Algorithms for the Symbolic Integration of Hyperlogarithms with Applications to Feynman Integrals

Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals

Erik Panzer Humboldt-Universität zu Berlin, Institut für Physik, Newton Straße 15, 12489 Berlin, Germany

Abstract We provide algorithms for symbolic integration of hyperlogarithms multiplied by rational functions, which also include multiple polylogarithms when their arguments are rational functions. These algorithms are implemented in Maple and we discuss various applications. In particular, many Feynman integrals can be computed by this method. Keywords: Feynman integrals, hyperlogarithms, polylogarithms, computer algebra, symbolic integration, ε-expansions

Program summary Unusual features: The complete program works strictly symbolically and the obtained results are exact. Whenever a Feyn- Manuscript Title: Algorithms for the symbolic integration of man graph is linearly reducible, its ε-expansion can be com- hyperlogarithms with applications to Feynman integrals puted to arbitrary order (subject only to time and memory Authors: Erik Panzer restrictions) in ε, near any even dimension of space-time and Program Title: HyperInt for arbitrarily ε-dependent powers of propagators with integer Journal Reference: values at ε = 0. Also the method is not restricted to scalar Catalogue identifier: integrals only, but arbitrary tensor integrals can be computed Licensing provisions: GNU General Public License, version 3 directly. Programming language: Maple [1], version 16 or higher Additional comments: Further applications to parametric in- Computer: Any that supports Maple tegrals, outside the application to Feynman integrals in the Operating system: Any that supports Maple Schwinger parametrization, are very likely. RAM: Highly problem dependent; from a few MiB to many Running time: Highly dependent on the particular problem GiB through the number of integrations to be performed (edges of Number of processors used: a graph), the number of remaining variables (kinematic invari- Supplementary material: Example worksheet Manual.mw ex- ants), the order in ε and the complexity of the geometry (topol- plaining most features provided and including plenty of exam- ogy of the graph). Simplest examples finish in seconds, but the ples of Feynman integral computations. time needed increases beyond any bound for sufficiently high Keywords: hyperlogarithms, polylogarithms, symbolic integra- orders in ε or graphs with many edges. tion, computer algebra, Feynman integrals, ε-expansions Classification: 4.4 Feynman diagrams, 5 Computer Algebra External routines/libraries: Subprograms used: References Nature of problem: Feynman integrals and their ε-expansions [1] Maple 161. Maplesoft, a division of Waterloo Maple Inc., Water- in dimensional regularization can be expressed in the Schwinger loo, Ontario. parametrization as multi-dimensional integrals of rational func-

arXiv:1403.3385v1 [hep-th] 13 Mar 2014 tions and logarithms. Symbolic integration of such functions therefore serves a tool for the exact and direct evaluation of 1. Introduction Feynman graphs. Solution method: Symbolic integration of rational linear com- An important class of special functions is given by mul- binations of polylogarithms of rational arguments is obtained tiple polylogarithms [1, 2] using a representation in terms of hyperlogarithms. The algorithms exploit their iterated integral structure. k X z 1 ··· zkr Restrictions: To compute multi-dimensional integrals with this : 1 r Lin1,...,nr (z1, . . . , zr) = n1 nr (1.1) k1 ··· kr method, the integrand must be linearly reducible, a criterion 0

Email address: [email protected] (Erik Panzer) 1MapleTM is a trademark of Waterloo Maple Inc.

Preprint submitted to Computer Physics Communications March 14, 2014 r = 1) studied for example in [3]. Many properties and Unfortunately, none of these programs was made pub- relations of these multivalued functions can be formulated licly available so far. This might partly be due to the and studied conveniently in terms of combinatorial struc- fact that the exposition in [11] does not provide a sim- tures, which renders them suitable for symbolic algorithms ple method to obtain certain integration constants in a that can be implemented on a computer. crucial intermediate step of the algorithm. In fact, [17] re- This is mainly a consequence of their representation as sorts to numeric evaluations to guess these constants and a special class of iterated integrals [4] and our preferred a similar approach is common to many applications of the basis are the classic hyperlogarithms [5] of symbol- and coproduct-calculus [18–20]. Also within the method of differential equations [21], boundary conditions Definition 1.1. Given a finite set Σ ⊂ C containing occur that must be obtained separately, e.g through phys- 0 ∈ Σ, each word w = ω . . . ω ∈ Σ× (ω denotes the σ1 σn σ ical reasoning or separate computations of expansions in letter for σ ∈ Σ) defines the hyperlogarithm Lw by setting logn z certain limits. Lωn (z) := and otherwise recursively applying 0 n! We close this gap and provide a complete implementa- Z z 0 tion of the method [11] of symbolic integration using hy- dz 0 L 0 := L 0 (z ). (1.2) perlogarithms in the computer algebra system Maple [22]. ωσ w z0 − σ w 0 This program was used in [23, 24] to compute several non- We also abbreviate L := L and write σ(n) trivial Feynman integrals (including divergences and com- σ1,...,σn ωσ1 ,...,ωσn for a sequence σ, . . . , σ of n letters σ. plicated kinematics) and we hope that it will prove helpful in further applications by physicists and mathematicians These functions are also referred to as generalized har- alike. monic polylogarithms (with linear weights) or Goncharov Since our foremost goal was to supply a tool for the polylogarithms, since they relate to (1.1) via computation of Feynman integrals, we did not aim for a r most general computer algebra framework to handle hy- (−1) Li~n (~z) = L (nr −1) (n2−1) (n1−1) (z) (1.3) 0 ,σr ,...,0 ,σ2,0 ,σ1 perlogarithms but instead focussed on this particular ap- r plication. Still, the algorithms were implemented for very where ~n = (n1, . . . , nr) ∈ N and σ1, . . . , σr 6= 0 are such σ2 σ3 z Qr −1 general situations and may be used for different problems that ~z = , ,..., , equivalently σi = z z . σ1 σ2 σr k=i k as well. Particle physicists observed special classes of hyperloga- For completeness let us mention that while we focus rithms in results of Feynman integral calculations. The on polylogarithms as iterated integrals, the representation most famous example is the case when Σ ⊆ {−1, 0, 1}, (1.1) as nested sums opens the door to completely different called harmonic polylogarithms in [6], and practical tools strategies like [25] with implementations readily available to compute with these are available like [7, 8]. Some algo- [26, 27]. A lot of progress is being made on symbolic ma- rithms for general hyperlogarithms are also implemented nipulation of sums and we like to point out [9] and the in [9]. However, the full power of definition 1.1 can be used numerous references therein. However, we will not pursue not only to express the result of Feynman integrals, but this approach in our work. actually to compute them in the first place. Namely, the study [10] of periods of moduli spaces of 1.1. Plan of the paper curves of genus zero computed multiple integrals In section 2 we present our algorithms to symbolically Z ∞ Z ∞ Z ∞ manipulate hyperlogarithms in sufficient detail so as to fk := fk−1(zk) dzk = dz1 ··· dzk f0 (1.4) 0 0 0 make an implementation straightforward. We follow the ideas of [11] where the reader might find illuminating ex- of certain polylogarithms f0(~z) such that each of the par- amples and details. Our main original contribution is sec- tial integrals fk is a hyperlogarithm in the next integration 2.5 where we solve the problem of determination of tion variable zk+1. This criterion on f0 is called linear integration constants mentioned above. reducibility in [11], where the symbolic integration algo- The Maple implementation HyperInt is presented in rithm of such functions is explained and applied theoret- section 3 and includes examples of its application to inte- ically to some finite scalar single-scale Feynman integrals gration problems and for transformations of arguments of (massless propagators). In [12] it was further shown that polylogarithms. linear reducibility is actually fulfilled for an infinite family To apply these methods to multiple integrals (1.4), we of non-trivial Feynman integrals, but still explicit results review the property of linear reducibility in section 4 and were missing. explain how to exploit the polynomial reduction algorithm This technique has then practically been used in [13] to contained in HyperInt. compute off-shell three-point functions and in [14–16] to Section 5 is devoted to our original motivation and calculate operator insertions into propagator graphs con- main application: the calculation of Feynman integrals. taining a single non-zero mass scale. A further application In HyperInt we supply a couple of commands to facili- to phase-space integrals related to Higgs production can tate the work with Feynman graphs. Detailed examples be found in [17]. 2 and demonstrations are contained in the attached Maple until 1 ¡ w = w ¡ 1 = w where 1 denotes the empty word worksheet Manual.mw. which is the identity of T (Σ). The coproduct ∆ : T (Σ) −→ To ensure correctness of our program, we performed T (Σ) ⊗ T (Σ) of interest is the deconcatenation plenty of tests. Some of them are summarized in Ap- n pendix A and provided in the file HyperTests.mpl. X ∆ (ω . . . ω ) := ω . . . ω ⊗ ω . . . ω . (2.3) Some combinatorial proofs were delegated to Appendix B σ1 σn σ1 σk σk+1 σn k=0 and we supply a short reference of functions and options provided by HyperInt in Appendix C. These combinatorial structures precisely capture the analytic properties of the iterated integrals [4, 31] 2. Algorithms for hyperlogarithms Z Z 1 dγ(z) Z ωσ1 . . . ωσn := ωσ2 . . . ωσn (2.4) We already mentioned references on hyperlogarithms, γ(z) − σ1 γ 0 γ|[0,z] multiple polylogarithms and iterated integrals. In section 2.1 we collect some standard results and fix our nota- dz 1 C of differential one-forms ωσ = z−σ ∈ Ω ( \ Σ) of a single tions. R variable z. With γ 1 := 1, (2.4) defines homotopy invari- Afterwards we follow the ideas of [11] for the integra- ant functions of a path γ : [0, 1] → C \ Σ of integration tion of hyperlogarithms and explain in detail how each step which are often identified with the resulting multivalued can be implemented combinatorially. In short, to com- C R ∞ analytic functions of the endpoint z = γ(1) ∈ \ Σ. pute fk = 0 dzkfk−1(zk) for some polylogarithm fk−1, Observe that in (1.2) we chose the singular base point we proceed in three steps: γ(0) = 0 ∈ Σ which is the reason why we had to define

