aa r X i v : . [ h e p - ph ] N ov December 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 1
Mathematical aspects of scattering amplitudes
Claude Duhr
Center for Cosmology, Particle Physics and Phenomenology,Universit´e catholique de Louvain,2, Chemin du Cyclotron, 1348 Louvain-La-Neuve, Belgium,Email: [email protected]
In these lectures we discuss some of the mathematical structures thatappear when computing multi-loop Feynman integrals. We focus on aspecific class of special functions, the so-called multiple polylogarithms,and discuss introduce their Hopf algebra structure. We show how thesemathematical concepts are useful in physics by illustrating on severalexamples how these algebraic structures are useful to perform analyticcomputations of loop integrals, in particular to derive functional equa-tions among polylogarithms.
1. Introduction
The Standard Model of particles physics has been extremely successful indescribing experimental data at an unprecedented level of precision. Whencomputing predictions for physical observables in the Standard Model, orin any other quantum field theory, a key role is played by scattering am-plitudes, which, loosely speaking, encode the differential probability for acertain scattering process to happen. In perturbation theory scatteringamplitudes can be expanded into a sum over Feynman diagrams, whichat each order involve Feynman graphs with a fixed number of loops. Thevirtual particles inside the loops are unobservable, and so we need to in-tegrate over their momenta. The computation of perturbative scatteringamplitudes beyond tree level therefore necessarily involves the computationof loop integrals.Despite the importance of loop Feynman diagrams for precision predic-tions in quantum field theory, the computation of the corresponding loop ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 2 C. Duhr integrals is often still a bottleneck. The reasons for this are manifold. Forexample, loop integrals are Lorentz-invariant functions of the momenta ofthe external particles in the process, and so multi-leg amplitudes give riseto functions depending on a large number of variables. Moreover, thesefunctions will in general not be elementary functions (say, rational or alge-braic), but the functions have a complicated branch cut structure dictatedby unitarity and describing intermediate virtual particles going on shell.The main purpose of these lectures is to study loop integrals from apurely mathematical and algebraic point of view. To be more concrete, weconsider in these lectures scalar Feynman integrals of the form I = Z L Y j =1 e γ E ǫ d D k j iπ D/ N ( { p i , k j } )( q − m + i ν . . . ( q N − m N + i ν N , (1)where ν i ∈ Z are integers and m i ≥
0, 1 ≤ i ≤ N denote the massesof the propagators. We denote the loop momenta by k i , 1 ≤ i ≤ L andthe external momenta by p i , 1 ≤ i ≤ E . Note that momentum must beconserved, and in the following we always assume all external momentain-going, i.e., P Ei =1 p i = 0. The momenta flowing through the propagatorscan then be expressed in terms of the loop and external momenta, q i = L X j =1 α ij k j + E X j =1 β ij p j , α ij , β ij ∈ {− , , } . (2)In the following, we will not write the dependence of the propagators on the+ i N ( { p i , k j } )is a polynomial in the scalar products between loop and/or external mo-menta. We stress that by including numerator factors we also include tensorintegrals into the discussion (where all the Lorentz indices have been con-tracted with suitable external momenta).We work in dimensional regularisation in D = D − ǫ dimensions, D a positive integer. We will only consider the case D = 4, although mostof what we are going to say also applies to Feynman integrals in otherdimensions. It can be shown that I is a meromorphic function of ǫ , i.e., I has at most poles in the complex ǫ -plane (but no branch cuts!). We willonly be interested the Laurent expansion of I close to ǫ = 0, and our mainobjects of interest will be the coefficients in the Laurent series, I = X k ≥ k I k ǫ k = I k ǫ k + I k +1 ǫ k +1 + I k +2 ǫ k +2 + . . . , k ∈ Z . (3) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 3 If k < I is divergent in D = D dimensions. Note that we includea factor e γ E ǫ / ( iπ D/ ) per loop, where γ E = − Γ ′ (1) = 0 . . . . is theEuler-Mascheroni constant. The reason for including this normalisationfactor will become clear in the next section.Feynman integrals, like the one in Eq. (1), are the main topic of these lec-tures. More precisely, we will be concerned with the mathematical structureand the properties of the numbers and functions appearing in the analyticexpressions for the coefficients of the Laurent expansion (3) in dimensionalregularisation. The first trivial observation is that Feynman integrals canonly depend on Lorentz-invariant quantities, like the scalar products p i · p j .However, as already pointed out at the beginning of this section the coef-ficients I k in Eq. (3) may not be simple elementary functions, but rathercomplicated special functions. These special functions and their propertiesare the focus of these lectures. The main questions we will ask are: • Can any kind of number / function appear in analytic expressions forFeynman integrals? • What are the algebraic properties of these functions (functional equa-tions, branch cuts, basis,. . . )? • Can we make general statements about the algebraic and analytic prop-erties of the Laurent coefficients? • Can we turn these purely mathematical properties of the functions intoconcrete tools for Feynman integral computations?These lectures are organised as follows: In Sec. 2 we give a broad clas-sification of the kind of special numbers and functions that can appear inloop computations. In Sec. 3 we introduce the main actors of these lectures,the multiple polylogarithms, and we discuss some of their basic properties.In Sec. 4 and Sec. 5 represent the core of the lectures, and we discuss alge-braic and number-theoretical properties underlying these special functions.In Sec. 6 we give a flavour of how these concepts can be used in loop compu-tations. We include an appendix where we summarise some of the algebraicconcepts used throughout the lectures.
2. Transcendentality and periods
In this section we investigate which classes of numbers and functions canappear in the Laurent coefficients I k . As a warm up, let us look at the ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 4 C. Duhr following two simple one-loop examples: B ( p ) = e γ E ǫ Z d D kiπ D/ k ( k + p ) , (4) T ( p , p , p ) = e γ E ǫ Z d D kiπ D/ k ( k + p ) ( k + p + p ) . (5)The integrals are easy to compute, and we get B ( p ) = 1 ǫ + 2 − log( − p ) (6)+ ǫ (cid:20)
12 log ( − p ) − − p ) − ζ + 4 (cid:21) + O ( ǫ ) ,T ( p , p , p ) = 2 √ λ (cid:20) Li ( z ) − Li (¯ z ) − log( z ¯ z ) log 1 − z − ¯ z (cid:21) + O ( ǫ ) , (7)where the variables z and ¯ z are defined by p p = z ¯ z and p p = (1 − z )(1 − ¯ z ) , (8)and λ ≡ λ ( p , p , p ) denotes the K¨allen function, λ ( a, b, c ) = a + b + c − ab − ac − bc . (9)Let us look more closely at Eqs. (6) and (7): as anticipated, we see thatrational functions are insufficient to write down the answer. First, we seethe appearance of zeta values , i.e., the Riemann ζ function at integer values, ζ n = ∞ X k =1 k n , n > . (10)Note that for n = 1 the series diverges, and for even n , ζ n is proportionalto π n , ζ n = ( − n +1 B n (2 π ) n n )! , (11)where B n denote the Bernoulli numbers, B = 16 , B = − , B = 145 , . . . (12)Next, we see that the answer contains (powers of) logarithms log z = Z z dtt . (13) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 5 Moreover, we need generalisations of the logarithm, like the dilogarithm Li ( z ), or more generally, the classical polylogarithms , defined recursivelyby Li n ( z ) = Z z dtt Li n − ( t ) = ∞ X k =1 z k k n , (14)where the starting point of the recursion is the ordinary logarithm, Li ( z ) = − log(1 − z ). Note that the series converges for | z | <
1. Comparing Eq. (10)and Eq. (14), we see thatLi n (1) = ζ n , n > . (15)Moreover, we see from Eq. (7) that the arguments of the (poly)logarithm arenot simple (ratios of) scalar products, but they can be rather complicatedfunctions of the latter.We see from the previous examples that analytic results for Feynmanintegrals can easily get pretty involved, already for small numbers of loopsand external legs. It is therefore no surprise that for more complicatedexamples even more complicated functions may arise. In particular, thefunctions defined in Eq. (13) and Eq. (14) are functions of a single argu-ment. In more complicated cases also multi-variable generalisations of thelogarithm appear. These functions will be studied in detail in Sec. 3. Anatural question to ask is: does the complexity of the functions involvedgrow indefinitely, i.e., can every function a priori appear in some Laurentcoefficient of some Feynman integral? To be more concrete, we may askthe following questions:(1) Can arbitrarily complicated functions appear, e.g., trigonometric func-tions, exponentials, etc?(2) Can the arguments of these functions be arbitrarily complicated, e.g.,could log(log p ) appear?(3) The definition of the integrals in Eq. (4) and (5) involves the numbers e , γ E and log π (via the Taylor expansion of π − D/ ), but they do notappear in the results for the integrals. Is this an accident?In the rest of this section, we will give a complete answer to these questions(and the answers will be negative in all cases).We have already observed that the results for Feynman integrals containnumbers that are not rational (cf. ζ = π / ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 6 C. Duhr
Definition 1.
A complex number is called algebraic (over Q ) if it is theroot of some polynomial with rational coefficients. A complex number that is not algebraic is called transcendental (over Q ) .The set of all algebraic numbers is denoted by Q . Remarkably, the set ofall algebraic numbers forms a field , i.e., the inverse of an algebraic numberis algebraic, as well as the sum and the product of two algebraic numbers a .We can extend this notion from algebraic and transcendental numbers to functions : A function is algebraic if it is a root of a polynomial with coef-ficients that are rational functions in the variables. Example 1. (i) If q is a rational number, then q is also algebraic, because it is the rootof the polynomial z − q . In other words, we have Q ⊂ Q .(ii) Every n -th root of q ∈ Q is algebraic, because n √ q is a root of z n − q .(iii) In particular, all roots of unity are algebraic, including the imaginaryunit i . In other words, Q contains also complex numbers.(iv) The inverse of √ / √ √ / z − p x + y is an algebraic function because P ( x, y, p x + y ) = 0, with P ( x, y, z ) = x + y − z .We have seen examples of algebraic number, but can be also give exam-ples of transcendental numbers? It is easy to see that not every complexnumber can be algebraic. Indeed, the set of rational numbers is countable,and so there is a countable number of polynomials with rational coefficients.Since every polynomial has a finite number of roots, the set Q is countable,while the set of all complex numbers is not. In practise it is very difficultto prove that a complex number is transcendental. One of the main resultsabout transcendental numbers is the theorem of Hermite-Lindemann: Theorem 1 (Hermite-Lindemann).
Let z be a non-zero complex num-ber. Then either z or e z is transcendental. The theorem of Hermite-Lindemann allows one to prove that many numbersappearing in loop computations are in fact transcendental.
Example 2. (i) e is transcendental, because e = e , and 1 is algebraic. a See Appendix 8 for a review of various algebraic structures. ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 7 (ii) π is transcendental, because − e iπ and i are algebraic.(iii) π n , and thus ζ n , are transcendental for all n . Indeed, if π n was al-gebraic, then there would be a polynomial P ( z ) with rational coeffi-cients with P ( π n ) = 0. But then π would be a root of the polynomial Q ( z ) ≡ P ( z n ), which is excluded because π is transcendental.(iv) log q is transcendental for all q ∈ Q , because q = e log q is algebraic.Looking back at our examples (6) and (7), we see that the Laurent co-efficients indeed contain transcendental numbers. Note that the theoremof Hermite-Lindemann does not allow us to determine whether the diloga-rithm, and in general polylogarithms, are transcendental or not. They are,nevertheless, commonly assumed to be transcendental (see Sec. 3). If wespecialise to ζ values, then we have shown above that all even zetas aretranscendental (because they are proportional to π n ). For odd zetas, onlyvery few transcendentality results are known. In particular, the only oddzeta value that is proven to be irrational is ζ [1]. We can therefore at bestemit the following Conjecture 1.
All classical polylogarithms as well as all zeta values aretranscendental.
