germany

Received 7

December 2015; received in revised form 2 March 2016; accepted 3 March 2016

Available

online 8 March 2016

Editor: Tommy

Ohlsson

Abstract

determine closed and compact expressions for the -expansion of certain Gaussian hypergeomet-

ric

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 singu-

lar

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.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

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 inte

grals 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

Corresponding author.

E-mail

address: stephan.stieberger@mpp.mpg.de (S. Stieberger).

http://dx.doi.org/10.1016/j.nuclphysb.2016.03.008

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

G. Puhlfürst, S. Stieberger / Nuclear Physics B 906 (2016) 168–193 169

neither rational nor algebraic but give rise to a branch cut structure following from unitarity and

the fact that virtual particles may go on-shell.

The class of functions describing Fe

ynman 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 D (typically

D = 4 −2). In the parameter space of the underlying higher transcendental functions this gi

ves

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 inte

grals or the evaluation of phase space integrals may give rise to half-integer

values, cf. e.g. [1].

The module of h

ypergeometric 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, ﬁnd-

ing 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 solv

ed 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]. Fo

a subclass of the latter the coefﬁcients of their expansions generically represent multiple polylog-

arithms (MPLs), which give rise to periods of mixed Tate motives in algebraic geometry. Then,

expansions around rational numbers p/q naturally yield q-th roots of unity exp(2πip/q) in the

arguments of the MPLs [6,7].

Amplitudes in ﬁeld-theory ve

ry 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 0, 1 and ∞. Fo

r a speciﬁc sub-

class (of at least

hypergeometric functions to be speciﬁed later) at rational values p/q of

parameters (real parameters shifted by p/q) after a suitable coordinate transformation the under-

lying ﬁrst order differential equations become a Fuchs system with q +2 regular singular points

at 0, exp(2πir/q) and ∞ with r = 1, ..., q (Schlesinger system). By properly assigning the

Lie algebra and monodromy representations of this linear system of dif

ferential 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 q 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 -e

xpansion

of certain generalized hypergeometric functions into the form dictated by the underlying Lie al-

gebra 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

q =2, which is rele

vant to a Feynman integral to be discussed in this work and comment on the

generic case q =2.

Feynman inte

grals are classiﬁed according to their topologies. A topology includes all inte-

grals, 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 dif

ferential

equations [8] (for reviews, see [10]). The idea of this method is to take derivatives of MIs w.r.t.

170 G. Puhlfürst, S. Stieberger / Nuclear Physics B 906 (2016) 168–193

kinematic invariants and masses. The results are combinations of integrals of the same topol-

ogy, 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  = 0. If the dif

ferential equations take a suitable form the coefﬁcient functions

of the -expansion can be given in a straightforward iterative form [11]. By replacing integra-

tions with integral operators the iterative solutions can be written as recurrence relations with

non-commutative coefﬁcients. Solving these recurrences yields the all-order -e

xpansions for

Feynman integrals. This approach allows to give the Laurent expansions as inﬁnite series explic-

itly 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 Fe

ynman integral at  = 0 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



a,b

c+



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 hypergeomet-

ric function



a,b

c+



. 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 solu-

tion for any given order in . This strate

gy 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 underly-

ing Fuchs system involving four regular singular points describing the hypergeometric function



a,b



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

a, b and c. One of these solutions will be matched with the relevant -expansion of the hyperge-

ometric function



a,b

c+



. As a consequence the coefﬁcients of the 

-order in the power

series expansion are given by a set of the MPLs of degree k entering the fundamental solution

supplemented by matrix products of k matrices. Furthermore, we construct the two regularized

zeta series generic to the underlying Fuchs system. The latter gives rise to two independent as-

sociators whose ratio will be related to the h

ypergeometric function



a,b



at the special

point z = 1. 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 h

yper-

geometric 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 D =4 −2

dimensions but also in general dimensions D. Finally, in section 4 we present for the hyperge-

ometric function



a,b

1−



the underlying Fuchs system involving q +2 regular singular

points. We comment on its generic solutions and the underlying q associators.

2. Hypergeometric function with half-integer parameters

The Gaussian hypergeometric function

is given by the power series [2]



a,b



∞



m=0

(a)

(b)

(c)

, (2.1)

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