1. Express fk−1 as a hyperlogarithm in zk. Lω0 (z) := log z specially and not by the divergent integral R z dz 2. Find a primitive F (z ) such that ∂ F = f . . k−1 k zk k−1 k−1 0 z R We extend the maps w 7→ w and w 7→ Lw linearly 3. Evaluate the limits fk = limzk→∞ Fk − limzk→0 Fk. γ to the whole shuffle algebra T (Σ). Then the fundamental Many of these operations are straightforward or explained properties of iterated integrals become with examples in [11]. All the work actually lies in step R R 0 R ¡ 0 R 1. above, which is the subject of the reviewing section 2.4 1. γ w · γ w = γ (w w ), i.e. γ is multiplicative, and our additional algorithm of section 2.5 to symbolically 2. By Chen’s lemma, concatenation γ ? η of two paths compute constants of integration. with η(1) = γ(0) gives for every word w = ωσ1 . . . ωσn The original work [10] in the setting of moduli spaces n contains a lot of details, worked examples and geometric Z X Z Z w = ω . . . ω · ω . . . ω . (2.5) interpretations of the ideas employed. In particular we σ1 σi σi+1 σn γ?η i=0 γ η recommend sections 5 and 6 therein which develop the theory of hyperlogarithms tailored to our setup, including h R i 3. Q(z) ⊗Q T (Σ) 3 f ⊗ w 7→ γ 7→ f(γ(1)) · w is logarithmic regularization in detail. γ Let us remark that instead of tracking a sequence (1.4) injective, so iterated integrals associated to different of one-dimensional iterated integrals, the natural approach words are linearly independent with respect to rational (actually even for algebraic) prefactors f. would be to consider every fk as an iterated integral in several variables zk+1, zk+2,... simultaneously. This idea The analogous properties hold for the hyperlogarithms is pursued in [28] and their authors are currently finalizing w 7→ Lw of (1.2). These functions Lw : C \ Σ −→ an implementation of this method as well. C are single-valued once we restrict to the simply con- nected domain where 0 < |z| < min {|σ|: 0 6= σ ∈ Σ} and 2.1. The tensor algebra and iterated integrals z∈ / (−∞, 0], after fixing log to the principal branch with The algebraic avatar of iterated integrals is the shuffle log 1 = 0. In the sequel we will only consider such hy- algebra perlogarithms f(z) = Lw(z) that allow for an analytic continuation to all of (0, ∞). This is necessary to give the ∞ R ∞ M M ⊗n integrals f(z) dz we want to compute a well-defined T (Σ) := Qw = T (Σ),T (Σ) := (QΣ) (2.1) 0 n n value. w∈Σ× n=0 spanned by all words over the alphabet Σ; some references 2.2. Integration and differentiation for this Hopf algebra are [29, 30]. It is graded by the × We consider the algebra L(Σ) := OΣ [Lw(z): w ∈ Σ ] weight n = |w| counting the number of letters in a word spanned by hyperlogarithms with rational prefactors whose (w = ω . . . ω ∈ Σn). Apart from the non-commutative σ1 σn denominators factor linearly with zeros in Σ only: concatenation product, it is equipped with the commutative shuffle product defined recursively by h 1 i O := Q z, : σ ∈ Σ . (2.6) Σ z − σ 0 0 0 (ωσw) ¡ (ωτ w ) := ωσ(w ¡ ωτ w ) + ωτ (ωσw ¡ w ) (2.2) 3 By construction we have ∂ L (z) = 1 L (z) Definition 2.1. For disjoint sets A, B ⊂ Σ the projection z ωσ1 ...ωσn z−σ ωσ2 ...ωσn 1 B such that L(Σ) is closed under ∂z, while for any f ∈ L(Σ) regA : T (Σ) −→ T (Σ) is determined by the requirements + we can find primitives F ∈ L (Σ), ∂zF (z) = f(z), in the B ¡ 0 B ¡ B 0 0 × + + × 1. regA(w w ) = regA(w) regA(w ) ∀w, w ∈ Σ , enlarged algebra L (Σ) := OΣ [{Lw : w ∈ Σ }] where B 2. regA(w) = w = ωσ1 . . . ωσn if σ1 ∈/ B and σn ∈/ A, 1 3. regB(w) = 0 for all 1 6= w ∈ A× ∪ B×. O+ := O Σ ∪ : σ, τ ∈ Σ and σ 6= τ . (2.7) A Σ Σ σ − τ τ {τ} We write regσ := reg{σ} and suppress empty sets in the Namely, a primitive for g(z)Lw(z) can be constructed by notation, e.g. reg = reg∅ and regτ = regτ . partial fractioning the rational prefactor σ σ ∅ This shuffle-regularization is a combinatorial operation X X Aσ,n X g(z) = + A zn ∈ O , (2.8) that projects onto words that neither begin with a letter (z − σ)n n Σ σ∈Σ n∈N n∈N0 in A nor end with a letter from B. Every word w ∈ T (Σ) decomposes uniquely as Lw(z) setting F = Lωσ w(z) as a primitive of z−σ and repeated use of the partial integration formulae X X ¡ ¡ (a,b) w = a b wA,B (2.14) Z dz L (z) L (z) Z dz ∂ L (z) a∈A× b∈B× w = − w + z w , (2.9) (z − σ)n+1 n(z − σ)n n(z − σ)n (a,b) B Z n+1 Z n into such A-B-regularized words wA,B ∈ im regA and thus n z · Lw(z) dz z (1,1) dz z Lw(z) = − ∂zLw(z) regB(w) = w . To compute (2.14) we can use n + 1 n + 1 A A,B (2.10) Lemma 2.2. For w = uωσa with a = ωa1 . . . ωan , to reduce the problem of finding a primitive to the case n Lωσ ...ωσ (z) X where the hyperlogarithm ∂ L (z) = 2 n ¡ ¡ z ωσ1 ...ωσn z−σ w = [u (−ωai ) ... (−ωa1 )] ωσ ωai+1 . . . ωan . (2.15) is of lower weight. This recursion terminates when w be- i=0 comes the empty word. Hence computation of a conver- × gent integral R ∞ f(z)dz for f ∈ L(Σ) reduces to obtaining So when σ∈ / A is the last letter of w not in A, thus a ∈ A , 0 n + we deduce reg (w) = (−1) (u ¡ ω . . . ω ) ω . a primitive F ∈ L (Σ) of f as described and evaluating A an a1 σ B ¡ × the limits Analogously reg (bωσu) = ωσ (u S(b)) for b ∈ B ∞ and σ∈ / B, setting S(b1 . . . bk) := (−bk) ... (−b1). Z B B B f(z) dz = lim F (z) − lim F (z). (2.11) Finally note regA = regA ◦ reg = reg ◦ regA. 0 z→∞ z→0 P i ¡ For A = {0} and B = ∅, (2.14) reads w = i ω0 wi 2.3. Divergences and logarithmic regularization where wi do not end in ω0. Since Lwi (z) is holomorphic The singularities of L (z) at z → τ ∈ Σ ∪ {∞} are P i i w at z → 0 and Lw(z) = i log z · Lwi (z) reveal f0,w(z) = at worst logarithmic, namely for any w ∈ T (Σ) there is a Lwi (z) from (2.12), we can compute the limit decomposition × Reg Lw(z) = Lreg (w)(0) = 0 when w ∈ Σ \{1} . (2.16) |w| ( i z→0 0 X (i) log z, τ = ∞ Lw(z) = f (z) · (2.12) w,τ logi(z − τ), τ 6= ∞ i=0 In fact our definition (1.2) is deliberately tuned such that the empty word w = 1 7→ Lw(z) = 1 is the only word in (i) × with functions fw,τ (z) uniquely defined upon the require- Σ with non-vanishing Regz→0 Lw(z). ment of being holomorphic at z → τ; for t = ∞ this means P i (i) 1 Lemma 2.3. Let w = ω ¡ wi ∈ L(Σ) for reg (wi) = holomorphy of fw,∞ at z → 0. Note that Lw(z) is fi- i 0 0 z P logi z nite for z → τ∈ / {0, ∞} whenever w does not begin with wi not ending on ω0. Then Lw(z) = i i! Lwi (z) and the letter ωτ . Lwi (z) are holomorphic at z → 0 and their series expan- P n The regularized limits are defined for any τ as sion Lwi (z) = n≥0 anz can be directly computed (recursively) from the iterated integral representation: Starting (0) Reg Lw(z) := fw,τ (τ), (2.13) with the empty word L (z) = 1, let L (z) = P a zn. z→τ 1 w n≥0 n Then such that limz→τ Lw(z) = Reg Lw(z) whenever this z→τ ∞ limit is finite. The advantage is then that by linearity, X an L (z) = zn and for any σ ∈ Σ \{0} , (2.17) ω0w n X X n=1 lim f(z) = λw Reg Lw(z) for f(z) = λwLw(z) z→τ z→τ ∞ w∈Σ× w∈Σ× 1 X an L (z) = zn+m+1. (2.18) ωσ w −σ σm(n + m + 1) can be computed for each word w separately and is thus n,m=0 well suited for an implementation, even though the limits limz→τ Lw(z) might diverge individually. 4 For expansions up at infinity, we first introduce an in- 2.4. Regularized limits as hyperlogarithms termediary point u ∈ (0, ∞) to split up the integration When we follow (1.4), after taking the limits (2.11) using Chen’s lemma (2.5), and then let u → ∞: we will from (2.26) have a representation of the partial n z integral Fk in terms of expressions Lreg∞(w)(∞) that de- X Z 0 L (z) = Reg ω . . . ω · Reg L (u). pend on the next integration variable t := z implicitly w σ1 σk ωσ ...ωσn k+1 u→∞ u→∞ k+1 k=0 u through the letters in the word w. To proceed with the in- (2.19) tegration process, we must rewrite Fk as a hyperlogarithm in t. Definition 2.4. For a word w = ωσ1 . . . ωσn , let n So let w = ωσ1 . . . ωσn (σn 6= 0) with letters σi(t) de- ∞ X k−1 ¡ pending on a parameter t, then we can take the derivative reg (w) := (ωσk − ω−1) (−ω−1) ωσk+1 . . . ωσn k=1 ∂tLw(z) in the integrand of the iterated integral Lw. Par- (2.20) tial fractioning and partial integration suffice to prove denote the projection of T (Σ) on words beginning with dif- n−1 ferences (ω − ω ) that annihilates reg∞(ωn ) = 0 for X ∂t(σi(t) − σi+1(t)) σ −1 −1 ∂tLw(z) = L...6ω ...−...6ω ...(z) ∞ ∞ σ (t) − σ (t) σi+1 σi any n > 0. Further set reg0 := reg ◦ reg0. i=1 i i+1 If w = (ω − ω )w0 ∈ im(reg∞), then −∂ σ ∂ σ σ −1 0 + t 1 L (z) − t n L (z) ωσ2 ...ωσn ωσ1 ...ωσn−1 Z z 0 z − σ1 σn (1 + σ)dz 0 0 Lw(z) = 0 0 Lw (z ) (2.21) 0 (z − σ)(z + 1) where 6 ωσi means to delete the letter ωσi from w. Applying Regz→∞ and exploiting Regz→∞ ∂t = ∂t Regz→∞ yields reveals that Lw(∞) = Regz→∞ Lw(z) is finite as an ab- 0 solutely convergent integral since Lw0 (z ) grows at worst ∂ Reg L (z) = − [∂ ln σ (t)] · Reg L (z) (2.27) t w t n ...6ωσn logarithmically by (2.12). We therefore conclude z→∞ z→∞ n−1 Reg L (z) = L ∞ (∞) for any w ∈ T (Σ) (2.22) X w reg0 (w) z→∞ + [∂t ln(σi − σi+1)] · Reg L...6ωσ ...−...6ωσ ...(z). z→∞ i+1 i i=1 from Reg Lω−1 (z) = 0 = Reg Lω0 and z→∞ z→∞ We assume that σ1, . . . , σn ∈ Q(t) are rational, such that 0 ∞ ¡ 0 Q λτ 2 Lemma 2.5. For any w, w ∈ L(Σ), reg (w w ) = any σi(t)−σj(t) = c τ (t−τ) factors linearly and thus ∞ ¡ ∞ 0 P λτ reg (w) reg (w ) is multiplicative and for any word ∂t ln [σi(t) − σj(t)] = τ t−τ together with (2.27) prove w = ωσ1 . . . ωσn , the following identity holds: Lemma 2.7. For rational letters Σ = {σ1(t), . . . , σN (t)} ⊂ n Q(t)\(0, ∞) without positive constants (such that the limit X k ¡ ∞ w = ω−1 reg ωσk+1 . . . ωσn . (2.23) Regz→∞ Lw(z) is well-defined for general t) and w ∈ T (Σ), k=0 The second ingredient to compute (2.19) lies in Reg Lw(z) ∈ L(Σt)(t) ⊗ Reg Reg L(Σ)(t) (2.28) z→∞ t→0 z→∞ Lemma 2.6. For any Möbius transform f(z) = az+b , cz+d is itself a hyperlogarithm in t with algebraic letters df −1(z) dz dz = − (2.24) Y Q f −1(z) − σ z − f(σ) z − f(∞) Σt := zeros of σi(t) − σj(t) ⊂ . (2.29) ipower series 2with zeros τ ∈ Q in the algebraic closure and integer multiplici- 1 Z in z . This suffices to compute limz→∞ F (z) in (2.11). ties λ ∈ 5 2.5. Regularized limits of regularized limits at infinity Therefore we can conclude that indeed,