The division into algebraic and transcendental numbers are still toocrude to give concrete answers to the questions we asked ourselves at thebeginning of this section. It is possible to define a class of number thatlies ‘in between’ the algebraic and transcendental numbers. These are theso-called periods [2]:
Definition 2.
A complex number is a period if both its real and imaginaryparts can be written as integrals of an algebraic function with algebraiccoefficients over a domain defined by polynomial inequalities with algebraiccoefficients.
We will see in the example below that every algebraic number is a period,but not every period is algebraic. Moreover, not every transcendental num-ber is a period. Indeed, there is a countable number of periods (becausethey are defined using algebraic numbers, and there is only a countablenumber of those), but there is an uncountable number of transcendentalnumbers. In other words, there are ‘more’ transcendental numbers thanthere are periods. Moreover, it can be shown that periods form a ring , i.e.,sums and products of periods are still periods. Inverses of periods are ingeneral not periods. If we denote the ring of periods by P , then we have ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 8 C. Duhr the inclusion Q ⊂ Q ⊂ P ⊂ C . (16) Example 3. (i) Every algebraic number q is a period, because q = R q dx .(ii) The logarithm of an algebraic number q is a period, because log q = R q dtt .(iii) π = R x + y ≤ dx dy is a period.(iv) The dilogarithm (and similarly all polylogarithms and all zeta values)are periods for algebraic arguments, becauseLi ( z ) = Z ≤ t ≤ t ≤ z dt dt t (1 − t ) . (17)In fact, it turns out that most of the numbers ‘we know’ are periods, andit is rather difficult to prove that a number is not a period! Numbers thatare conjectured not to be periods are e , γ E , 1 /π , log π ,. . . .We can now state the main result of this section, which will give theanswers to all the questions at the beginning of this section: Theorem 2 (Bogner, Weinzierl [3]).
In the case where all scalar prod-uct p i · p j are negative or zero, all internal masses positive, and all ratios ofinvariants algebraic, the coefficients of the Laurent expansion of a Feynmanintegral are periods. The proof of the theorem is presented in ref. [3]. The idea of the proof is,loosely speaking, that every Feynman integral admits a Feynman parameterrepresentation, I = e Lγ E ǫ ( − ν Γ (cid:18) ν − L D (cid:19) (18) × Z N Y j =1 dx ′ j x ν j − j Γ( ν j ) ! δ − X j ∈ S x j U ν − ( L +1) D/ ( −F ) ν − LD/ , with ν = P Ni =1 ν i and S is any non-empty subset of { , . . . , n } , and U and F are homogenous polynomials in the Feynman parameters that arecompletely determined by the topology of the Feynman graph. The mainobservation is that after expansion in ǫ (by means of sector decompositionin the case of divergent integrals) Eq. (18) indeed defines order-by-order ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 9 an integral of a rational function over some domain defined by rationalinequalities, and thus a period. There is one caveat, however: Eq. (18) stillexplicitly depends on γ E , which is expected not to be a period. This factoris exactly cancelled by a similar factor coming from the ǫ expansion of theΓ function appearing in Eq. (18). Indeed, using the recursion for the Γfunction, Γ(1 + z ) = z Γ( z ), as well as the formulaΓ(1 + Lǫ ) = exp − Lγ E ǫ + ∞ X k =2 ǫ k ( − L ) k k ζ k ! , (19)it is easy to see that the factor exp( Lγ E ǫ ) cancels.It is easy to check that the theorem is true for the examples in Eq. (6)and (7). Moreover, the theorem allows us to answer the three questions weasked at the beginning of the section:(1) Trigonometric and exponential functions cannot appear in results forFeynman integrals, because e is (expected) not (to be) a period. Notethat inverse trigonometric functions are allowed!(2) The arguments of the polylogarithms should not be arbitrarily com-plicated: for example, log(log p ) would not be a period for algebraicvalues of p .(3) It is not a coincidence that the dependence on γ E and log π cancelled. Infact, this normalisation was introduced precisely to make the theoremtrue. Note that this normalisation factor is related the one absorbedinto the renormalised coupling constant in the MS-scheme.
3. Multiple polylogarithms
In the previous section we have seen that (the Laurent coefficients of)Feynman integrals evaluate to a restricted set of numbers and functionscalled periods, and we have given concrete examples of periods that appearin Feynman integral computations: zeta values and polylogarithms. Formulti-loop multi-leg integrals depending on many scales it is known thatmore complicated generalisations of the logarithm function may appear.In this section we define and study one of these generalisation (bearingin mind that this is not yet the end of the story!), the so-called multiplepolylogarithms. ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 10 C. Duhr
Definitions
Similar to the classical polylogarithms defined in Eq. (14), multiple poly-logarithms (MPLs) can be defined recursively, for n ≥
0, via the iteratedintegral [4, 5] G ( a , . . . , a n ; z ) = Z z dtt − a G ( a , . . . , a n ; t ) , (20)with G ( z ) = G (; z ) = 1 and with a i ∈ C are chosen constants and z is acomplex variable. In the following, we will also consider G ( a , . . . , a n ; z ) tobe functions of a , . . . , a n . In the special case where all the a i ’s are zero,we define, using the obvious vector notation ~a n = ( a, . . . , a | {z } n ), a ∈ C , G ( ~ n ; z ) = 1 n ! log n z , (21)consistent with the case n = 0 above. The vector ~a = ( a , . . . , a n ) iscalled the vector of singularities of the MPL and the number of elements n ,counted with multiplicities, in that vector is called the weight of the MPL.Note that the definition of MPLs makes it clear that they are periods (foralgebraic values of the arguments). In general, it is not known if they aretranscendental, but in the following we will always assume that they are.Equation (21) shows that MPLs contain the ordinary logarithm and theclassical polylogarithms as special cases. In particular, we have G ( ~a n ; z ) = 1 n ! log n (cid:16) − za (cid:17) and G ( ~ n − , z ) = − Li n ( z ) . (22)In the case where the a i ’s are constant, the above definition was alreadypresent in the works of Poincar´e, Kummer and Lappo-Danilevsky [6] as“hyperlogarithms”, as well as implicitly in the 1960’s in Chen’s work oniterated integrals [7]. Note that the notation for MPLs in the mathematicsliterature differs slightly from the one used in the physics literature, I ( a ; a , . . . , a n ; a n +1 ) = Z a n +1 a dtt − a n I ( a ; a , . . . , a n − ; t ) , (23)and I ( a ; ; a ) = 1. The functions defined by Eq. (20) and Eq. (23) arerelated by (note the reversal of the arguments) G ( a n , . . . , a ; a n +1 ) = I (0; a , . . . , a n ; a n +1 ) . (24)The iterated integrals defined in Eq. (23) are slightly more general than theones usually defined by physicists, as they allow us to freely choose the basepoint of the integration. It is nevertheless easy to convert every integral ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 11 with a generic base point a into a combination of iterated integrals withbase point 0. Example 4.
It is easy to see that at weight one we have I ( a ; a ; a ) = I (0; a ; a ) − I (0; a ; a ) = G ( a ; a ) − G ( a ; a ) . (25)Starting from weight two the relation is more complicated because theintegrations are nested, e.g., I ( a ; a , a ; a ) = Z a a dtt − a I ( a ; a ; t )= Z a a dtt − a [ I (0; a ; t ) − I (0; a ; a )]= I (0; a , a ; a ) − I (0; a , a ; a ) (26) − I (0; a ; a )[ I (0; a ; a ) − I (0; a ; a )]= G ( a , a ; a ) − G ( a , a ; a ) − G ( a ; a )[ G ( a ; a ) − G ( a ; a )] . In Eq. (14) we gave two definitions for the classical polylogarithms: arecursive integral definition and a series definition, and the MPLs so faronly generalise the integral definition. There is also a way to generalise theseries definition [4]:Li m ,...,m k ( z , . . . , z k ) = X
Basic properties of MPLs
In this section we discuss some basic properties of MPLs. A large collectionsof properties (including elementary proofs) can be found in ref. [5].It can easily be checked from the integral representation (20) ofMPLs that G ( a , . . . , a n ; z ) is divergent whenever z = a . Similarly, G ( a , . . . , a n ; z ) is analytic at z = 0 whenever a n = 0. Note that thisis consistent with the series representation, Eq. (27).If we consider the a i ’s constant, then, due to the singularities at z = a i in the integral representation, multiple polylogarithm will have a verycomplicated branch cut structure. In particular, G ( a , . . . , a n ; z ) has branchcuts in the complex z at most extending from z = a i to z = ∞ . Note thatif the a i ’s are allowed to vary, the branch cut structure becomes much morecomplicated. Example 5. (i) G ( ~a n ; z ) = n ! log n (cid:0) − za (cid:1) has a single branch cut in the complex z plane, extending from z = a to z = ∞ .(ii) G (0 , z ) = − Li ( z ) has a branch cut extending in the complex z planefrom z = 1 to z = ∞ . The branch cut starting at z = 0 is absent inthis case.If the (rightmost) index a n of ~a is non-zero, then the function G ( ~a ; x )is invariant under a rescaling of all its arguments, i.e., for any k ∈ C ∗ wehave G ( k ~a ; k z ) = G ( ~a ; z ) , a n = 0 . (29)Furthermore, multiple polylogarithms satisfy the H¨older convolution [8],i.e., whenever a = 1 and a n = 0, we have, ∀ p ∈ C ∗ , G ( a , . . . , a n ; 1)= n X k =0 ( − k G (cid:18) − a k , . . . , − a ; 1 − p (cid:19) G (cid:18) a k +1 , . . . , a n ; 1 p (cid:19) . (30)In the limiting case p → ∞ , this identity becomes, G ( a , . . . , a n ; 1) = ( − n G (1 − a n , . . . , − a ; 1) . (31)The previous examples make it clear that there are many relations amongMPLs. Such relations among special functions of the same type are called functional equations . While many functional equations among classicalpolylogarithms can be found in the literature, almost no results are known ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 13 for functional equations among MPLs. In Sec. 5 we present a way to derive,or rather to circumvent, functional equations among MPLs. The shuffle algebra
In this section we derive one of the main properties of MPLs (actually, ofiterated integrals in general), namely we will see that the product of twoMPLs defined with the same integration limits can be written as a linearcombination of MPLs.Let us illustrate this in detail on some example, and let us consider theproduct of two MPLs of weight one. Using the integral representation, wecan write G ( a ; z ) G ( b ; z ) = Z z dt t − a Z z dt t − b = Z Z (cid:3) dt dt ( t − a ) ( t − b ) , (32)where in the last step we used Fubini’s theorem to combine the two inte-grals into a double integral over the square with corners (0 , , z ), ( z, z, z ). We can split the square along the diagonal into a sum of twotriangles, and we obtain, G ( a ; z ) G ( b ; z )= Z Z ≤ t ≤ t ≤ z dt dt ( t − a ) ( t − b ) + Z Z ≤ t ≤ t ≤ z dt dt ( t − a ) ( t − b )= Z z dt t − a Z t dt t − b + Z z dt t − b Z t dt t − a = G ( a, b ; z ) + G ( b, a ; z ) . (33)We see that the product of two MPLs of weight one becomes a linear com-bination of MPLs of weight two. We can repeat exactly the same argumentfor MPLs of higher weights: we interpret the product of the two integralsas an integral over a hypercube, and recursively split along the diagonalsto decompose the integral over the hypercube into iterated integrals. Notethat, just like in the example above, it is important that the limits of in-tegration are the same. In the end, we see that a product of MPLs withweights n and n can always be written as a sum of MPLs with weight n + n , G ( a , . . . , a n ; z ) G ( a n +1 , . . . , a n + n ; z )= X σ ∈ Σ( n ,n ) G ( a σ (1) , . . . , a σ ( n + n ) ; z ) , (34) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 14 C. Duhr where Σ( n , n ) denotes the set of all shuffles of n + n elements, i.e., thesubset of the symmetric group S n + n defined by (cf. ref. [7])Σ( n , n ) = { σ ∈ S n + n | σ − (1) < . . . < σ − ( n )and σ − ( n + 1) < . . . < σ − ( n + n ) } , (35)i.e., the subset of S n + n that preserves the ordering inside the vectors( a , . . . , a n ) and ( a n +1 , . . . , a n + n ). This property turns the set of allMPLs into a shuffle algebra , i.e., a vector space equipped with the shufflemultiplication. Note that the shuffle product preserves the weight of theMPLs. We say in this case that the algebra is graded . Example 6. G ( a, b ; z ) G ( c ; z ) = G ( a, b, c ; z ) + G ( a, c, b ; z ) + G ( c, a, b ; z ) , (36) G ( a, b, c ; z ) G ( d ; z ) = G ( a, b, c, d ; z ) + G ( a, b, d, c ; z ) (37)+ G ( a, d, b, c ; z ) + G ( d, a, b, c ; z ) ,G ( a, b ; z ) G ( c, d ; z ) = G ( a, b, c, d ; z ) + G ( a, c, b, d ; z ) (38)+ G ( c, a, b, d ; z ) + G ( a, c, d, b ; z )+ G ( c, a, d, b ; z ) + G ( c, d, a, b ; z ) . Example 7.