We now explain how to compute the regularized limits (0) X i (i) Reg fw,∞(∞) = Reg (α log t) · fw0,∞(∞) (2.32) symbolically, without a need for numeric evaluations t→0 t→0 (which are for example used in [17] as explained in its i (0) appendix D). The ideas we present in the first half of this = Reg fw0,∞(∞) = Reg Reg Lw0 (z). section were very recently also sketched in [16]. t→0 t→0 z→∞ 0 Note that the limits C of (2.32) are constant only with A suitable such rescaling ensures finiteness of all σi(t) respect to the variable t, while in our applications these at t → 0 and thus resolves problem I above. will in general still depend on further variables. × Example 2.10. Applying lemmata 2.9, 2.8, equation (2.22) Let w ∈ Σ be a word with letters Σ ⊂ Q(t) \ (0, ∞) z and a Möbius transformation 2.6 with f(z) = 1+z shows depending rationally on t. We can restrict to w = reg0(w) not ending with ω0 since Reg Lω (z) = 0. The sim- z→∞ 0 Reg Reg Lω−1ω−1/t (z) = Reg Reg Lω−tω−1 (z) plest possible case is t→0 z→∞ t→0 z→∞ Z 1 Q Lemma 2.8. When limt→0(Σ) ⊂ \ (0, ∞) is finite and = Reg Lω0ω−1 (z) = L(ω0−ω−1)ω−1 (∞) = − ω0ω1 = ζ2. z→∞ 0 w = ωσ1 . . . ωσn ends in a letter with limt→0(σn) 6= 0, then To address the issue II when σn(0) = 0, we make Reg Reg Lw(z) = Reg L lim w(z). (2.33) t→0 z→∞ z→∞ t→0 Definition 2.11. For any 0 6= σ(t) ∈ Q(t), the Laurent P∞ n Proof. Using (2.22) it suffices to investigate differences series σ(t) = n=N t an at t → 0 with aN 6= 0 defines a vanishing degree deg (σ) := N ∈ Z and a leading coef- w = (ωσ1 − ω−1)ωσ2 . . . ωσn and consider Lw(∞) as the t −N absolutely convergent integral (2.21) with integrand ficient leadt(σ) := aN = limt→0 σ(t) · t . For a word

w = ωσ1 . . . ωσn set degt(w) := min {degt(σi): 1 ≤ i ≤ n}. 1 + σ (t) 1 1 1 ··· . (2.34) Whenever the final letter of a word w = ωσ . . . ωσ is (z1 + 1)(z1 − σ1(t)) z2 − σ2(t) zn − σn(t) 1 n of smallest vanishing degree degt(σn) = degt(w), rescaling We can apply the theorem of dominated convergence to 0 − degt(w) 0 σk := σk · t ensures σn(0) = leadt(σn) 6= 0 and show limt→0 Lw(∞) = Llimt→0 w(∞), essentially because lemma 2.8 becomes applicable. Hence let us define the limiting integrand is absolutely integrable itself. So reg w := lim ω 0 . . . ω 0 if deg (w) = deg (ω ), (2.36) σ σn t t σn t→0 t→0 1 Reg Reg Lw(z) = Reg Lreg∞(w)(∞) = L lim reg∞(w)(∞) t→0 z→∞ t→0 t→0 e.g. regt→0(ω−1ω−1/t) = ω0ω−1 in example 2.10. 0 holds for all w as allowed in 2.8. Looking at (2.20), we But if degt(σn) > degt(w), the rescaled σn(t) will van- ∞ may swap reg and limt→0 since the latter just substitutes ish at t → 0. In this case let letters ωσ 7→ ωσ(0). Finally apply (2.22) again. k := max {i: degt(σi) = degt(w)} < n This naive method can fail for three different reasons: denote the last place in w with minimal vanishing degree. I Some σ(t) ∈ Σ diverges in the limit t → 0. P Using (2.15) we can rewrite w = wi ¡ ai such that II σn(0) = 0, because the limiting integrand (2.34) at z each wi ends in ωσk and ai is a suffix of ωσk+1 . . . ωσn , i.e. t = 0 is not integrable: R n−1 dzn diverges. 0 zn |ai| ≤ n − k < n. Applying this procedure recursively to III Some letter has a limit σ (0) ∈ (0, ∞) on the positive i each ai finally results in a representation real axis, wherefore (2.34) acquires a singularity at X ¡ ¡ zi = σi(0) inside the domain of integration. w = (wi,1 ... wi,ri ) (2.37) We consider our main contribution as the algorithm to i deal with these cases, which we present below. First note of w in the shuffle algebra into elements wi,j each ending in some σ with minimum vanishing degree deg (σ ) = Z i,j t i,j Lemma 2.9 (Scaling invariance). Given some α ∈ and deg (w ). Q 0 t i,j a word w = ωσ1 . . . ωσn in letters σi ∈ (t), let w := 0 α ωσ0 . . . ωσ0 for σ (t) := t · σi(t). Then Example 2.12. For w = ω−1ω−t this decomposition reads 1 n i ¡ w = ω−1 ω−t − ω−tω−1. So with Regz→∞ Lω−1 (z) = 0, Reg Reg Lw(z) = Reg Reg Lw0 (z). (2.35) t→0 z→∞ t→0 z→∞ Reg Reg Lω−1ω−t (z) = − Reg Lω0ω−1 (z). t→0 z→∞ z→∞ Proof. Since Reg L (z) = Reg L (z) we can z→∞ w z→∞ reg0(w) 0 00 −α Definition 2.13. For any alphabet Σ ⊂ Q(t) let restrict to σn 6= 0 and rescale z = z t in (1.2) to con- α clude that Lw(z) = Lw0 (zt ). In regard of the regulariza- leadt(Σ) := {0} ∪˙ {leadt(σ): σ ∈ Σ \ 0} . (2.38) tion (2.12) at τ = ∞, this shows that Further we denote by regt→0 : T (Σ) −→ T (leadt(Σ)) the (i) X j j−i α (j) α unique morphism of shuffle algebras that extends (2.36) f (z) = log (t ) · f 0 (zt ) . w,∞ i w ,∞ j≥i with regt→0(ω0) = 0. 6 iÊ t This combinatorial regularized limit of words is a pro- it − 1 3 jection. For w ∈ T (Σ) decomposed as (2.37), it is just 1+ t X ¡ ¡ reg w = reg wi,1 ... reg wi,ri . (2.39) t→0 t→0 t→0

i −1 Ê Putting together the lemmata 2.9 and 2.8 with the lin- 0 1 2 −t earity and multiplicativity of Regt→0, Regz→∞ and w 7→ 1 Lw(z) we can compute regularized limits combinatorially t using 2 − t Corollary 2.14. For an alphabet Σ ⊂ C(t) \ (0, ∞) such that leadt(Σ) ⊂ C \ (0, ∞), any w ∈ T (Σ) fulﬁlls Figure 1: This graph shows the limits of Σ in example 2.17 when t → 0 with positive real part and small positive imaginary part. Reg Reg L (z) = Reg L (z). (2.40) w regt→0(w)