In the previous section we have seen that G ( a , . . . , a n ; z ) isanalytic at z = 0, provided that a n = 0. If a n = 0, it is always possible touse the shuffle algebra to write G ( a , . . . , a n ; z ) in terms of functions whoserightmost index of the vector of singularities is non-zero (apart from objectsof the form G ( ~ n ; x )), e.g., if a = 0, G ( a, , z ) = G (0 , z ) G ( a ; z ) − G (0 , , a ; z ) − G (0 , a, z )= G (0 , z ) G ( a ; z ) − G (0 , , a ; z ) − [ G (0 , a ; z ) G (0; z ) − G (0 , , a ; z )]= G (0 , z ) G ( a ; z ) + G (0 , , a ; z ) − G (0 , a ; z ) G (0; z ) . (39) The stuffle algebra
In the previous section we showed that MPLs form a shuffle algebra. Thisalgebra structure is a consequence of the iterated integral definition (20).In this section we show that there is another algebra structure defined onMPLs, this time induced by the sum representation of MPLs as nestedsum, (27). ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 15 Let us consider the product of two MPLs of depth one. We getLi ( z ) Li ( z ) = ∞ X n =1 z n n ∞ X n =1 z n n = X n ,n ≥ z n z n n n = X n >n ≥ + X n = n ≥ + X n >n ≥ z n z n n n = Li , ( z , z ) + Li , ( z , z ) + Li ( z z ) . (40)Products of MPLs of higher depths can be handled in a similar way. Thealgebra generated in this way is called a stuffle algebra or quasi-shufflealgebra . Just like the shuffle product, the stuffle product preserves theweight. However, it does not preserve the depth, but rather the depth ofthe product is bounded by the sum of the depths. We talk in this caseof an algebra filtered by the depth. We emphasise that the stuffle algebrastructure is completely independent of the shuffle algebra. Example 8. Li m ,m ( z , z ) Li m ( z ) = Li m ,m ,m ( z , z , z ) (41)+Li m ,m ,m ( z , z , z ) + Li m ,m ,m ( z , z , z )+Li m ,m + m ( z , z z ) + Li m + m ,m ( z z , z ) , Li m ,m ,m ( z , z , z ) Li m ( z ) = Li m ,m ,m ,m ( z , z , z , z ) (42)+Li m ,m ,m ,m ( z , z , z , z ) + Li m ,m ,m ,m ( z , z , z , z )+Li m ,m ,m ,m ( z , z , z , z ) + Li m ,m ,m + m ( z , z , z z )+Li m ,m + m ,m ( z , z z , z ) + Li m + m ,m ,m ( z z , z , z ) , Li m ,m ( z , z ) Li m ,m ( z , z ) = Li m ,m ,m ,m ( z , z , z , z ) (43)+Li m ,m ,m ,m ( z , z , z , z ) + Li m ,m ,m ,m ( z , z , z , z )+Li m ,m ,m ,m ( z , z , z , z ) + Li m ,m ,m ,m ( z , z , z , z )+Li m ,m ,m ,m ( z , z , z , z ) + Li m + m ,m + m ( z z , z z )+Li m ,m ,m + m ( z , z , z z ) + Li m ,m + m ,m ( z , z z , z )+Li m ,m ,m + m ( z , z , z z ) + Li m ,m + m ,m ( z , z z , z )+Li m + m ,m ,m ( z z , z , z ) + Li m + m ,m ,m ( z z , z , z ) . ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 16 C. Duhr
Special instances of MPLs
Multiple polylogarithms are a very general class of functions that containmany other functions as special cases. In particular, there are several classesof special functions introduced by physicists in the context of specific Feyn-man integral computations that can be expressed through MPLs. In thissection we give a brief review of these functions, which commonly appearin loop computations. (1) Harmonic polylogarithms (HPLs) [9]:
HPLs correspond to thespecial case where a i ∈ {− , , } . For historical reasons, harmonicpolylogarithms only agree with MPLs up to a sign, and are denoted by H rather than G . The exact relation between HPLs and MPLs is H ( ~a ; z ) = ( − p G ( ~a ; z ) , (44)where p is the number of elements in the vector ~a equal to (+1). Be-cause of the importance of HPLs for phenomenology, they have beenimplemented into various computer codes that allow one to evaluateHPLs numerically in a fast and reliable way [10–14]. (2) Two-dimensional harmonic polylogarithms (2dHPLs) [15]: a i ∈ { , , − y, − − y } ,for y ∈ C . They appear in the computation of four-point functionswith three on-shell and one off shell leg [15–17], and can be evaluatednumerically using the techniques of ref. [13, 18]. (3) Generalized harmonic polylogarithms (GHPLs) [19]: GHPLsare defined as iterated integrals involving square roots of quadraticpolynomials as integration kernels, e.g., G ( − r, ~a ; z ) = Z z dt p t (4 + t ) G ( ~a ; t ) , (45)whenever the integral converges. These integrals appear in loop ampli-tudes that present a two-particle threshold at s = 4 m ( z = − s/m ).In ref. [20] it was shown that GHPLs can always be expressed in termsof MPLs via the change of variable z = (1 − ξ ) ξ , ξ = √ z − √ z √ z + √ z . (46)Letting t = (1 − η ) /η in Eq. (45), we find G ( − r, ~a ; z ) = − Z ξ dηη G (cid:18) ~a ; (1 − η ) η (cid:19) . (47) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 17 If we assume recursively that the G -function in the right-hand side canbe expressed through MPLs of the form G ( . . . ; η ), then it is easy to seethat the remaining integral will lead to MPLs. Example 9.
Let us consider G ( − r, − z ). Performing the change ofvariables (46), we get, G ( − r, − z ) = − Z ξ dηη G (cid:18) −
1; (1 − η ) η (cid:19) . (48)We have G (cid:18) −
1; (1 − η ) η (cid:19) = log (cid:18) − η ) η (cid:19) = log(1 − η + η ) − log η = log(1 − c η ) + log(1 − ¯ c η ) − log η = G (¯ c ; η ) + G ( c ; η ) − G (0; η ) , (49)where c = exp( iπ/
3) and ¯ c = exp( − iπ/
3) are two primitive sixth rootsof unity. So we get G ( − r, − z )= − G (0 , ¯ c ; ξ ) − G (0 , c ; ξ ) + G (0 , ξ ) + G (0 , ¯ c ; 1) + G (0 , c ; 1) . (50) (4) Cyclotomic harmonic polylogarithms (CHPLs) [21]: CHPLs area generalisation of HPLs defined by the iterated integrals C a~lb~m ( z ) = Z z dt f ab ( t ) C ~l~m ( t ) , (51)with f ( z ) = 1 z and f lm ( z ) = z l Φ m ( z ) , ≤ l ≤ ϕ ( m ) , (52)where φ ( m ) is Euler’s totient function and Φ m ( z ) denotes the m -thcyclotomic polynomial,Φ ( z ) = z − , Φ ( z ) = z + 1 , Φ ( z ) = z + z + 1 , . . . (53)By definition, the roots of cyclotomic polynomials are roots of unity.Thus, if we factor the cyclotomic polynomials and us partial fractioning,we can express all CHPLs in terms of MPLs where the a i ’s are roots ofunity. ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 18 C. Duhr
Example 10.
Let us consider C ( z ). we have C ( z ) = Z z dt Φ ( t ) = Z z dt − t + t = Z z dt ( t − c )( t − ¯ c )= 1 c − ¯ c [ G ( c ; z ) − G (¯ c ; z )] = − i √ G ( c ; z ) − G (¯ c ; z )] , (54)with c = exp( iπ/
3) and ¯ c = exp( − iπ/
4. Multiple zeta values4.1.
Definition of MZVs
In Eq. (15) we saw that there is a connection between classical polyloga-rithms and the values of the Riemann zeta function at positive integers.It is natural to ask how to generalise these relations to the more generalpolylogarithmic functions defined in Sec. 3.In this section we discuss multiple zeta values (MZVs) , a ‘multi-index’extension of the ordinary zeta values defined in Eq. (10).
Ordinary zetavalues are the values at 1 of classical polylogarithms, and so it is naturalto define MZVs as the values at 1 of MPLs,
Definition 3.
Let m , . . . , m k be positive integers. ζ m ,...,m k = Li m k ,...,m (1 , . . . ,
1) = X n >...>n k > n m . . . n m k k . (55)Note that if m = 1 in Eq. (55), then ζ ,m ,...,m k is divergent. In thefollowing we will only consider convergent series. The weight and the depthof an MZV are defined in the same way as for MPLs. The reason to studythese numbers (just like ordinary zeta values, MZV will be numbers, notfunctions!) is twofold: First, they are ubiquitous in both mathematics andin multi-loop computations, and so they deserve a deeper study. Second,they allow us to introduce some of the concepts and the way of thinkingthat we will use in subsequent sections, but in a simpler and more controlledframework. As such, this section also serves as a preparation for subsequentsections.Using Eq. (28), we see that MZVs also admit a definition in termsof iterated integrals, and the integral is convergent whenever the MZVis. Note that the existence of this integral representation implies that allMZVs are periods (‘integrals of rational functions’). In Sec. 2 we already ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 19 mentioned that, apart from ζ , it is not known if a given odd zeta valueis transcendental or not, and it is therefore not surprising that basicallynothing is known about the transcendentality of MZVs. In the followingwe will assume the ‘usual folklore’ that Conjecture 2.
All MZVs are transcendental.
Relations among MZVs
In Sec. 2 we have seen that all ordinary even zeta values are proportionalto powers of π , or in other words, all even ordinary zeta values are related, ζ n = c ζ n , for some c ∈ Q . The main question we will ask ourselves inthe rest of this section is whether there are more such relations amongMZVs. Actually, we already have at our disposal a machinery to generateinfinite numbers of relations! We know that MPLs satisfy shuffle and stufflerelations, and so by listing systematically all the shuffle and stuffle relationsamong (convergent) MZVs, we can generate lots of relations. Example 11. (i) Using the stuffle algebra, we can write ζ = Li (1) = 2Li , (1 ,
1) + Li (1) = 2 ζ , + ζ . (56)Similarly, using the shuffle algebra, ζ = G (0 ,
1; 1) = 4 G (0 , , ,
1; 1) + 2 G (0 , , ,
1; 1)= 4 ζ , + 2 ζ , . (57)(ii) Using the stuffle algebra, we can write ζ ζ = Li (1) Li (1) = Li , (1 ,
1) + Li , (1 ,
1) + Li (1)= ζ , + ζ , + ζ . (58)Similarly, using the shuffle algebra, ζ ζ = G (0 ,
1; 1) G (0 , ,
1; 1)= 6 G (0 , , , ,
1; 1) + 3 G (0 , , , ,
1; 1) + G (0 , , , ,
1; 1)= 6 ζ , + 3 ζ , + ζ , . (59)We see that, because the shuffle and stuffle products preserve the weight,we can only generate relations among MZVs of the same weight in thisway. Note that the first time we can generate shuffle or stuffle identities ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 20 C. Duhr is at weight four, because at lower weight all products involve divergentMZVs. There are, however, relations among MZVs of weight three ζ , = ζ . (60)In other words, there must be more exotic relations among MZVs than theshuffle and stuffle relations among convergent MZVs we have considered sofar. This brings up the following interesting questions:(1) Are there relations among MZVs of different weight?(2) Can we characterise all the relations that exist between MZVs?(3) Can we describe a ‘basis’ for MZVs, at least for fixed weight?Amazingly, all of these questions can be answered, at least at the level ofconjectures (that have been tested numerically to hundreds of digits forrather high weights).In order to formulate these conjectures, and also to prepare the groundfor the following sections, let us rephrase these questions in mathematicallanguage. First, we need to be slightly more precise and define what kind ofrelations we are looking for. In the following, we mean by ‘relations amongMZVs’ a relation of the type P ( Z , . . . , Z n ) = 0, where Z i are MZVs and P is a polynomial with rational coefficients. The first conjecture states that Conjecture 3.