t→0 z→∞ z→∞ iÊ Example 2.15. For a, b, c ∈ C\[0, ∞), the decomposition 1 of the form (2.37) for the word w = ω ω ω 2 is a bt ct Ê −1 0 2 w = ωa ¡ωbt ¡ωct2 −ωa ¡ωct2 ωbt −ωbtωa ¡ωct2 +ωct2 ωbtωa γ and shows Regt→0 Regz→∞ Lw(z) = Regz→∞ Lu(z) for ¡ ¡ ¡ ¡ Figure 2: The letters {1 + t, 2 − t} ⊂ Σ in example 2.17 induce a u = ωa ωb ωc − ωa ω0ωb − ω0ωa ωc + ω0ω0ωa. deformation of the real integration path [0, ∞) towards γ, which + avoids the positive limits in passing below Σ0 = {1} and above We still need to address problem III on our agenda: − Σ0 = {2}. What happens when some leadt(σk) ∈ (0, ∞) approaches a positive value? In this situation, Regt→0 Regz→∞ Lw(z) can have a discontinuity along t ∈ [0, ∞). For example, Now consider a word w = ωσ1 . . . ωσn with degt(σn) = deg (w). Only the letters Σ := {σ : deg (σ ) = deg (w)} z − t 1 t w k t k t 0 − degt(w) Reg Lωt (z) = Reg log = log = ln |t| − i arg(−t) play a role since after rescaling σk(t) = σk(t) · t by z→∞ z→∞ −t −t 0 lemma 2.9, all other letters have degt(σk) = degt(σk) − C deg (w) > 0 and therefore approach 0 ∈/ (0, ∞) in the is deﬁned only when t ∈ \[0, ∞); otherwise Lωt (z) is not t limit t → 0 of (2.33). By homotopy invariance, well-deﬁned for real z ≥ t. So to make sense of Regt→0 H± C we must tie t ∈ := {z ∈ : ± Im z > 0} to either the Z upper or lower half-plane resulting in Reg Reg Lw(z) = reg(w) (2.44) t→0+iε z→∞ γ t→0

Reg Reg Lωt (z) = −iπ and Reg Reg Lωt (z) = iπ. t→0+iε z→∞ t→0−iε z→∞ is the iterated integral along a smooth deformation γ of the originally real integration contour [0, ∞). It avoids Definition 2.16. Choosing t ∈ H+ partitions the alphabet the positive letters among leadt(Σ) as follows: + − C Σ = Σe ∪˙ Σ ∪˙ Σ ⊂ (t) \ (0, ∞) (2.41) − degt(w) < 0: Σw\Σe ⊂ Σ , γ passes above all leadt (Σw), into non-positive leadt(Σ)e ∩ (0, ∞) = ∅ and letters with + ± ± ± degt(w) > 0: Σw\Σe ⊂ Σ , γ passes below all leadt (Σw), leadt(Σ ) ⊂ (0, ∞). These are separated by Σ ⊂ H for sufficiently small Re t and infinitesimal Im t > 0. In − + degt(w) = 0: limt→0(Σw \ Σ)e ⊂ Σ0 ∪ Σ0 and γ passes particular we note that whenever leadt(σ) ∈ (0, ∞), − + above Σ0 and below Σ0 as illustrated in figure 2 for example 2.17. We must require that Σ+ ∩ Σ− = ∅ as deg (σ) < 0 ⇒ σ ∈ Σ− and deg (σ) > 0 ⇒ σ ∈ Σ+. 0 0 t t otherwise γ is pinched between letters from Σ+ and (2.42) Σ in the limit t → 0. This situation did not occur We denote the finite positive limits by − in our applications but could be incorporated in the ± n ± o future. Σ0 := lim σ(t): σ ∈ Σ and degt(σ) = 0 . (2.43) t→0 In order to keep the implementation simple, we express 1 (2.44) again in the form Regz→∞ Lv(z) with v not contain- Example 2.17. For Σ = −1 + it, −t, 3t, 1 + t, 2 − t, t , the limit t → 0 + iε is shown in figure 1 and (2.41) reads ing any positive letters (such that Lv(z) is single-valued on z ∈ (0, ∞) and does not need additional specification of the 1 contour γ). This is achieved by splitting up the contour Σe = {−1 + it, 0, −t} , Σ+ = {1 + t, 3t} , Σ− = 2 − t, . t γ = ηu ? γu at u > 0 with the straight path ηu from 0 to u.

7 iÊ 3. The implementation HyperInt ηu u1 z 3.1. General remarks

We implemented the algorithms of section 2 in the Ê 0 2 computer algebra system Maple [22]. Even though these γu procedures are very flexible, we did not intend to provide a general purpose package supporting arbitrary symbolic calculations with hyper- and polylogarithms. Instead, we were driven by our aim to compute Feyn- Figure 3: The contour γ of example 2.17 shown in figure 2 is homo- topic to the splitting ηu ? γu. man integrals as we comment on in section 5. Therefore other applications are not as well supported, but we will give examples showing how HyperInt can be used for quite So for w = ωσ1 . . . ωσn with σn 6= 0, Chen’s lemma (2.5) general calculations with polylogarithms. takes the form Note that we did not include facilities for numeric eval- n uations of hyper- and polylogarithms, because first of all Z X Z w = ωσ1 . . . ωσk · Lσk+1,...,σn (u) (2.45) these are not necessary for the algorithms and secondly γ k=0 γu there are already established programs available for this task, e.g. [32, 33]. where u < τ := min ({σ , . . . , σ } ∩ (0, ∞)) approaches 1 n The program uses the remember option of Maple, which the first potential branch point of L (z) on the positive w creates lookup tables to avoid recomputations of func- real axis, see figure 3. In the limit u → τ we obtain tions. But some of these functions depend on global pa- × rameters as explained for instance in section 3.6. There- Lemma 2.18. Let w = ωσ1 . . . ωσn ∈ Σ with σn 6= 0 and denote by τ := min (Σ ∩ (0, ∞)) the first (smallest) fore, whenever such a parameter is changed, the function potential branch point of Lw(z) on the positive real axis. forgetAll() must be called to invalidate those lookup ta- The analytic continuation past τ < z = γ(1) along γ is bles. Otherwise the program might behave inconsistently.

n X Z z 3.2. Installation and files Lw(z) = (ωσ . . . ωσ ) · L τ (τ) (2.46) 1 k reg (ωσk+1 ...ωσn ) k=0 τ The program requires no installation. It is enough to load it during a Maple-session by invoking R z where τ denotes the iterated integral (2.4) whenever σk 6= > read "HyperInt.mpl"; τ. It is extended to all words by imposition of R z(w ¡v) = τ if the file HyperInt.mpl is located in the current direc- R z(w) · R z(v) and letting γ determine the branch of τ τ tory or another place in the search paths of Maple. If can be found, it will be loaded automat- Z z Z z − τ periodLookups.m ω := ω = ±iπ + log for z > τ. (2.47) ically which is of great benefit as explained in section 3.4. τ τ τ τ γ All together, we supply the following main files: Example 2.19. Take the dilogarithm L (z) = − Li (z) ω0ω1 2 HyperInt.mpl and the path γ passing below τ = 1 shown in figure 3. Since Contains our implementation of the algorithms in reg1(ω ) = 0 and L (1) = −ζ , (2.46) reduces to 1 ω0ω1 2 section 2 as well as supplementary procedures to Z z Z z Z z Z z handle Feynman graphs and Feynman integrals. Li2(z)−ζ2 = − (ω0ω1) := − ω0 · ω1 + (ω1ω0). 1 1 1 1 periodLookups.m This table stores a reduction of multiple zeta values Inserting R z ω := R ω = iπ + log(z − 1)and z = 1 + z0, 1 1 γ 1 up to weight 12 to a (conjectured) basis and simi- 0 we obtain for z = z − 1 > 0 (z > 1) the representation larly for alternating Euler sums up to weight 8. It is not required to run the program, but necessary for Li (z) = ζ + L (z0) − (iπ + log z0)L (z0) 2 2 ω0ω−1 ω−1 efficient calculations involving high weights. Details 0 0 = ζ2 − Li2(−z ) − (iπ + log z ) log z. follow in section 3.4. R z In (2.46) we can resolve τ into explicit factors ±iπ Manual.mw R z from (2.47) as dictated by γ and the iterated integrals τ . This Maple worksheet explains the practical usage Recursive splitting of these at the next positive let- of HyperInt. In particular it includes plenty of ex- 0 ter τ := min ({σ1, . . . , σn}\ τ ∩ (0, ∞)) finally expresses plicit Feynman integral computations. Many expla- 0 R R τ 0 R τ 00 nations, details and comments are provided here. γ (w) in powers of ±iπ and integrals 0 (v ), τ (v ),... which are simply defined by the straight line integration HyperTests.mpl paths 0 → τ, τ → τ 0 and so on. Through Möbius transfor- A series of various tests of the program. Calling mations (2.25), these may all be transformed to 0 → ∞. 8 Maple with maple HyperTests.mpl has to run with- form = i out any error messages. Please report immediately if same as form = Hlog, but produces the notation errors occur. Note that these tests only work when periodLookups.m can be found by HyperInt. i[0, σn, . . . , σ1, z] := Hlog (z, [σ1, . . . , σn]) Due to the many different tests, including some Feyn- which is used in zeta_procedures [34]. The result man integrals, the reader might find it instructive to can then be evaluated numerically in that program, read this file. See also section Appendix A. e.g. using evalz (·).

3.3. Representation of polylogarithms and conversions Example 3.1. The dilogarithm Li2(z) has representations Internally, polylogarithms are represented as lists > convert(polylog(2,z), Hlog); − Hlog (1, [0, 1/z]) f = [[g1, [w1,1, . . . , w1,r1 ]], [g2, [w2,1, . . . , w2,r1 ]],...] (3.1) of pairs of rational prefactors gi and lists of words wi,j = > convert(polylog(2,z), HlogRegInf); reg0 (wi,j) not ending on ω0. These encode the function [[1, [[−1 + z, −1]]], [−1, [[−1, −1]]]] r X Yi ∞ Due to the many functional relations, a general polyloga- f = gi · Lreg (wi,j )(∞) (3.2) i j=1 rithm f(~z) has many different representations. In particular, the representation (3.2) is far from being unique. and we decided not to use (2.2) to combine those words It is therefore crucial to be able to express polyloga- ¡ into the linear combination jwi,j for two reasons: rithms in a basis in order to simplify results and to detect 1. Empirically, this expansion of shuffle products tends relations. As was demonstrated in [11], lemma 2.7 provides to increase the number of terms considerably. such a basis through 2. Our algorithm of section 2.5 to compute Regt→0 produces products of words with different sets of letters. Corollary 3.2. Let f(~z) = Regz→∞ Lw(z) for w ∈ L(Σ) C Mixing these letters due to a shuffle introduces spu- with rational letters Σ ⊂ (~z) and choose an order ~z = rious letters in following integration steps which we (z1, . . . , zn). Then there is a unique way to write want to avoid. X f(~z) = Lwi,1 (z1) · ... · Lwi,n (zn) · ci (3.4) To encode hyperlogarithms of a particular variable z, we i use a list notation without explicit products: as a linear combination of products of hyperlogarithms of X f = [[g1, w1], [g2, w2],...] := gi(z)Lwi (z). (3.3) words wi,j ∈ T (Σi) with letters in some algebraic alpha- i bets Σi ⊂ C(zi+1, . . . , zn), which may only depend on the These representations make the implementation of the al- following variables. The factors ci in (3.4) are constants gorithms of section 2 straightforward, but for easier, human- (with respect to ~z), namely readable input and output we allow the notations ci ∈ Reg ... Reg Reg L(Σ)(z). (3.5) zn→0 z1→0 z→∞ Hlog (z, [σ1, . . . , σr]) := Lσ1,...,σr (z) and

Mpl ([n1, . . . , nr], [z1, . . . , zr]) := Lin1,...,nr (z1, . . . , zr) Its implementation constitutes the essential function for hyperlogarithms (1.2) and multiple polylogarithms (1.1). fibrationBasis (f, [z1, . . . , zr],F ) , HyperInt extends the native function convert(f, form) to transform an expression f containing any of the functions which writes a polylogarithm f (preferably in the list notation (3.1), otherwise it will be converted ﬁrst) in the form {log, ln, polylog, dilog, Hlog, Mpl, Hpl} (3.4) with respect to the order ~z = [z1, . . . , zr] of variables into one of the possible target formats (when ~z is omitted, ~z = [] is used). If the optional table F is supplied, the result will be stored as F = c . form = HlogRegInf: [wi,1,...,wi,n] i transforms f into the list representation (3.1). Example 3.3. This function can be used to obtain functional relations between polylogarithms. For example, form ∈ {Hlog, Mpl}: expresses f in terms of L or Li, using (1.3). > fibrationBasis(polylog(2,1-z), [z]); > convert(%, Mpl); = form Hpl − Hlog (z, [1, 0]) + ζ translates hyperlogarithms Hlog(z, w) with words w ∈ 2 × {−1, 0, 1} into the compressed notation of harmonic − Mpl ([2] , [z]) + ln(z) Mpl ([1] , [z]) + ζ2

polylogarithms that was introduced in [6]. Concretely, reproduces the classic identity Li2(1 − z) = ζ2 − Li2(z) − (z) := L (z) where 0 := ω and for Hpln1,...,nr n1,...,nr 0 log z log(1−z). Similarly, we obtain the inversion relation ζ N n−1 1 1 5 2 3 7 2 any n ∈ , ±n := ∓ω0 ω±1. for Li5 − x = 120 ln x + 6 ln x + 10 ζ2 ln x + Li5(−x): 9 × > fibrationBasis(polylog(5, -1/x), [x]): for some cases of u ∈ 0, µ: µN = 1 with N-th roots of > convert(%, Mpl); unity µ, see [37]. 1 1 7 HyperInt can load lookup tables to benefit from such ζ ln(x)3 + ln(x)5 + Mpl ([5] , [−x]) + ζ2 ln(x) 6 2 120 10 2 relations and we supply the file periodLookups.m which As an example involving multiple variables, the five-term provides the reductions that were proven in the data mine relation of the dilogarithm is recovered as project [38] using standard relations. It includes multiple zeta values up to weight 12 and alternating Euler sums > polylog(2,x*y/(1-x)/(1-y))-polylog(2,x/(1-y) (u ∈ {−1, 0, 1}×) up to weight 8 in the notation )-polylog(2,y/(1-x)): > fibrationBasis(%, [x, y]); n n ζ := Li 1 ,... r , (3.7) n1,...,nr |n1|,...,|nr | Hlog (y, [0, 1]) + Hlog (x, [0, 1]) − Hlog (x, [1]) Hlog (y, [1]) |n1| |nr| Z Note that for more than one variable, each choice ~z with indices n1, . . . , nr ∈ \{0}, nr 6= 1. When u ∈ × × of order defines a different basis and a function may take {0, a, 2a} ∪{−a, 0, a} , Möbius transformations are used a much simpler form in one basis than in another. For to express Lu(1) in terms of alternating Euler sums and 1 log a. example, Li1,2(y, x) + Li1,2( y , xy) is just > f:=Mpl([1,2], [y,x])+Mpl([1,2], [1/y,y*x]): Example 3.4. HyperInt automatically attempts to load > fibrationBasis(f, [x,y]); periodLookups.m, but can run without it. With its help, Hlog (x, [0, 1/y, 1]) + Hlog (x, [0, 1, 1/y]) > fibrationBasis(Mpl([3], [1/2])); but in another basis takes the form 1 3 1 7 ln(2) − ln(2)ζ2 + ζ3 > fibrationBasis(f, [y,x]); 6 2 8 is reduced to MZV and ln 2. But if periodLookups.m is Hlog (y, [0, 1, 1/x]) + Hlog (y, [0, 1/x]) Hlog (x, [1]) not available, we obtain merely − Hlog (y, [0, 0, 1/x]) − Hlog (y, [0, 1]) Hlog (x, [1]) > fibrationBasis(Mpl([3], [1/2])); We like to emphasize that every order ~z defines a true basis 1 −ζ − ζ − ζ + ln(2)3 without relations. In particular this means that f = 0 −3 2,−1 1,−2 6 if and only if fibrationBasis(f, ~z) returns 0, no matter which order ~z was chosen. The user can define a different basis reduction or pro- Analytic continuation in a variable z is performed along vide bases for periods involving higher weights3, or addi- a straight path, therefore the result can be ambiguous tional letters. These must be defined as a table, when this line contains a point where the function is not analytic. In this case, an auxiliary variable zeroOnePeriods[u] := Lu(1), (3.8) ( +1 when z ∈ H+, and saved to a file f. To read it call loadPeriods(f). δz = (3.6) −1 when z ∈ H− Example 3.5. Polylogarithms Li~n(~z) at fourth roots of unity ~z ∈ {±1, ±i}|n| up to weight |n| ≤ 2, like will appear to distinguish the branches above and below the real axis. From example 2.19 consider > f := Mpl([1,1],[I,-1])+Mpl([1,1],[-1,I]): > fibrationBasis(f); > fibrationBasis(polylog(2, 1+z), [z]); Hlog (1, [−I,I]) + Hlog (1, [−1,I]) Iπδz Hlog (z, [−1]) − Hlog (z, [−1, 0]) + ζ2 are tabulated in periodLookups4thRoots.mpl in terms of ln 2, i, π and Catalan’s constant Im Li (i): 3.4. Periods 2 > loadPeriods("periodLookups4thRoots.mpl"): Our algorithms express constants like (3.5) through it- × > fibrationBasis(f); erated integrals Reg L (z) of words w ∈ Q with 0→∞ w 1 1 1 algebraic letters. These are transformed into iterated in- ζ + ln(2)2 − Iπ ln(2) + I Catalan 8 2 2 4 tegrals Lu(1) by u = zeroInfPeriod(w). Such special values of multiple polylogarithms satisfy a huge number of relations and it is clearly highly desirable to express them 3 in a basis over Q. For MZV and alternating sums, [38] provides reductions up to weights 22 and 12, respectively. The case u ∈ {0, 1}× of multiple zeta values (MZV) is by now perfectly understood on the motivic level [35], such that conjectural Q-bases are available at arbitrary weight and [36] even provides a reduction algorithm that was implemented in [34]. Similar results can also be found

10 3.5. Integration of hyperlogarithms 3.5.1. Singularities in the domain of integration The most important function provided by HyperInt is The integration (3.9) requires that f(z) ∈ L(Σ)(z) is a Z ∞ hyperlogarithm without any letters Σ+ := Σ ∩ (0, ∞) = ∅ integrationStep(f, z) := f(z) dz (3.9) inside the domain of integration, which ensures that f(z) 0 is analytic on (0, ∞). and computes the integral of a polylogarithm f, which Otherwise f(z) can have poles or branch points on Σ+ must be supplied in the form (3.1). First it explicitly and the integration is then performed along a deformed rewrites f(z) ∈ L(Σ)(z) following lemma 2.7 as a hyper- contour γ as discussed in section 2.5. The dependence on logarithm in z. Then a primitive F = integrate(f, z) γ (see figure 3) is encoded in the variables is constructed as explained in section 2.2 and finally ex- ( panded at the boundaries z → 0, ∞. +1 when γ passes below σ, δz,σ = (3.12) R ∞ Li1,1(−x/y,−y) −1 when γ passes above σ. Example 3.6. To compute 0 y(1+y) dy, type > convert(Mpl([1,1],[-x/y,-y])/y/(y+1), 1 Example 3.9. The integrand f(z) = 1−z2 has a simple HlogRegInf): integrationStep(%, y): pole at z → 1 and is not integrable over (0, ∞). Instead, > fibrationBasis(%, [x]); HyperInt computes the contour integrals ζ2 Hlog (x, [1]) + Hlog (x, [1, 0, 1]) − Hlog (x, [0, 0, 1]) > hyperInt(1/(1-z^2), z): fibrationBasis(%); A more convenient and flexible form is the function Warning, Contour was deformed to avoid potential singularities at {1}. hyperInt (f, [z1 = a1..b1, . . . , zr = ar..br]) 1 b " b # Z r Z 1 (3.10) − · Iπδz,1 := ··· f dz1 ··· dzr 2 ar a1 Note even when positive letters Σ+ occur, f(z) can be which computes multi-dimensional integrals by repeated analytic on (0, ∞) nonetheless. In this case the dependence application of (3.9) in the order z1, . . . , zr as specified. It on any δz,σ drops out in the result. automatically transforms the domains (ak, bk) of integration to (0, ∞) and furthermore, f can be given in any form ln(z) Example 3.10. The integrand f(z) = 1−z2 is analytic at that is understood by convert (·, HlogRegInf). z → 1 and thus on all of (0, ∞). It integrates to Example 3.7. A typical integral studied in the origin [10] > hyperInt(ln(z)/(1-z^2), z): of the algorithm is I2 of equation (8.6) therein: > fibrationBasis(%); > I2 := 1/(1-t1)/(t3-t1)/t2: Warning, Contour was deformed to avoid > hyperInt(I2, [t1=0..t2, t2=0..t3, t3=0..1]): potential singularities at {1}. > fibrationBasis(%); 3 − ζ2 2ζ3 2