There are no relations among MZVs of different weights.
This implies that all the terms in the polynomial P have the same weight.Note that our definition of ‘relation’ relies crucially on our Conjecture 2that all MZVs are transcendental! Indeed, suppose that there is an MZV Z of weight n that is a rational number. Then for any other MZV Z ofweight n we could write P ( Z , Z ) = 0, where P ( x, y ) = Z − xy − y = 0is a polynomial with rational coefficients. In other words, we would haveobtained a relation among MZVs of weight n + n and n .Let us now denote the vector space of all convergent MZVs of weight n > Z n . By definition, we put Z = Q and Z = { } (because thereare no convergent MZVs of weight one). Furthermore, we define the vectorspace of all MZVs to be the direct sum of all the Z n , Z = ∞ M n =0 Z n = Q ⊕ Z ⊕ Z ⊕ . . . . (61)This definition might look innocent at first glance, but it contains a lotof deep mathematical statements! In particular, it embodies already theconjectures 2 and 3. Indeed, the fact that the sum is direct implies that ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 21 Z m ∩ Z n = { } , for m = n . If Z ∈ Z n , n = 0, is rational, then Z ∈Z ∩ Z n = { } , and so Z = 0. In other words, there is no rational MZV.Similarly, assume that there is a relation between MZVs of different weights,say m and n . This means that there are elements Z ∈ Z m and Z ∈ Z n such that Z + Z = 0, and so Z = − Z . But then Z , Z ∈ Z m ∩Z n = { } ,and so Z = Z = 0.Next, note that Z is actually not only a vector space, but it is analgebra, because the MZVs can be equipped with a product (say, the shuffleproduct). We have already seen in Sec. 3 that the shuffle product preservesthe weight, and we called such an algebra graded . We can now formalise thisby saying that whenever Z ∈ Z m and Z ∈ Z n , we have Z Z ∈ Z m + n .We can now formulate the previous questions in our new language ofvector spaces:(1) What are the dimensions of the vector spaces Z n ?(2) Can we write down an explicit basis for each of the Z n ?We can answer these questions (at least conjecturally) using the celebrated double-shuffle conjecture . Loosely speaking, the conjecture states that, ifwe formally also include the divergent MZVs, then the only relations amongthe convergent
MZVs are those that can be obtained via shuffle and stuffleidentities.
Example 12.
If we formally include all divergent MZVs, we can write thefollowing stuffle relation at weight three: ζ ζ = Li (1) Li (1) = Li , (1 ,
1) + Li , (1 ,
1) + Li (1)= ζ , + ζ , + ζ . (62)Similarly, we can write the shuffle relation ζ ζ = G (1; 1) G (0 ,
1; 1) = G (1 , ,
1; 1) + 2 G (0 , ,
1; 1)= ζ , + 2 ζ , . (63)Note that these relations are purely formal, because both sides of the equal-ities are divergent. However, both in Eq. (62) and Eq. (63) the only diver-gent quantity in the right-hand side is ζ , . If we take the difference of thetwo equations, then all the divergent quantities cancel, and we are left witha concrete relation among convergent MZVs:0 = − ζ , + ζ , (64)i.e., we find obtained the ‘exotic’ relation (60). In other words, we haveobtained a relation between convergent MZVs as the difference between ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 22 C. Duhr (formal) shuffle and stuffle identities. Such a relation is called a regularisedshuffle relation . Conjecture 4.
The only relations among MZVs are shuffle, stuffle andregularised shuffle relations.
Example 13.
We have already derived shuffle and stuffle relations atweight four in Eq. (56) and (57). We can now add regularised shufflerelations. We start by writing formal stuffle relations ζ ζ = ζ , + ζ , + ζ , (65) ζ ζ , = ζ , , + 2 ζ , , + ζ , + ζ , , (66)and shuffle relations ζ ζ = ζ , + 2 ζ , + ζ , , (67) ζ ζ , = ζ , , + 3 ζ , , . (68)Taking the difference, we obtain two regularised identities among conver-gent MZVs of weight four,0 = − ζ , + ζ − ζ , , (69)0 = − ζ , , + ζ , + ζ , . (70)Combining these relations with Eq. (56) and (57), we have obtained fourrelations among MVZs of weight 4. The solution is ζ = ζ , , = 25 ζ , ζ , = 110 ζ , ζ , = 310 ζ , (71)i.e., all MZVs of weight four are proportional to π !The double-shuffle conjecture answers the two-questions we asked ear-lier, because we can, at least in principle, solve the double-shuffle relationsfor each weight, and in this way we can construct an explicit basis for each Z n . This has been done explicitly up to high weights in ref. [22]. Moreover,there is a conjecture about the dimensions d n = dim Q Z n : d = 0 , d = d = 1 , d k = d k − + d k − , k > . (72)In Table 1 we show an explicit basis of MZVs up to weight eight. Note thatthe first time a generic MZV can no longer be written as a polynomial inordinary zeta values is at weight eight. Moreover, one can give an explicitbasis for every weight [23, 24]: the MZVs of the form ζ m ,...,m k with m i ∈{ , } are expected to form a basis of all MZVs. ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 23 ζ ζ ζ ζ ζ , ζ ζ , ζ ζ ζ , ζ ζ , ζ ζ , ζ ζ , ζ ζ , ζ ,
5. The Hopf algebra of MPLs5.1.
Functional equations among MPLs
In the previous section we have seen that it is possible, at least at the levelof conjectures, to describe all the relations among MZVs, and to give acomplete basis of MZVs at each weight. In this section we will generalisethis idea to the framework of MPLs. It is clear that finding all the relationsamong MPLs is a monumental task, which is much more complicated thanin the case of MZVs. In particular, it is clear that for MPLs there must benew relations that go beyond shuffle and stuffle relations, because we nowhave to deal with functions rather than numbers, and so we will also needto take into account relations among MPLs with different arguments. Inthe rest of these lectures we refer to such relations as functional equations .The main question we will try to answer in this section is thus: Is there away to describe functional equations among MPLs?In order to get a feeling for functional equations, let us look at somesimple representatives: At weight one, there is precisely one fundamentalfunctional equation, namelylog( ab ) = log a + log b . (73)All other functional equations for the logarithm are just a consequence ofthis relation. At weight two, we have for example the following functionalequations for the dilogarithmLi (1 − z ) = − Li ( z ) − log(1 − z ) log z + ζ , (74)Li (cid:18) − z (cid:19) = − Li (1 − z ) −
12 log z , (75)as well as the five-term relationLi (cid:18) x − y (cid:19) + Li (cid:18) y − x (cid:19) − Li (cid:18) xy (1 − x )(1 − y ) (cid:19) = Li ( x ) + Li ( y ) + log(1 − x ) log(1 − y ) . (76)Note that these identities are only valid for specific values of the variables.Many more identities can be found in the literature (see, e.g., ref. [25, 26]). ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 24 C. Duhr
Functional equations like the ones above are not only interesting fromthe purely mathematical point of view, but they also play an importantrole when computing Feynman integrals. For example, polylogarithms havebranch cuts, and functional equations can be used to analytically continuethe functions. In the previous sections we argued that also MPLs appear inFeynman integral computations. It is then not surprising that functionalequations for MPLs are also needed in physics. Unfortunately, not manyexamples of functional equations for MPLs are known in the mathemat-ics literature. For this reason, there is a substantial literature in physicswhere functional equations for special classes of MPLs have been studied(cf. Sec. 3.5), e.g., ref. [9–13, 15, 18–21, 27, 28]. All of the methods pre-sented in these references are tailored to specific special classes of functions,and usually require the manipulation of the integral or series representa-tions of the functions (e.g., an identity derived via some change of variablesin the integral representation).The purpose of this section is to present a method that allows oneto derive functional equations among MPLs (or at least some classes offunctional equations). The main differences to the special cases consideredso far in the physics literature are(1) the method is completely generic and applies to arbitrary MPLs, andis not tailored to specific special classes of iterated integrals.(2) the method is completely algebraic and combinatorial in nature, andit is completely agnostic of the underlying integral or series represen-tations.We will use an algebraic framework similar to the one used for MZVsin the previous section. In particular, let us define A n as the vector spacespanned by all ‘polylogarithmic functions’ of weight n , and we put A = Q .Note that A n includes all MZVs of weight n , Z n ⊂ A n , but unlike in theMZV-case, A = { } , because A contains all ordinary logarithms. We alsodefine A to be the direct sum of the vector spaces A n , A = ∞ M n =0 A n . (77)Just like in the case of MZVs, this definition only makes sense if we assumethe following Conjecture 5.
All MPLs are transcendental functions, and there are norelations among MPLs of different weights. ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 25 Obviously, A is an algebra, given by the multiplication of functions (cf.the shuffle and stuffle products), and we know already that this algebrais graded by the weight. Unlike the MZV-case, where all relations aregiven by shuffle and stuffle identities, there are much more complicatedrelations among MPLs, and those relations cannot be recovered from shuffleand stuffle relations alone. In particular, all the functional equations thatchange the arguments of the functions cannot be covered by double-shuffles.We therefore need a much more general and flexible algebraic framework ifwe want to find all the relations among MPLs of a given weight. Coalgebras and Hopf algebras
In this section we briefly review the algebraic concepts that we will needto formulate our framework. We will not give a detailed account of all themathematical definitions, and content ourselves to give the basics that areneeded to follow the discussions in the remaining sections. More detaileddefinitions can be found in Appendix 8.We have already seen that A is an algebra, i.e., a vector space with amultiplication that has a unit element and is associative, ( ab ) c = a ( bc ),and distributive, a ( b + c ) = ab + ac and ( a + b ) c = ac + bc . In particular,there is a map, the multiplication, which assigns to a pair of elements ( a, b )their product ab . It will be useful to see the multiplication as a map µ from A ⊗ A to A , and the pair ( a, b ) will be denoted by a ⊗ b . We will not makeuse of all the properties of the tensor product A ⊗ A . Here it suffices tosay that a ⊗ b behaves just like a pair of elements, subject to the bilinearityconditions( a + b ) ⊗ c = a ⊗ c + b ⊗ c , a ⊗ ( b + c ) = ( a ⊗ b ) + ( a ⊗ c ) , (78)( k a ) ⊗ b = a ⊗ ( k b ) = k ( a ⊗ b ) , (79) ∀ a, b, c ∈ A and k ∈ Q . Moreover, if A is an algebra, then A ⊗ A is analgebra as well, and the multiplication is defined ‘component-wise’,( a ⊗ b )( c ⊗ d ) = ( ac ) ⊗ ( bd ) . (80)We need an additional algebraic structure: a coalgebra is a vector space A equipped with a comultiplication , i.e., a linear map ∆ : A → A⊗A whichassigns to every element a ∈ A its coproduct ∆( a ) ∈ A ⊗ A . Moreover, thecomultiplication is required to be coassociative , (∆ ⊗ id)∆) = (id ⊗ ∆)∆.The meaning of the coassiciativity is the following: the coproduct is aprescription that assigns to every element a ∈ A a ‘pair’ of elements (or ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 26 C. Duhr rather, a linear combination of pairs). We can schematically write a ∆( a ) = X i a (1) i ⊗ a (2) i . (81)If we have a prescription to split an element a into two, then we can iteratethis prescription to split a into three. However, we can do this in twodifferent ways, a X i a (1) i ⊗ a (2) i X i ∆( a (1) i ) ⊗ a (2) i = X ij a (1 , ij ⊗ a (1 , ij ⊗ a (2) i , (82) a X i a (1) i ⊗ a (2) i X i a (1) i ⊗ ∆( a (2) i ) = X ij a (1) i ⊗ a (2 , ij ⊗ a (2 , ij . (83)Coassociativity states that these two expressions are the same, i.e., theorder in which we iterate the coproduct is immaterial. In other words,their is are unique prescriptions to split an object into two, three, four, etc.pieces.Finally, if A is equipped with both a multiplication and a comultiplica-tion, we require them to be compatible in the sense that the coproduct ofa product is the product of the coproducts,∆( ab ) = ∆( a ) ∆( b ) , (84)where in the right-hand side the multiplication should be interpreted ac-cording to Eq. (80). A vector space with compatible multiplications andcomultiplications is called a bialgebra . If the bialgebra is graded as an al-gebra, we require the coproduct to respect the weight as well, i.e., the sumof the weights of the two factors in the coproduct of a equals the weight of a . Example 14.
Consider a set of letters, say { a, b, c } , and let us consider thevector space A spanned by all linear combinations of words (with rationalcoefficients) in these letters. There is a natural multiplication on A , givenby concatenation of words, e.g., ( ab ) ⊗ c abc . Note that A is graded, andthe weight is given by the length of the word. Next, let us define a linearmap ∆ : A → A ⊗ A in the following way:(i) on letters, ∆ acts like ∆( x ) = 1 ⊗ x + x ⊗ x ∈ { a, b, c } .(ii) we extend the definition to words of length ≥ ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 27 For example, we have∆(1) = 1 ⊗ , (85)∆( ab ) = ∆( a )∆( b ) = (1 ⊗ a + a ⊗ ⊗ b + b ⊗
1) (86)= 1 ⊗ ( ab ) + ( ab ) ⊗ a ⊗ b + b ⊗ a . ∆( abc ) = ∆( ab )∆( c ) (87)= (1 ⊗ ( ab ) + ( ab ) ⊗ a ⊗ b + b ⊗ a )(1 ⊗ c + c ⊗ ⊗ ( abc ) + ( abc ) ⊗ ab ) ⊗ c + b ⊗ ( ac ) + c ⊗ ( ab )+( ac ) ⊗ b + a ⊗ ( bc ) + ( bc ) ⊗ a . Note that the coproduct respects the weight, i.e., the length of a word. Letus explicitly check coassociativity. We can now iterate the coproduct of abc . If we iterate in the first entry, we get(∆ ⊗ id)∆( abc ) = ∆(1) ⊗ ( abc ) + ∆( abc ) ⊗ ab ) ⊗ c + ∆( b ) ⊗ ( ac ) + ∆( c ) ⊗ ( ab ) + ∆( ac ) ⊗ b + ∆( a ) ⊗ ( bc )+ ∆( bc ) ⊗ a = 1 ⊗ ⊗ ( abc ) + 1 ⊗ ( abc ) ⊗ abc ) ⊗ ⊗ ab ) ⊗ c ⊗ b ⊗ ( ac ) ⊗ c ⊗ ( ab ) ⊗ ac ) ⊗ b ⊗ a ⊗ ( bc ) ⊗
1+ ( bc ) ⊗ a ⊗ ⊗ ( ab ) ⊗ c + ( ab ) ⊗ ⊗ c + a ⊗ b ⊗ c + b ⊗ a ⊗ c + 1 ⊗ b ⊗ ( ac ) + b ⊗ ⊗ ( ac ) + 1 ⊗ c ⊗ ( ab )+ c ⊗ ⊗ ( ac ) + 1 ⊗ ( ac ) ⊗ b + ( ac ) ⊗ ⊗ b + a ⊗ c ⊗ b + c ⊗ a ⊗ b + 1 ⊗ a ⊗ ( bc ) + a ⊗ ⊗ ( bc ) + 1 ⊗ ( bc ) ⊗ a + ( bc ) ⊗ ⊗ a + b ⊗ c ⊗ a + c ⊗ b ⊗ a . (88)Similarly, if we iterate in the second entry, we get(id ⊗ ∆)∆( abc ) = 1 ⊗ ∆( abc ) + ( abc ) ⊗ ∆(1) + ( ab ) ⊗ ∆( c )+ b ⊗ ∆( ac ) + c ⊗ ∆( ab ) + ( ac ) ⊗ ∆( b ) + a ⊗ ∆( bc )+ ( bc ) ⊗ ∆( a )= 1 ⊗ ⊗ ( abc ) + 1 ⊗ ( abc ) ⊗ ⊗ ( ab ) ⊗ c + 1 ⊗ b ⊗ ( ac )+ 1 ⊗ c ⊗ ( ab ) + 1 ⊗ ( ac ) ⊗ b + 1 ⊗ a ⊗ ( bc ) + 1 ⊗ ( bc ) ⊗ a + ( abc ) ⊗ ⊗ ab ) ⊗ ⊗ c + ( ab ) ⊗ c ⊗ b ⊗ ⊗ ( ac )+ b ⊗ ( ac ) ⊗ b ⊗ a ⊗ c + b ⊗ c ⊗ a + c ⊗ ⊗ ( ab )+ c ⊗ ( ab ) ⊗ c ⊗ a ⊗ b + c ⊗ b ⊗ a + ( ac ) ⊗ ⊗ b + ( ac ) ⊗ b ⊗ a ⊗ ⊗ ( bc ) + a ⊗ ( bc ) ⊗ a ⊗ b ⊗ c + a ⊗ c ⊗ b + ( bc ) ⊗ ⊗ a + ( bc ) ⊗ a ⊗ . (89) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 28 C. Duhr
We see that Eq. (88) and Eq. (89) give the same result, i.e., the coproductis coassociative, and so A is a bialgebra.A Hopf algebra is a bialgebra together with an additional structure,called antipode , that we do not need in the following. We will thereforeskip the definition of the antipode here and identify Hopf algebras andbialgebras. We conclude this section by introducing some definitions:(1) An element x in a Hopf algebra is called primitive if ∆( x ) = 1 ⊗ x + x ⊗ x is primitive if it cannot be decomposed in any non-trivial way.(2) The reduced coproduct is defined by ∆ ′ ( x ) = ∆( x ) − ⊗ x − x ⊗ i ,...,i k which assignto an element x the the part of the iterated coproduct where the factorsin the coproduct have weights ( i , . . . , i k ). Example 15.
Using the definitions from Example 14, we see that all lettersare primitive elements. The reduced coproducts are∆ ′ ( ab ) = a ⊗ b + b ⊗ a , (90)∆ ′ ( abc ) = ( ab ) ⊗ c + b ⊗ ( ac ) + c ⊗ ( ab ) + ( ac ) ⊗ b + a ⊗ ( bc ) (91)+( bc ) ⊗ a . The different components of the coproduct are∆ , ( ab ) = a ⊗ b + b ⊗ a , (92)∆ , ( abc ) = ( ab ) ⊗ c + ( bc ) ⊗ a + ( ac ) ⊗ b , (93)∆ , ( abc ) = a ⊗ ( bc ) + b ⊗ ( ac ) + c ⊗ ( ab ) , (94)∆ , , ( abc ) = a ⊗ b ⊗ c + b ⊗ c ⊗ a + c ⊗ a ⊗ b (95)+ a ⊗ c ⊗ b + b ⊗ a ⊗ c + c ⊗ b ⊗ a . The Hopf algebra of MPLs
In this section we show that MPLs form a Hopf algebra, and we definethe coproduct on MPLs [29]. The construction and the definition of thecoproduct is a bit subtle, and it will be carried out in three stages: (1) The coproduct in the generic case:
We start by defining a co-product on MPLs of the form I ( a ; a , . . . , a n ; a n +1 ), where the a i are generic , i.e., the a i do not take special values and a i = a j if i = j . (2) Shuffle regularisation: In a second step, we extend the definitionto the non-generic case, where for example some of the a i are allowed ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 29 to be equal. This introduces additional singularities that need to beregularised. (3) Inclusion of even zeta values: Finally, we show how to consistentlyinclude the even zeta values.
The coproduct in the generic case.
In this section we define thecoproduct on I ( a ; a , . . . , a n ; a n +1 ), where the a i are generic [29]. It ismore convenient to work with the I -notation rather than the G -notation,because it makes some of the formulas more transparent. The coproducton MPLs is defined by [29]∆( I ( a ; a , . . . , a n ; a n +1 ))= X i
Let us consider the coproduct of a generic MPL of weightone, I ( a ; a ; a ). There are only two different ways to select points on thehalf circle: ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 30 C. Duhr ⊗ I ( a ; a ; a ) I ( a ; a ; a ) ⊗ a , a , a ;in that case there is no complementary polygon, and so the second factorin the coproduct is just 1. The dashed polygon denotes a complementarypolygon with vertices a , a , a (it is complementary to the trivial polygonwith vertices a and a ). We see that MPLs of weight one are primitiveelements, i.e., they cannot be decomposed further. Example 17.
Let us consider the coproduct of a generic MPL of weighttwo, I ( a ; a , a ; a ). First, there are the two trivial ways to inscribe apolygon into the semi-circle:1 ⊗ I ( a ; a , a ; a ) I ( a ; a , a ; a ) ⊗ I ( a ; a ; a ) ⊗ I ( a ; a ; a ) I ( a ; a ; a ) ⊗ I ( a ; a ; a ) Example 18.
At weight three, we have the usual two trivial terms: ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 31 ⊗ I ( a ; a , a , a ; a ) I ( a ; a , a , a ; a ) ⊗ I ( a ; a ; a ) ⊗ I ( a ; a , a ; a ) I ( a ; a ; a ) ⊗ I ( a ; a , a ; a ) I ( a ; a , a ; a ) ⊗ I ( a ; a ; a ) I ( a ; a , a ; a ) ⊗ I ( a ; a ; a ) I ( a ; a , a ; a ) ⊗ I ( a ; a ; a )In addition, we have for the first time a contribution from a term with twocomplementary polygons (indicated by dashed and dotted lines): ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 32 C. Duhr I ( a ; a ; a ) ⊗ [ I ( a ; a ; a ) I ( a ; a ; a )]The worked out case at weight four can be found in ref. [30]. Shuffle regularisation.