Example 3.8. The “Ising-class” integrals En were deﬁned 3.5.2. Detection of divergences in [39]: For n ≥ 2 let u := Qk t , u := 1 and set k i=2 i 1 By default, the option _hyper_check_divergences =  2 true is activated and triggers, after each integration, a Z 1 Z 1 Y uj − uk test of convergence. The primitive F (z) is expanded as En := 2 dt2 ... dtn   . (3.11) 0 0 uj + uk 1≤j 0 or j < 0 are explic- are rational linear combinations of alternating Euler sums. itly checked to vanish F i, j = 0 using fibrationBasis; We included a simple procedure IsingE(n) to evaluate the limit z → ∞ is treated analogously. This method them in the attached manual. In particular we can conﬁrm is time-consuming and we recommend to deactivate this the conjecture on E5 made in [39]: option for any involved calculations, expecting that the > IsingE(5); convergence is granted by the problem at hand. 2 2ζ3 (−37 + 232 ln(2)) − 4ζ2 31 − 20 ln(2) + 64 ln (2) Example 3.11. An endpoint divergence at z → ∞ is de- 318 512 R ∞ ln z 2 2 4 tected for 0 1+z dz = limz→∞ Lω−1ω0 (z): − ζ2 + 42 − 992ζ1,−3 − 40 ln(2) + 464 ln (2) + ln (2) 5 3 > hyperInt(ln(z)/(1+z), z); For illustration further exact results for En up to n = 8 can be found in IsingE.mpl. Time- and memory-requirements Error, (in integrationStep) Divergence at z = of these computations are summarized in table 1. infinity of type ln(z)^2 11 n 1 2 3 4 5 6 7 8 time 10 ms 41 ms 52 ms 235 ms 2.0 s 40.6 s 29.3 min 28 h RAM 35 MiB 51 MiB 51 MiB 76 MiB 359 MiB 1.6 GiB 1.9 GiB 30 GiB

R TM Table 1: Resources consumed during computation of the Ising-type integrals En of (3.11) running on Intel Core i7-3770 CPU @ 3.40 GHz. The column with n = 1 (when En := 1) requires no actual computation and shows the time and memory needed to load periodLookups.m.

The expansions (3.13) are only performed up to i, j ≤ fails because factorization is initially only attempted over _hyper_max_pole_order (default value is 10). If higher the rationals K = Q. Instead we can allow for an al- order expansions are needed, an error is reported and this gebraic extension K = Q(R) by speciﬁcation of a set variable must be increased. R = _hyper_splitting_field of radicals: Note that the expansion (3.13) is only computed at > _hyper_splitting_field := {I}: the endpoints z → 0, ∞. Polar singularities inside (0, ∞) > integrationStep([[1/(1+z^2), []]], z); ∞ 1 1 are not detected, e.g. hyperInt (1−z)2 , z = 1−z = 1 > fibrationBasis(%); 0 calculates the integral along a contour evading z = 1 just 1 1 I, [[−I]] , − I, [[I]] as discussed in section 3.5.1. One can split the integration 2 2 1 k Z ∞ X Z τi+1 π f(z) dz = f(z) dz (3.14) 2 0 i=0 τi We can also go further and factorize over the full algebraic closure K = Q(~z) by setting _hyper_algebraic_roots := at such critical points Σ+ = {τ1 < . . . < τk} with τ0 := 0, true. Over K, all rational functions Q(~z) factor linearly τ := ∞ with the eﬀect that all singularities now lie at k+1 such that we can integrate any f ∈ Regt→∞ L(Σ)(t) as endpoints and will be properly analyzed by the program. long as we start with rational letters Σ ⊂ Q(~z). A problem arises if calculations involve periods for which This feature is to be considered experimental and only no basis reduction is known to HyperInt, because the van- applied in transformWord which implements lemma 2.7: ishing Fi,j = 0 of a potential divergence might not be de- Given an irreducible polynomial P ∈ Q[~z] and a distin- tected. One can then set _hyper_abort_on_divergence := guished variable z, the symbolic notation false to continue with the integration. All Fi,j of (3.13) X are stored in the table _hyper_divergences. ω := ω : P | = 0 (3.15) Root(P,z) z0 z=z0

Example 3.12. When periodLookups.m is not loaded, sums the letters corresponding to all the roots of P . > hyperInt(polylog(2,-1/z)*polylog(2,-z)/z,z); Example 3.13. A typical situation looks like this: Error, (in integrationStep) Divergence at z = infinity of type ln(z) > f,g:=Hlog(x,[-z,x+x^2]),Hlog(x,[x+x^2,-z]): > fibrationBasis(f+g, [x, z]); inadvertently ﬁnds a divergence. Namely, F1,0 of (3.13) is Error, (in linearFactors) z+x+x^2 does not > entries(_hyper_divergences, pairs); factor linearly in x 1 4 (z = ∞, ln (z)) = 4ζ1,3 + 2ζ2,2 − π To express f + g as a hyperlogarithm in x, the roots R = 36 n √ o (P, x) = − 1± 1−4z of P = z + x + x2 seem neces- and its vanishing corresponds to an identity of MZV. Root 2 sary. After allowing for such algebraic letters, we obtain: We like to remark that through this observation, the > _hyper_algebraic_roots := true: computation of an integral which is known to be ﬁnite in > fibrationBasis(f+g, [x, z]); fact implies some relations among periods. − Hlog (x, [−1, −z]) − Hlog (x, [−z, −1]) 3.6. Factorization of polynomials + Hlog (x, [−z, 0]) + Hlog (x, [0, −z])

Since we are working with hyperlogarithms through- Since this result actually does not involve ωR at all one out, it is crucial that all polynomials occurring in the cal- might wonder why it was necessary in the ﬁrst place. The culation factor linearly with respect to the integration vari- reason is that the individual contributions f and g indeed 4 able z. For example, need ωR. Only in their sum this letter drops out: > integrationStep([[1/(1+z^2), []]], z);

Error, (in partialFractions) 1+z^2 is not 4In this extremely simple example this is clear since by (2.5), x+z x linear in z f + g = Lω−z (x) · Lωx(x+1) (x) factorizes into log z · log 1+x . We thus see why our representation (3.1) is preferable to one where all products of words are multiplied out (as shuffles). 12 > alias(R = Root(z+x+x^2, x)): straightforward: Multiple instances of Maple can each com- > fibrationBasis(f, [x, z]); pute a different piece of an integral whose results can be Hlog (x, [R, −z]) + Hlog (x, [R, −1]) − Hlog (x, [R, 0]) added up afterwards. Some example scripts are provided and discussed in the manual. + Hlog (x, [−z, 0]) − Hlog (x, [−z, −1]) Also note that the product representation (3.1) inher- − Hlog (x, [−1, −z]) − Hlog (x, [0, −z]) ently allows for different representations of the same words, because a product can either be represented symbolically Note that further processing of functions with such al- or as the corresponding sum of shuffles. We argued that 5 gebraic letters is not supported by HyperInt, because shuffling out every product is not desirable, so a better their integrals are in general not hyperlogarithms anymore. solution could be to choose an order on the alphabet Σ, However, the case of example 3.13 occurs frequently where- which then gives rise to a polynomial basis of the shuffle fore the option _hyper_ignore_nonlinear_polynomials algebra T (Σ) in terms of Lyndon words [40]. (default value is false) is available to ignore all algebraic letters in the first place. That is, all words containing such a letter are immediately dropped when it is set to true. 4. Polynomial reduction and linear reducibility In the example above this gives the correct result for In order to compute multi-dimensional integrals (1.4) f+g, but will provoke false answers when fibrationBasis by iterated integration using the algorithms of section 2, is applied to f or g alone. Hence this option should only be we must require that for each k, the partial integral used when linear reducibility is granted; preferably using the methods of section 4. fk ∈ L (Σk)(zk+1) where Σk ⊂ C (zk+2, . . . , zn) (4.1)