The definition of the coproduct on MPLs givenin Eq. (96) only holds in the case of generic a i . In order to understand why,let us consider I (0; 1 , z ). Looking at Example 17, we see that we have aterm I (0; 1; z ) ⊗ I (1; 1; z )The MPL in the second factor of this term in the coproduct is divergent!Indeed, passing to the G -notation, we have I (1; 1; z ) = G (1; z ) − G (1; 1), andwe have seen in Sec. 3 that G ( a , . . . , a n ; z ) is divergent whenever z = a .In other words, the coproduct (96) does not make sense for I (0; 1 , z ),because it contains divergent quantities (even though the original function I (0; 1 , z ) is well-definite).The idea is now to replace the MPLs that appear inside the coproductby suitably regularised versions of MPLs [5, 29]. There are different waysone can define the regularised versions of MPLs. In the following we presentthe so-called shuffle regularisation . The idea is similar to what we did inorder to define the regularised shuffle relations: We formally keep all thedivergent quantities G ( z, a , . . . , a n ; z ) in a first step. Then, we proceed ina way similar to what we did in Example 7, and we use the shuffle algebrato express all the divergent MPLs in terms of convergent ones, except forMPLs of the form G ( z, . . . , z ; z ). We then define the shuffle-regularisedversion of the MPLs (denoted by G reg and I reg in the following) by puttingall the divergent quantities G ( z, . . . , z ; z ) to zero. Example 19. ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 33 (i) Consider the divergent quantity G ( z, . . . , z ; z ). By definition, itsshuffle-regularised value is zero, G reg ( z, . . . , z ; z ) = 0.(ii) Next consider the divergent quantity G ( z, a ; z ), a = z . We can write G ( z, a ; z ) = G ( z ; z ) G ( a ; z ) − G ( a, z ; z ) . (97)The regularised value is then G reg ( z, a ; z ) = − G ( a, z ; z ) . (98)(iii) Finally, consider the divergent quantity G ( z, z, a ; z ), a = z . We canwrite G ( z, z, a ; z ) = G ( z, z ; z ) G ( a ; z ) − G ( z, a, z ; z ) − G ( a, z, z ; z )= G ( z, z ; z ) G ( a ; z ) − [ G ( z ; z ) G ( a, z ; z ) − G ( a, z, z ; z )] − G ( a, z, z ; z )= G ( z, z ; z ) G ( a, z ) − G ( z ; z ) G ( a, z ; z ) + G ( a, z, z ; z ) . (99)The regularised version is then G reg ( z, z, a ; z ) = G ( a, z, z ; z ) . (100)The coproduct in the non-generic case is now defined by replacing I by I reg everywhere in the right-hand side in Eq. (96). One might wonder whythis prescription works (e.g., why it does not spoil any of the other definingconditions of the coproduct). The reason for this is that the regularisedversions satisfy the same algebraic properties as the unregularised MPLs.In particular, it is easy to see that the unregularised MPLs agree withthe regularised ones whenever they converge. Moreover, the regularisationprocedure preserves the multiplication,[ G ( ~a ; z ) G ( ~b ; z )] reg = G reg ( ~a ; z ) G reg ( ~b ; z ) . (101) Example 20.
Let us check this last property explicitly on some example:[ G ( z, a ; z ) G ( b ; z )] reg = [ G ( z, a, b ; z ) + G ( z, b, a ; z ) + G ( b, z, a ; z )] reg = G reg ( z, a, b ; z ) + G reg ( z, b, a ; z ) + G reg ( b, z, a ; z )= − G ( a, z, b ; z ) − G ( a, b, z ; z ) − G ( b, a, z ; z )= − G ( a, z ; z ) G ( b ; z )= G reg ( z, a ; z ) G reg ( b ; z ) . (102) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 34 C. Duhr
Since the regularised and the unregularised MPLs have the same algebraicproperties, we will often not make the distinction explicitly, and alwaysassume that inside the coproduct all MPLs have been replaced by theirshuffle-regularised versions.
Example 21.
We can now give the coproduct of the classical polyloga-rithms, and we will be able to give a closed formula for the coproductof Li n ( z ). To motivate this formula, let us first look at some low weightexamples:(i) Let us start by looking at the coproduct of I (0; 1 , z ) = − Li ( z ).Besides the two trivial ones, we have the following two terms I reg (0; 1; z ) ⊗ I reg (1; 0; z ) I reg (0; 0; z ) ⊗ I reg (0; 1; 0)= − Li ( z ) ⊗ log z = 0Combining all the pieces, we get∆(Li ( z )) = 1 ⊗ Li ( z ) + Li ( z ) ⊗ ( z ) ⊗ log z . (103)(ii) Next, let us look at the coproduct of I (0; 1 , , z ) = − Li ( z ). Besidesthe two trivial ones, we have the following six terms I reg (0; 1; z ) ⊗ I reg (1; 0 , z ) I reg (0; 0; z ) ⊗ I reg (0; 1 ,
0; 0)= Li ( z ) ⊗ log z ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 35 I reg (0; 1 , z ) ⊗ I reg (0; 0; z ) I reg (0; 0 , z ) ⊗ I reg (0; 1; 0)= Li ( z ) ⊗ log z = 0 I reg (0; 1 , z ) ⊗ I reg (1; 0; 0) I reg (0; 0; z ) ⊗ [ I reg (0; 1; 0) I reg (0; 0; z )]= 0 = 0Putting all the terms together, we find∆(Li ( z )) = 1 ⊗ Li ( z ) + Li ( z ) ⊗
1+ Li ( z ) ⊗ log z + Li ( z ) ⊗ log z . (104)(iii) From the previous examples, it is easy to discern a pattern for all clas-sical polylogarithms: the only non-vanishing contributions are thosewhere the inscribed polygon contains the vertex labeled by ‘1’ andwhere there is exactly one complementary polygon. These contribu-tion are easy to evaluate, and we find∆(Li n ( z )) = 1 ⊗ Li n ( z ) + n − X k =0 Li n − k ( z ) ⊗ log k zk ! . (105) Inclusion of even zeta values.
We know now how to compute the co-product of arbitrary MPLs. In particular, this implies that we also knowhow to compute the coproduct of MZVs, by writing them as MPLs eval-uated at 1. For example, letting z = 1 in Eq. (105), we see that ordinaryzeta values are primitive,∆( ζ n ) = ζ n ⊗ ⊗ ζ n . (106) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 36 C. Duhr
This, however, is problematic for even zeta values. Indeed, we find∆( ζ ) = 25 ∆( ζ ) = ζ ⊗ ⊗ ζ + 45 ζ ⊗ ζ , (107)i.e., we have obtained a contradiction with Eq. (106) for n = 4. It is easyto see that a similar problem arises for iπ (= log( − ζ and iπ (i.e.,‘ ζ = iπ = 0’), and indeed, A is, strictly speaking, not a Hopf algebra. Ifwe define H to be the algebra A modulo iπ , then H is a Hopf algebra withthe coproduct given in Eq. (96) [29]. It is clear, however, that this situationis not satisfactory from a practical point of view.In the following we discuss how to remedy this problem without havingto work modulo iπ , and we follow very closely ideas introduced by Brownin ref. [31] in the context of MZVs. First, we note that we can triviallywrite A = Q [ iπ ] ⊗ H , (108)where Q [ iπ ] denotes the ring of polynomials in iπ with rational coefficients.The meaning of this is simply that by passing from A to H , we have removedall powers of iπ , and we can compensate for this by allowing the coefficientsin front of elements of H to be polynomials in iπ rather than just rationalnumbers. Next, we define the coproduct b to be a map ∆ : A → A ⊗ H insuch a way that it acts by Eq. (96) on elements of H , while on iπ it actsby [30, 31] ∆( iπ ) = iπ ⊗ . (109)Practically speaking, this means that we have to put iπ to zero everywherebut in the left-most factor of the coproduct. This obviously resolves theaforementioned contradiction in a trivial way, because∆( ζ ) = ζ ⊗ ζ ⊗ ζ ) . (110)This completes our review of the Hopf algebra of MPLs. We note thatthe fact that the coproduct maps A to A ⊗ H introduces an ‘asymmetry’between the left and right factors, and we may ask what the meaning ofthis asymmetry is. This will be discussed in the rest of this section. b Strictly speaking, ∆ is no longer a coproduct, but a coaction, and A is a comodule ratherthan a Hopf algebra. Since this distinction is only purely technical for our purposes, wewill continue to call A a ‘Hopf algebra’, keeping in mind that we need this specialtreatment of iπ . ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 37 Let us start by analysing the rightmost factor of the coproduct. Itturns out that this factor encodes the behaviour of the functions underdifferentiation. More precisely, we have∆ (cid:18) ∂∂z F (cid:19) = (cid:18) id ⊗ ∂∂z (cid:19) ∆( F ) , (111)i.e., derivatives only act in the rightmost factor of the coproduct. Notethat this gives a convenient way to compute the derivatives of MPLs withrespect to arbitrary variables. Indeed, if F denotes a function of weight n ,we have ∂∂z F = µ (cid:18) id ⊗ ∂∂z (cid:19) ∆ n − , ( F ) , (112)where µ ( a ⊗ b ) = ab denotes multiplication. Example 22.
Let us illustrate Eq. (111) on the simple example of thedilogarithm. The left-hand side of Eq. (111) gives∆ (cid:18) ∂∂z Li ( z ) (cid:19) = ∆ (cid:18) z Li ( z ) (cid:19) = 1 z (1 ⊗ Li ( z ) + Li ( z ) ⊗ . (113)The right-hand side gives (cid:16) id ⊗ ∂∂z (cid:17) ∆(Li ( z ))= 1 ⊗ ∂∂z Li ( z ) + Li ( z ) ⊗ ∂∂z ( z ) ⊗ ∂∂z log z = 1 ⊗ Li ( z ) z + Li ( z ) ⊗ z = 1 z (1 ⊗ Li ( z ) + Li ( z ) ⊗ , (114)and we indeed obtain the same answer. Example 23.
Assume that you want to compute the derivative of G (1 , y ; z ) with respect to y . Using Eq. (112), we obtain ∂∂y G (1 , y ; z ) = µ (cid:18) id ⊗ ∂∂z (cid:19) ∆ , ( G (1 , y ; z ))= G (1; z ) ∂∂y G (1 + y ; 1) − G (1 + y ; z ) ∂∂y G (1; 1 + y )+ G (1 + y ; z ) ∂∂y G (1; z ) . (115) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 38 C. Duhr
The remaining derivatives are just derivatives of ordinary logarithms, ∂∂y G (1 + y ; 1) = ∂∂y log (cid:18) −
11 + y (cid:19) = 1 y (1 + y ) , (116) ∂∂y G (1; 1 + y ) = ∂∂y log( − y ) = 1 y , (117) ∂∂y G (1; z ) = 0 . (118)Thus, ∂∂y G (1 , y ; z ) = 1 y (1 + y ) G (1; z ) − y G (1 + y ; z ) . (119)Just like the rightmost factor of the coproduct encodes the derivativesof a function, the leftmost factor encodes its discontinuities. Note that thisis consistent with the fact that ∆( iπ ) = iπ ⊗
1. If Disc denotes the operatorthat takes the discontinuity of a function across some branch cut, then wehave ∆ (Disc F ) = (Disc ⊗ id) ∆( F ) . (120) Example 24.