3.7. Additional functions is a hyperlogarithm in the next integration variable zk+1. In the manual we describe some further procedures pro- The alphabet Σk is restricted to rational functions of the C vided by HyperInt (note that all algorithms of section 2 remaining variables, in particular Σk ⊂ (zk+2), because are were implemented), like the extension of the commands only then lemma 2.7 guarantees that its integral fk+1 ∈ diff and series to compute differentials and series ex- L ((Σk)z+2)(zk+2) is a hyperlogarithm in zk+2. pansions of hyperlogarithms. Definition 4.1. We call f0(~z) linearly reducible if for some ordering z1, . . . , zn of its variables, sets Σk exist such 3.8. Performance that (4.1) holds for all 0 ≤ k < n. During programming we focussed on correctness and we are aware of considerable room for improvement of the For illustration let us suppose we want to integrate efficiency of HyperInt. But we hope that our code and the 1 f (x, y, z) := (4.2) details provided in section 2 will inspire further, stream- 0 ((1 + x)2 + y)(y + z2) lined implementations, even outside the regime of com- over x and y. To integrate x, we must include in Σ the puter algebra systems. This is possible since apart from √ x the factorization of polynomials (which can be performed algebraic zeros −1 ± i y to get a hyperlogarithm f0(x) ∈ before the actual integration, see the next section), all op- L (Σx)(x) in x. But then the integral erations boil down to elementary manipulations of words √ Z ∞ arctan y (lists) and computations with rational functions. f0 dx = √ 2 (4.3) Ironically, often just decomposing into partial fractions 0 y(y + z ) becomes a severe bottleneck in practice, as was also noted is not a hyperlogarithm in y at all6. On the other hand, in [16]. This happens when an integrand contains denom- since f (y) ∈ L −(1 − x)2, −z2 (y) for letters rational inator factors to high powers or very large polynomials in 0 in x, integration of y results in a hyperlogarithm the numerator. We observed that Maple consumes a lot of main mem- Z ∞ log(1 + x) − log z ory, in very challenging calculations the demand grew be- f0 dy = 2 (4.4) 0 (x + 1 + z)(x + 1 − z) yond 100 GiB. Often this turns out to be the main limita- tion in practice. in x over letters {−1, −1 ± z}. So in the order z1 := y, Our program uses some functions that are not thread- z2 := x linear reducibility is given and we can integrate safe and can therefore not be parallelized automatically. Z ∞ Z ∞ Lω1ω0 (z) − Lω−1ω0 (z) However, since the integration procedure considers every dx dy f0 = , (4.5) hyperlogarithm individually, a manual parallelization is 0 0 z which is a harmonic polylogarithm in z. 5These are sometimes referred to as generalized harmonic polylogarithms with nonlinear weights. √ 6But it is a hyperlogarithm in t := y, so in this simple case a change of variables would help us out. 13 In principle, we can try to integrate f for some arbi- 0 We see that the results for S and S match with the trary order and verify, after each step, that Σ is rational {y} {x,y} k letters of (4.4) and (4.5), but S is not computed because (or otherwise abort and try a different order). But for- {x} S is not linear in x. tunately this is not necessary since there are means to ∅ analyse the singularities of the integrals fk in advance. Our implementation can use the knowledge of such re- Namely, polynomial reduction algorithms were presented ductions in two places (examples are given in the manual): in [11] and [12]. These compute, for each subset I ⊂ E := {z1, . . . , zn} of variables, a set SI ⊂ Q[E \ I] of irreducible • When a table S is supplied as the (optional) fourth polynomials that provide an upper bound of the Landau parameter to fibrationBasis, then all words wi,k varieties as introduced in [12]. In particular this means in (3.4) containing letters not in Σk of (4.6) are re- that if there exists an ordering z1, . . . , zn of the variables moved from the result. such that all p ∈ S are linear in z , for any 0 ≤ k < n ∞ Ik k+1 • In the first step of integrating R f dz, the inte- and I := {z , . . . , z }, then the linear reducibility (4.1) is 0 k 1 k grand f is rewritten as a hyperlogarithm in z using granted with the rational alphabets P transformWord(f, z) = w Lw(z) · cu. Setting [ Σk := {0} ∪ {zeros of p in w.r.t. zk+1} . (4.6) > _hyper_restrict_singularities := true:

p∈SIk > _hyper_allowed_singularities := S: Readers familiar with the symbol calculus will realize that ensures that any word w containing a letter that is the polynomials SIk provide an upper bound of the entries not a zero of some polynomial p(z) ∈ S is dropped. of the symbol of fk. Explicit examples of such reductions are worked out in [11, 12] and the appendix of [24]. 4.3. Spurious polynomials and changes of variables

Bear in mind that the sets SI only provide upper bounds 4.1. Performance on the alphabet. In course of our calculations we regularly A polynomial reduction can signiﬁcantly speed up com- observed that, with the number |I| of integrated variables putations of integrals (1.4): During the step when fk is increasing, more and more polynomials in SI tend to be rewritten as a hyperlogarithm in zk+1 following section 2.4, spurious. In extreme cases it happens that a reduction all words that contain a letter not in Σk can be dropped, contains surplus non-linear polynomials in every variable, since the knowledge of (4.6) proves that all such contri- while f0 actually is linearly reducible. butions must in total add up to zero (see example 3.13, But even when linear reducibility strictly fails, it is where the algebraic roots ωRoot(P,z) drop out for f + g). sometimes possible to change variables such that the in- Note that the dimension of the space of hyperloga- tegrand becomes linearly reducible in these new variables. rithms over an alphabet Σk grows exponentially with the We explain this in [24] using the example of a divergent, weight. Therefore, a polynomial reduction is absolutely massive four-point box integral. Similar transformations crucial for problems of high complexity and cutting down are also employed in [16] to calculate generating functions the number of polynomials in Σk is highly desirable. In of operator insertions into ﬁnite one-scale integrals. Also practice this means that after computation of a polyno- note the discussion [41] of alphabets containing square root mial reduction, one should look for a sequence z1, . . . , zn letters that are typical for applications in particle physics of variables not only ensuring that SIk are linear in zk+1, and can be rationalized through simple changes of vari- but also minimizing the number of zk+1-dependent poly- ables. nomials in SIk .

4.2. Implementation in HyperInt 5. Application to Feynman integrals HyperInt implements the compatibility graph method In section 2 we investigated hyperlogarithms on their [12] of polynomial reduction and provides it as the com- own, but the algorithms were originally developed in [11] mand cgReduction (L). The entries LI = [SI ,CI ] of the for the computation of Feynman integrals. Important re- table L are pairs of polynomials S and edges C ⊂ SI I I 2 sults on their linear reducibility (including counterexam- between them. ples) and the geometry of Feynman graph hypersurfaces Example 4.2. The reduction of the integrand (4.2) starts were obtained in [12]. In [23, 24] we successfully applied with the complete graph on the factors of its denominator: our implementation to compute many non-trivial exam- > S:={x^2+2*x+1+y,y+z^2}: L[{}]:=[S, {S}]: ples, including massless propagators up to six loops and > cgReduction(L): also divergent integrals depending on up to seven kine- 7 > L[{x}][1]; L[{y}][1]; L[{x,y}][1]; matic invariants. All results presented in these papers were computed using this prgram HyperInt. L{x}1 {x + 1, x + 1 + z, x + 1 − z} 7These can be downloaded from http://www.math.hu-berlin. {1 + z, z − 1} de/~panzer/. 14 Some further discussions on multi-scale and subdiver- 2 gent integrals in the parametric representation are also given in [42–44]. 1 2 We hope that our implementation will be particularly 6 useful for applications to particle physics. 5 7 Remark 5.1. Our method applies only to the small class 3 of linearly reducible graphs, which is a subset of those 1 5 Feynman graphs that can be evaluated in terms of poly- 8 logarithms. By now it is however well known that quan- 4 3 tum field theory exceeds this space of functions not only in the massive case [45–47], but also in massless integrals 4 [48]. Even in supersymmetric theories, elliptic integrals and generalizations have been identified, e.g. [49, 50]. Figure 4: Four-loop massless propagator of section 5.4. In [23] this one is called M3,6. Edges are labelled in black, vertices in red. 5.1. Parametric representation and ε-expansion The popular method of Schwinger parameters [51] ex- 5.3. Additional functions in HyperInt presses Feynman integrals Φ(G) associated to Feynman graphs G by In Appendix C.4 we list the most important functions that support the calculation of Feynman integrals. These Z ∞ ae−1 − sdd Y α dαe ϕ entail simple routines to construct the graph polynomials Φ(G) = Γ(sdd) e · · δ(1 − α ) D/2−sdd eN ψ and ϕ. 0 Γ(ae) ψ e∈E For divergent integrals, the parametric integrands in (5.1) the representation (5.1) can be divergent. Such a situation in D space-time dimensions. To each edge e ∈ E of demands partial integrations, which effectively implement the graph corresponds a Schwinger variable α , and the e the analytic (dimensional) regularization and produce a corresponding scalar propagator may be raised to some convergent integral representation in the end. This pro- power a . The superficial degree of divergence is sdd := e cedure is defined and exemplified in [24] and implemented P a − |G| · D for the loop number |G| of G. The two e∈E e 2 into HyperInt as described in the manual. graph polynomials ψ and ϕ are for example defined in [52], the δ-distribution freezes an arbitrary α . eN 5.4. Examples 5.2. ε-expansion Plenty of examples are provided in the Maple worksheet Manual.mw, wherefore we only present a very brief case of For calculations in dimensional regularization8, we set a four-loop massless propagator here. D = 4 − 2ε and also the edge powers a = A + εν are e e e First we define the graph of figure 4 by its edges E and ε-dependent and expanded near an integer A ∈ Z. As- e specify two external momenta of magnitude one entering suming that (5.1) is convergent9 for ε = 0, we can expand the graph at the vertices 1 and 3. The polynomials ψ and the integrand in ε and obtain each coefficient cn of the P n ϕ can be computed with Laurent series Φ(G) = cnε as period integrals n > E:=[[1,2],[2,3],[3,4],[4,1],[5,1],[5,2], Z ∞ (n) (n) Y dαe P · f [5,3],[5,4]]: c = Γ(sdd) · δ(1 − α ) (5.2) n (n) eN > psi:=graphPolynomial(E): 0 Γ(ae) Q e∈E > phi:=secondPolynomial(E, [[1,1], [3,1]]): where P (n),Q(n) ∈ Q[~α] denote polynomials and f (n) ∈ This graph has vertex-width three [12] and is therefore (n) linearly reducible. Still let us calculate a polynomial re- Q[~α, log ~α, log ϕ, log ψ]. In particular f ∈ L(Σe)(αe) is duction to verify this claim: a hyperlogarithm in αe whenever ϕ and ψ are linear in (n) αe. If f even turns out to be linearly reducible, we can > L:=table(): S:=irreducibles({phi,psi}): integrate it with HyperInt. > L[{}]:=[S, {S}]: cgReduction(L): Afterwards we can investigate the polynomial reduction (for example with the procedure reductionInfo(L)) and 8A definition in momentum space can be found in [53], while in the parametric representation it is immediate. find a linearly reducible sequence ~z of variables. We rec- 9This can always be arranged for with the help of preparatory ommend to always check this with partial integrations as was shown in [24]. > z:=[x[1],x[2],x[6],x[5],x[3],x[4],x[7],x [8]]: > checkIntegrationOrder(L, z[1..7]):

15 1. (x[1]): 2 polynomials, 2 dependent • Equation (7.99), repeated as (44) in appendix A.2.7: 9 2 3 2. (x[2]): 5 polynomials, 4 dependent The second term − 4 π log (ξ) of the last line must 3. (x[6]): 8 polynomials, 4 dependent 3 2 3 be replaced with − 4 π log (ξ). 4. (x[5]): 7 polynomials, 4 dependent 5. (x[3]): 6 polynomials, 6 dependent • Equation A.3.5. (9): The terms −2 Li3 (1/x)+2 Li3(1) 6. (x[4]): 4 polynomials, 3 dependent should read + Li3(1/x) − Li3(1) instead. 7. (x[7]): 1 polynomials, 1 dependent 1 • In equation (7.132), a factor 2 in front of the sec- Final polynomials: n 1 ond summand Dp=0 p {· · · } is missing (it is correctly {} given in 7.131). The integrand is assembled according to (5.1) which in • Equation (8.80): (1 − v) inside the argument of the this case is already convergent as-is. We expand to second fourth Li -summand must be replaced by (1 + v), so order in ε with 2 that after including the corrections mentioned in the > sdd := nops(E)-(1/2)*4*(4-2*epsilon): following paragraph, the correct identity reads > f := series(psi^(-2+epsilon+sdd)*phi^(-sdd), epsilon=0): (1 + v)w −(1 − v)w 0 = Li2 + Li2 > f:=add(coeff(f,epsilon,n)*epsilon^n,n=0..2): 1 + w 1 − w Now we integrate out all but the last Schwinger parameter (1 − v)w −(1 + v)w + Li2 + Li2 (A.1) > hyperInt(f, z[1..-2]): 1 + w 1 − w 2 2 and reduce the result into a basis of MZV: −(1 − v )w 1 2 1 + w − Li2 2 + log . > fibrationBasis(f)*z[-1]: 1 − w 2 1 − w > collect(%, epsilon); 2 −2 • Equation (16.46) of [57]: x must read x . 2 3 168 2 2 254ζ7 + 780ζ5 − 200ζ2ζ5 − 196ζ3 + 80ζ2 − ζ2 ζ3 ε π4 π4 5 • Equation (16.57) of [57]: 40 must read 30 . 80 2 3 Some tests are constructed by calculation of parametric + −28ζ3 + 140ζ5 + ζ2 ε + 20ζ5. 7 integrals with known results in terms of polylogarithms Examples containing more external momenta, massive prop- and MZV. We used the expansion of Euler’s beta function agators and also divergences are included in Manual.mw. in the form ∞ Acknowledgments. I thank Francis Brown for his beautiful P ζ(n) n n n exp n (x + y − (x + y) ) Z ∞ −x articles and Dirk Kreimer for continuous encouragement. n=2 z dz = 2−x−y Oliver Schnetz kept me interested into graphical functions 1 − x − y 0 (1 + z) and kindly verified many of my computations with his and also checked the identity (z ≥ 0) very own methods, thereby providing a strong cross-check. Also Johannes Henn provided some ε-expansions to me Z ∞ 1 1 1 z − Li (−x − z) − Li − dx for tests, and he motivated the study of divergent inte- x x + z n x n x + 1 grals in the parametric representation. Many discussions 0 with Christian Bogner of concrete examples and problems = n Lin+1(−z), (A.2) greatly improved my understanding of iterated integrals. which is easily derived inductively for any n ∈ N. Figures were generated with JaxoDraw [56]. The two families of “bubble chain graphs” shown in figure A.5 can be calculated with standard techniques in Appendix A. Tests of the implementation momentum-space. Following the forest formula, we get ∂ We extensively tested our implementation with a va- : P (Bn,m) = 2 ΦR (Bn,m) = (n + m)! (A.3) riety of examples. Most of these are supplied in the file ∂q q2=1 HyperTests.mpl which must run without any errors. Since for the derivative of the Feynman integrals ΦR renormal- it contains many diverse applications of HyperInt, it might 2 also be useful as a supplement to the manual. ized by subtraction at external momentum q = 1. The Plenty of functional and integral equations of polylog- second family has generating function arithms, taken from the books [3, 57], are checked with X xnym HyperInt. These tests revealed a few misprints in [3]: ln (1 − x − y) P Bˆ n!m! n,m n,m≥0 • Equation (7.93): − 9 π2 log2(ξ) must be − 9 π2 log2(ξ). 4 12 X 2ζ = 2r+1 x2r+1 + y2r+1 − (x + y)2r+1 (A.4) 2r + 1 r≥1

16 n m Bn,m := Bˆn,m := n m

Figure A.5: Two series of one-scale graphs with subdivergences in four dimensions. They occur in φ4-theory as vertex graphs with two nullified external momenta, incident to the two three-valent vertices. and we used (A.3) and (A.4) to verify our results obtained Lemma 2.5. The statement is trivial for n = 1 and for from the parametric integral representations for these pe- n > 1 we apply (2.2) to (2.23) such that the right-hand riods derived in [58]. side becomes We furthermore tested some simple period integrals of n [10] and transformations of polylogarithms into hyperlog- X k ¡ i−k−1 ¡ (ωσi − ω−1) ω−1 (−ω−1) ωσi+1 . . . ωσn arithms given in [59]. Our results for the integrals En of 0≤k 0, the outer shuffle product in ν=0 the right-hand side of (2.15) decomposes with respect to which allow us to rewrite (∗) as the last letter into Z Z Z Z X µ τ ν a b regτ (uωτ ) · reg (ωτ v) · ωτ · ωτ . (n−1 ) γu ηu γu ηu X µ+ν+a+b=n [u ¡ (−ω ) ... (−ω )] ω ¡ ω . . . ω ω ai a1 σ ai+1 an−1 an The sum over a + b = n − µ − ν of the last two terms com- i=0 R n−µ−ν ( n ) bines to γ ωτ . Now the limit u → τ in the remaining ¡ X ¡ two factors is finite, such that (∗) then becomes + u (−ωai ) ... (−ωa1 ) ωai+1 . . . ωan ωσ. i=0 Z z Z z Z τ X µ n−µ−ν τ ν regτ (uωτ ) · ωτ · reg (ωτ v) The first contribution is uωσωa . . . ωa by the induction 1 n µ+ν≤n τ τ 0 hypothesis and the second contribution vanishes because it n z τ represents {u ¡ (S? id)(ω . . . ω )} ω for the antipode X Z Z a1 an σ = (uωµ) · regτ ωn−µv . S of the Hopf algebra T (Σ). τ τ µ=0 τ 0

17 R z R We used (B.1) again and the definition τ ωτ := γ ωτ . Appendix C.2. Maple functions extended by HyperInt convert(f, form) with form ∈ {Hlog, Mpl, HlogRegInf} Appendix C. List of functions and options provided Rewrites polylogarithms f in terms of hyper- or poly- by HyperInt logarithms using (1.3). Choosing form = HlogRegInf transforms f into the list representation (3.1). Appendix C.1. Options and global variables diff(f, z) (default: 1) _hyper_verbosity Computes the partial derivative ∂ f of hyperloga- The higher this integer, the more progress informa- t rithms Hlog (g(t), w(t)) or polylogarithms Mpl (~n,~z(t)) tion is printed during calculations. The value zero that occur in f. This works completely generally, i.e. means no such output at all. also when a word w(t) depends on t. _hyper_verbose_frequency (default: 10) series(f, z = 0) Sets how often progress output is produced during Implements the expansion of f = L (z) at z → 0. integration or polynomial reduction. w To expand at different points, use fibrationBasis _hyper_return_tables (default: false) first as explained in the manual. When true, integrationStep returns a table instead of a list. This is useful for huge calculations, Appendix C.3. Some new functions provided by HyperInt because Maple can not work with long lists. Note that there are further functions in the package, cf. the manual. _hyper_check_divergences (default: true) When active, endpoint singularities at z → 0, ∞ are integrationStep(f, z) R ∞ R ∞ detected in the computation of integrals 0 f(z) dz. Computes 0 f dz for f in the form (3.1).

_hyper_max_pole_order (default: 10) hyperInt(f, ~z) with a list ~z = [z1, . . . , zr] or single ~z = z1 R ∞ R ∞ Sets the maximum values of i and j in (3.13) for Computes 0 dzr ... 0 dz1f from right to left. Any which the functions fi,j are computed to check for variable can also specify the bounds zi = ai..bi to R bi potential divergences fi,j 6= 0. compute dzi instead. ai

_hyper_abort_on_divergence (default: true) fibrationBasis(f, [z1, . . . , zr],F,S) This option is useful when divergences are detected Rewrites f as an element of L(Σ1)(z1)⊗...⊗L(Σr)(zr). erroneously, as happens when periods occur for which Note that Σi ⊂ C(zi+1, . . . , zr) in general are alge- no basis is supplied to the program. braic functions of the following variables. A table F (with indexing function sparsereduced) can be sup- _hyper_divergences plied to store the result in compact form, otherwise A table collecting all divergences that occurred. Hlog-expressions are returned.

_hyper_splitting_field (default: ∅) For each defined key zi of S, the result is projected This set R of radicals defines the field k = Q(R) over S from L(Σi)(zi) onto L(Σi )(zi) restricting to letters which all factorizations are performed. S Σi := {zeros of p(zi): p ∈ Szi }. All words including other letters are dropped in the computation. _hyper_ignore_nonlinear_polynomials (default: false) Set to true, all non-linear polynomials (that would index/sparsereduced result in algebraic zeros as letters) will be dropped This indexing function corresponds to Maples sparse, during integration. This is permissible when linear but entries with value zero are removed. It is used reducibility is granted. to collect coefficients of hyperlogarithms.

_hyper_restrict_singularities (default: false) forgetAll() When true, the rewriting of f as a hyperlogarithm Clears cache tables for internal functions and should in z (performed during integration) projects onto the be called when options were changed. algebra L(Σ) of letters Σ speciﬁed by the roots of the set _hyper_allowed_singularities (default: ∅) of transformWord(w, t) irreducible polynomials. This can speed up the inte- Given a word w = [σ1, . . . , σn] as a list, this function gration. returns a list [[w1, u1],...] of pairs such that

_hyper_algebraic_roots (default: false) X Reg Lw(z) = Lwi (t) · Reg Lui (z) z→∞ z→∞ When true, all polynomials will be factored linearly i which can introduces algebraic functions. Further computations with such functions are not supported. and implements the algorithm of section 2.4. Note that ui is given in the product form (3.1). 18 reglimWord(w, t) findDivergences(f, P ) × Given a word w = [σ1, . . . , σn] ∈ Σ with rational For any pair J ∩ K = ∅ of disjoint sets of variables, C K letters Σ ⊂ (t), it implements our algorithm from the degree ωJ (f) of divergence when z → 0, ∞ (for section 2.5 to compute u (in the representation (3.1)) z ∈ J, K) is computed as defined in [24]. The result such that is a table indexed by the sets J ∪˙ K−1, holding the K values of ωJ (f) that are ≤ 0 when ε = 0. 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