We again illustrate this property on the example of thedilogarithm. Li ( z ) has a branch cut extending from z = 1 to z = ∞ , andthe discontinuity across the cut isDisc Li ( z ) = Li ( z + i − Li ( z − i
0) = 2 πi log z . (121)The right-hand side of Eq. (120) gives∆ (Disc Li ( z )) = 2 πi ⊗ log z + (2 πi log z ) ⊗ . (122)The left-hand side gives (cid:16) Disc ⊗ id (cid:17) ∆(Li ( z ))= Disc 1 ⊗ Li ( z ) + Disc Li ( z ) ⊗ ( z ) ⊗ log z = (2 πi log z ) ⊗ πi ⊗ log z , (123)where we used the fact that Disc Li ( z ) = 2 πi . ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 39 The symbol map
In the previous section we have seen that we can decompose an MPL ofweight n into smaller weights by acting with the coproduct, and coasso-ciativity allows us to iterate this decomposition in a unique way. Thisdecomposition will obviously stop at some point, namely when we have de-composed a function of weight n into an n -fold tensor product of functionsof weight one, i.e., ordinary logarithms. This maximal iteration of the co-product has a special status, and is often referred to as the symbol in theliterature [7, 29, 32–34], S ( F ) ≡ ∆ ,..., ( F ) mod iπ , (124)where we also put all iπ terms to zero. Note that, since all the factorsin the symbol are just ordinary logarithms, it is conventional to drop the‘log’-signs inside the factors of the tensor product, i.e., we write a ⊗ . . . ⊗ a n instead of log a ⊗ . . . ⊗ log a n . The entries a i in the symbol of F are oftenreferred to as the alphabet of F .In the following we give some properties of the symbol map S , mostof which are direct consequences of the corresponding properties of thecoproduct. First, it is obvious that S is linear. Second, the symbols of aproduct is mapped to the shuffle of the symbols, S ( F G ) = S ( F ) (cid:1) S ( G ) , (125)where (cid:1) denotes the shuffle product on tensors, e.g.,( a ⊗ b ) (cid:1) ( c ⊗ d ) = a ⊗ b ⊗ c ⊗ d + a ⊗ c ⊗ b ⊗ d + c ⊗ a ⊗ b ⊗ d + a ⊗ c ⊗ d ⊗ b + c ⊗ a ⊗ d ⊗ b + c ⊗ d ⊗ a ⊗ b . (126)Next, the additivity of the logarithm, log( ab ) = log a + log b translates intothe property . . . ⊗ ( ab ) ⊗ . . . = . . . ⊗ a ⊗ . . . + . . . ⊗ b ⊗ . . . , (127) . . . ⊗ a n ⊗ . . . = n ( . . . ⊗ a ⊗ . . . ) , (128)and the fact that we work modulo iπ leads to . . . ⊗ ρ ⊗ . . . = 0 , (129)where ρ is a root of unity, ρ n = 1 for some n . This last property impliesthat S has a non-trivial kernel. In particular, the kernel contains all MZVs, S ( ζ m ,...,m k ) = 0, but it contains additional non-trivial elements that arenot necessarily MZVs, e.g., S (cid:20) Li (cid:18) (cid:19) + 124 log (cid:21) = 0 . (130) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 40 C. Duhr
A collection of elements in the kernel of the symbol map is given in ref. [34].Finally, we may ask if every possible tensor S = X i ,...,i k c i ,...,i k a i ⊗ . . . ⊗ a i k , (131)can be the symbol of some function, i.e., whether we can find a function F such that S ( F ) = S . The answer to this question is negative in general,but one can show that such a function F exists if and only if S satisfies the integrability condition [7], for all 1 ≤ j ≤ k − X i ,...,i k c i ,...,i k d log a i j ∧ d log a i j +1 a i ⊗ . . . a i j − ⊗ a i j +2 ⊗ a i k = 0 , (132)where ∧ denotes the usual wedge product on differential forms. Functional equations of MPLs
In this section we discuss our main application of the Hopf algebra of MPLs:the derivation of functional equations for MPLs. The idea is simple: As-sume we are trying to proof an identity F = G , where F and G are expres-sions of weight n . We can decompose this expression into lower weights us-ing the coproduct, and prove a sequence of simpler identities instead, whichonly involve simpler functions (where ‘simpler’ means ‘lower weights’), forwhich we may assume that all identities are known. We will illustrate thisprocedure on some examples in the following. Note that in practise theexpressions can soon become rather big. The examples in the following arechosen because they are simple enough so that all the manipulations canbe carried out on a piece of paper. Example 25.
Throughout this section we assume that x is a real positivevariable to which we assign a small positive imaginary part. We proceedrecursively in the weight to build up the inversion relations for classicalpolylogarithms.For the classical polylogarithm of weight one, the inversion relation iseasy to obtain,Li (cid:18) x (cid:19) = − log (cid:18) − x (cid:19) = − log(1 − x ) + log( − x )= − log(1 − x ) + log x − iπ . (133)In order to obtain the inversion relation for weight two, we act with ∆ , ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 41 on Li (1 /x ) and insert the inversion relation for Li (1 /x ),∆ , (cid:20) Li (cid:18) x (cid:19)(cid:21) = − log (cid:18) − x (cid:19) ⊗ log (cid:18) x (cid:19) = log(1 − x ) ⊗ log x − log x ⊗ log x + iπ ⊗ log x = ∆ , h − Li ( x ) −
12 log x + iπ log x i . (134)Note that in the last step Eq. (109) played a crucial role. We conclude thatthe arguments in the left and right-hand sides are equal modulo primitiveelements of weight two. We thus make the ansatz,Li (cid:18) x (cid:19) = − Li ( x ) −
12 log x + iπ log x + c ζ , (135)for some rational number c . Specializing to x = 1, we immediately obtain c = 2, which is indeed the correct inversion relation for Li .At weight three, we act with ∆ , , on Li (1 /x ) and we obtain∆ , , (cid:20) Li (cid:18) x (cid:19)(cid:21) = − log (cid:18) − x (cid:19) ⊗ log (cid:18) x (cid:19) ⊗ log (cid:18) x (cid:19) = − log(1 − x ) ⊗ log x ⊗ log x + log x ⊗ log x ⊗ log x − iπ ⊗ log x ⊗ log x = ∆ , , h Li ( x ) + 16 log x − iπ x i . (136)Eq. (136) is not yet the correct inversion relation for Li . After subtractingthe terms we have found in Eq. (136), we look at the image of the differenceunder ∆ , or ∆ , . For example, we obtain∆ , " Li (cid:18) x (cid:19) − (cid:16) Li ( x ) + 16 log x − iπ x (cid:17) = −
12 log (cid:18) − x (cid:19) ⊗ log (cid:18) x (cid:19) + 12 log(1 − x ) ⊗ log x −
12 log x ⊗ log x + iπ ⊗ log x = 0 . (137)We see that acting with ∆ , does not provide any new information.This is not surprising, as the missing terms are of the form ζ log x , and∆ , ( ζ log x ) = 0. Acting with ∆ , and using the inversion relation for ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 42 C. Duhr Li , we obtain new non-trivial information,∆ , " Li (cid:18) x (cid:19) − (cid:16) Li ( x ) + 16 log x − iπ x (cid:17) = Li (cid:18) x (cid:19) ⊗ log (cid:18) x (cid:19) − Li ( x ) ⊗ log x −
12 log x ⊗ log x + ( iπ log x ) ⊗ log x = − h − Li ( x ) −
12 log x + iπ log x + 2 ζ i ⊗ log x − Li ( x ) ⊗ log x −
12 log x ⊗ log x + ( iπ log x ) ⊗ log x = − ζ ⊗ log x = ∆ , (cid:16) − ζ log x (cid:17) . (138)Thus,Li (cid:18) x (cid:19) = Li ( x ) + 16 log x − iπ x − ζ log x + αζ + β iπ . (139)Specializing to x = 1 gives α = β = 0, which is indeed the correct inversionrelation for Li . Proceeding in exactly the same way, we can now derivethe inversion relations for all the classical polylogarithms.
6. Applications to loop amplitudes6.1.
MPLs and Feynman integrals
In this section we give examples of how the concepts introduced in theprevious sections apply to loop integrals. Since the area of potential appli-cations of the Hopf algebraic techniques are very wide, we do by now meansintent to be exhaustive, but we only try to give a flavour of what kind ofapplications are possible. Note that we will restrict ourselves to classes ofloop integrals that can be expressed in terms of MPLs, keeping in mindthat this may not always be possible.The first question one may ask is how the loop integrals themselvesfit into the algebraic picture of the previous section. It is well-known thatFeynman integrals have discontinuities, and the locations of the branch cutsare solutions of the so-called Landau equations [35]. For example, in theparticular case of massless propagators all branch cuts start at points wherea Mandelstam invariant becomes zero or infinite. Thus, the position of thebranch points of Feynman integrals (seen as functions of Lorentz invariant ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 43 scalar products and masses) are not arbitrary, but dictated by unitarity.In Eq. (120) we have seen that discontinuities are captured by the leftmostfactor of the coproduct of a function. It then follows that the leftmost factorin the coproduct can only have discontinuities which are compatible withthe Landau equations! This condition, known as the first entry condition ,puts strong constraints on the analytic expressions for Feynman integrals.In particular, for massless propagators this implies that the first entry inthe symbol of such an integral can only be a Mandelstam invariant [36].The next question is whether one can make any kind of generic state-ments about the weight of loop amplitudes (at least in the case where theycan be expressed in terms of MPLs). Currently, only very few theorems areknown for specific classes of loop integrals [37, 38], but there are conjecturesabout the weight of generic Feynman integrals in four dimensions: Conjecture 6. In D = 4 − ǫ dimensions, the Laurent coefficient of ǫ k ofan L -loop amplitude contains terms of weight at most L + k . Note that the conjecture only gives an upper bound for the weight. Ingeneral, a loop amplitude will contain all weights up to the bound given bythe conjecture. In special quantum field theories, like for example the N = 4Super Yang-Mills (SYM) theory, this bound is expected to be saturated,i.e., in the coefficient of ǫ k of an L -loop amplitudes has exactly weight 2 L + k .We emphasise that these conjectures only make sense if the amplitude canbe expressed in terms of MPLs in the first place.The previous conjecture only allows one to set an upper bound on theweight of Feynman integrals, and so one may ask whether there is a morerefined version of it, e.g., is it possible to find ‘building blocks’ that are of uniform weight (i.e., where all the terms in a given Laurent coefficient ofhave the same weight) and out of which the amplitude can be constructed.It turns out that this is indeed the case, but in order to formulate thecorresponding conjecture we need to introduce a few more concepts.Let us start by defining more precisely what we mean by ‘buildingblocks’ of the loop amplitude. It is clear that the amplitude is a linearcombination of scalar integrals of the type (1), and let us concentrate ona specific subset of of these integrals that have the same set of propaga-tors (and we may interpret numerators as propagators raised to negativepowers). These integrals in general differ only by the powers ν i of the prop-agators c . Such a family of loop integrals is referred to as a topology . Moreprecisely, we can define a topology to be a family of scalar integrals differ- c We allow for some propagators to be absent, i.e., they are raised to the power zero. ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 44 C. Duhr ing only by the exponents ν i ∈ Z such that every scalar product betweentwo loop momenta, k i · k j , or between a loop and an external momentum, k i · p j , can be written as a linear combination of propagators. This lat-ter condition ensures that all the numerator factors can be interpreted aspropagators raised to non-positive powers. Example 26.
Let us consider the three integrals I = e γ E ǫ Z d D kiπ D/ k ( k + p ) , (140) I = e γ E ǫ Z d D kiπ D/ k · p k ( k + p ) ( k + p + p ) , (141) I = e γ E ǫ Z d D kiπ D/ k + 2 k · p k ( k + p ) ( k + p + p ) ( k − p ) . (142)All three integrals can in fact be embedded into the same box topology ,Box( ν , ν , ν , ν )= e γ E ǫ Z d D kiπ D/ k ] ν [( k + p ) ] ν [( k + p + p ) ] ν [( k − p ) ] ν . (143)Indeed, the two-point integral can be written as I = Box(1 , , , I , we can rewrite the numerator in terms of prop-agators k · p = 12 [( k + p ) − k − p ] , (144)and so we get I = 12 Box(1 , , , −
12 Box(0 , , , − p Box(1 , , , . (145)Similarly, the numerator of the four-point integrals I can written as k + 2 k · p = ( k + p + p ) − ( k + p ) + k + p − ( p + p ) , (146)and so I = Box(1 , , , − Box(1 , , , , , ,
1) + [ p − ( p + p ) ] Box(1 , , , . (147)Let us now consider a topology, and let us see it as a function of the powers ν i of the propagators. If we work in dimensional regularisation, then theintegral of a total derivative vanishes, i.e., we can write0 = Z L Y i =1 d D k i ! ∂∂k µj · v µ (cid:16) . . . (cid:17) , (148) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 45 where v µ ∈ { k µ , . . . , k µL , p µ , . . . , p µE − } can be either a loop momentum oran external momentum. If we act with the derivative on the propagators,we will essentially only shift the powers of the propagators, and so wewill eventually arrive at a set of linear recursion relations in the powers ν i of the propagators, known as integration-by-parts identities (IBPs) [39].These recursion relations can be solved algorithmically, e.g., using Laporta’salgorithm [40–47]. Since the recursion is linear, we can express all theintegrals in the topology in terms of a basis of the solution space of thelinear system. Such a basis is referred to as a set of master integrals , andthey are the ‘building blocks’ we were looking for. Example 27.
Consider the topology defined byBub( ν , ν ) = e γ E ǫ Z d D kiπ D/ k ] ν [( k + p ) ] ν . (149)We can write down two IBP relations,0 = ( D − ν − ν ) Bub( ν , ν ) − ν Bub( ν − , ν + 1) (150)+ ν p Bub( ν , ν + 1) , ν − ν ) Bub( ν , ν ) + ν p Bub( ν + 1 , ν ) (151) − ν p Bub( ν , ν + 1) − ν Bub( ν + 1 , ν − ν Bub( ν − , ν + 1) . If we solve the recursion, we see that there is only one single master integral,Bub(1 , ,
3) = − ( D − D − D − p ) Bub(1 , . (152)While it can be proven that the number of master integrals is always fi-nite [48], and their number can be predicted from the topology [49], thereare of course different ways of choosing the set of master integrals. Re-cently, it was conjectured there is a distinguished set of master integrals forevery topology [50]: Conjecture 7.
For every topology there is a set of uniformly transcendentalmaster integrals with unit leading singularity.
While the conjecture asserts the existence of this set of master integrals, nogeneric algorithm is known to find this basis (some partial results to findthe basis exist, see, e.g., ref. [50–53]). The main advantage of having this ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 46 C. Duhr uniformly transcendental basis is that, once this basis is known, the masterintegrals can easily be computed via differential equations. Indeed, it iswell-known that master integrals satisfy coupled systems of first-order dif-ferential equations [54–56]. If the basis of master integrals is uniformly tran-scendental, then the differential equations decouple order-by-order in ǫ andcan easily be solved [50]. In fact, the differential equations satisfied by uni-formly transcendental master integrals are special instances/generalisationsof the so-called Knizhnik-Zalmolodchikov equation. Although this topic istightly connected to iterated integrals and MPLs, this topic would lead usto far, and we will therefore not cover it in this set of lectures. Simplification of analytic results
Probably the most obvious application to loop amplitudes of the ideas pre-sented in these lectures is the simplification of the sometimes large andcomplicated results that arise from these computations. For example, ana-lytic results for two-loop multi-scale Feynman integrals and scattering am-plitudes may involve combinations of several thousands of MPLs, and so thequestion whether a given a expression can be rewritten in terms of ‘simpler’functions and/or in a more compact form is highly relevant. The relationbetween the ‘complicated’ and the ‘simple’ results may be seen as one bigfunctional equation relating the two expressions. We stress, however, that‘simplicity’ can sometimes be a purely subjective notion – mathematicallyit is always exactly the same analytic function!The first time a striking simplification of a loop amplitude was achievedin ref. [32], where also the symbol was introduced for the first time in thephysics literature. In ref. [57, 58] the so-called two-loop six-point remainderfunction in N = 4 SYM was evaluated, and the result was expressed as a17-page-long combination of MPLs of uniform weight four. In ref. [32] thesymbol map was used to rewrite the same function as a single line of classi-cal polylogarithms. By now these techniques have also found there way intoQCD computations, and have in particular been used to simplify the ana-lytic expressions for the two-loop amplitudes for a Higgs boson plus threepartons [30, 59, 60], light-quark contributions to top-pair production [61]as well as diboson production at two loops [51, 62]The first question we may ask is whether there are criteria to ensurethe existence of a simpler representation of a function. In Conjecture 6we have seen that the coefficient of ǫ k in the Laurent expansion of an L -loop Feynman diagram is conjectured to have weight at most 2 L + k . The ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 47 constant term of a two-loop integral can thus have weight at most four.It can be shown (cf. ref. [25, 63, 64]) that MPLs of weight at most threecan always be expressed in terms of classical polylogarithms, consistentwith the fact that one-loop integrals in four space-time dimensions canalways be expressed in terms of dilogarithms and ordinary logarithms. Thisstatement is no longer true for MPLs of weight four. Based on a conjecturein ref. [65], a necessary and sufficient condition was formulated in ref. [32]to determine whether a given combination of MPLs of weight four can beexpressed in terms of classical polylogarithms only. To state this criterion,it is convenient to introduce a linear operator δ acting on tensors of weightfour by δ ( a ⊗ a ⊗ a ⊗ a ) ≡ ( a ∧ a ) ∧ ( a ∧ a ) , (153)where we defined a ∧ b ≡ a ⊗ b − b ⊗ a . We then have the following Conjecture 8.
Let F be a combination of MPLs of weight four. Then F can be expressed in terms of classical polylogarithms only if and only if δ ( S ( F )) = 0 . Similar conjectures can be made for higher weights [66], and also to decidewhether a function can be expressed through product of lower weight func-tions only [34, 67, 68]. A detailed discussion of all of these criteria wouldhowever go beyond the scope of these lectures.We have now criteria to decide if a given can be written in a simplifiedform, but so far we have not answered the question how to actually find thisform. In practise, finding this simplified form can be very difficult, even ifwe know that it exists. The general idea is that if we manage to write downan ansatz for what the simplified form is in terms of MPLs whose coefficientsare some unknown rational numbers then we can fix the coefficients by usingthe Hopf algebra techniques of Sec. 5. The difficulty, however, often lies infinding this ansatz in the first place, and often this cannot be done withoutadditional input. One of the issues is related to finding the arguments of thepolylogarithms to write down an ansatz. In ref. [34] was presented to findrational arguments that can appear as arguments of polylogarithms witha prescribed alphabet. Once a set of possible arguments is identified, it issufficient to write down an ansatz of all polylogarithmic functions in thesearguments and to fix the coefficients by requiring for example the symbolsof the ansatz and the original function to agree. Note that there mightnot be a unique solution for the coefficients. Indeed, if there are residual ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 48 C. Duhr relations among the functions appearing in the ansatz, the coefficients canonly be fixed up to these relations.
Direct integration
The last application that we are going to discuss is the use of algebraictechniques to explicitly compute integrals. We often led to compute mutli-fold integrals over rational functions, and in many cases modern computeralgebra systems often fail to do the integrals. The reason is that, if we tryto do the integrals one-by-one, the integrand is more complicated after eachintegration, because every integration in principle increases the weight ofthe integrand by one unit:(1) The first integrations usually leads to a logarithms.(2) The second integration produces a dilogarithm, i.e., a function of weighttwo, with a complicated argument.(3) The third integration produces a function of weight three with a com-plicated argument.(4) etc.The idea is then to use at each step functional equations to rewrite the inte-grand in a form where the next integration can be done using the definitionof MPLs, Eq. (20). MPLs are defined by iterated integration of linear fac-tors, and so at each step in the integration process we need to find a variablein which all the denominators are linear. There are criteria that allow oneto determine a priori if there is an order of the integration variables suchthat this procedure succeeds [69, 70]. If so, it is possible to perform all theintegrations in an algorithmic way [69] (see also ref. [71–76]). We note thatfor this strategy to succeed, it is mandatory to have a convergent integral,so that we can expand in ǫ under the integration sign. Extensions of thismethod to divergent integrals were discussed in ref. [72]. Example 28.
Let us illustrate this method on the following integral: I ( ǫ ) = Z ∞ dx dx dx x ǫ (1 + x ) ǫ − x − ǫ (1 + x ) − ǫ − x ǫ (1 + x ) − ǫ − × (1 + x + x + x x ) − ǫ − . (154)Out goal is to compute the first few terms in the ǫ -expansion of the integral.It is easy to check that the integral is finite as ǫ →
0, and so we can ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 49 immediately expand in epsilon under the integration sign, I ( ǫ ) = I + I ǫ + I ǫ + I ǫ + O ( ǫ ) . (155)The first coefficient is trivial to compute, I = π − . (156)Next, let us turn to the coefficient I , given by the integral I = Z ∞ dx dx dx (1 + x ) (1 + x ) (1 + x ) (1 + x + x + x x ) × " G ( − x ) − G ( − x ) − G ( − x ) + G (0; x ) − G (0; x )+ 2 G (0; x ) − G ( − x − x ) − G (cid:18) − x − x − x ; x (cid:19) , (157)where we have already written all logarithms in terms of MPLs, e.g.,log(1 + x + x + x x )= log(1 + x ) + log (cid:18) x x (cid:19) + log (cid:18) x x x + x (cid:19) = G ( − x ) + G ( − − x ; x ) + G (cid:18) − x − x − x ; x (cid:19) . (158)It is easy to compute a primitive with respect to x for the integrand of I ,e.g., Z dx x + x + x x G ( − x ) = 1 x G (cid:18) − x − x − x , − x (cid:19) . (159)In the following we only concentrate on this single term (which is in factthe most complicated one) to illustrate the procedure. All other terms canbe dealt with in a similar way. We now need to take the limits x → x → ∞ of the primitive. The limit x → x → G (cid:18) − x − x − x , − x (cid:19) = 0 . (160)The limit x → ∞ is obtained by letting x = 1 / ¯ x and deriving theinversion relation for this MPL, which can be done using the techniques of ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 50 C. Duhr
Sec. 5. For a more algorithmic approach we refer to ref. [74]. We find G (cid:18) − , − x − x − x ; x (cid:19) = G (0 , x ) − G ( − x ) G ( − x )+ G ( − x ) G (0; x ) − G ( − , − x − x ) + G (0 , − x ) − G (0 , x ) − ζ + O (1 /x ) . (161)Note that the function has a logarithmic singularity for x → ∞ , which willcancel against similar contributions from other terms. The same steps caneasily be repeated for all the terms appearing in the primitive with respectto x .After having taken the limits, we can immediately compute the primitivewith respect to x , e.g., Z dx x + x G ( − x ) = G ( − − x , − x ) . (162)The limit x → x → ∞ can again becomputed by letting x = 1 / ¯ x and deriving the inversion relation andletting ¯ x →
0. We find G ( − − x , − x ) = G (0 , x ) − G (0 , − x ) + O (1 /x ) . (163)We are finally only left with the integral over x . The primitive involvesintegrals like Z dx x G ( − , x ) = G ( − , − , x ) . (164)Proceeding just like before to take the limits, we finally get I = − ζ + 2 π . (165)The higher terms in the ǫ expansion can be obtained in exactly the sameway. For this particular integral we find for example I = 149 π − ζ − π − , I = − ζ + 149 π
108 + 6076 ζ − π ζ + 29 π . (166) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 51
7. Conclusion
In these lectures we described mathematical and algebraic structures gov-erning multiple polylogarithms, a class of special functions through whichlarge classes of multi-loop Feynman integrals can be expressed. In particu-lar, we discussed functional equations for multiple polylogarithms and howthese relations are governed by the Hopf algebra structure underlying thesefunctions.Although progress in understanding the mathematics underlying multi-loop integrals has been fast over the last couple of years and many resultshave been obtained that were thought impossible only a few years ago,there is still a lot to do. It is known that starting from two loops notevery Feynman integrals can be expressed through multiple polylogarithmsalone, but generalisations of polylogarithms to elliptic curves appear [77–81]. Currently, only very little is known about these functions, both on thephysics and on the mathematics side. Understanding the structure of thesefunctions and how the structures presented in these lectures generalise tohigher genus is a fascinating topic, that will most likely lead to new resultsboth in physics and in number theory.
8. Appendix8.1.
Rings and fields A ring is a set R equipped with an addition + and a multiplication · suchthat(i) R is an additive commutative group.(ii) The multiplication is associative and has a unit element.(iii) The distributivity law holds: a · ( b + c ) = a · b + a · c , ( a + b ) · b = a · b + a · c . (167)Note that the multiplication may be commutative, but this is not manda-tory. Moreover, we do not require R to be a multiplicative group, i.e., notevery element has a multiplicative inverse. If every element has a multi-plicative inverse and the multiplication is commutative, then we call R a field . A ring homomorphism is a map φ between two rings such that thering structure is preserved, i.e., ∀ a, b ∈ R , φ ( a + b ) = φ ( a ) + φ ( b ) and φ ( a · b ) = φ ( a ) · φ ( b ) . (168) ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 52 C. Duhr
Tensor products and algebras
Consider two vector spaces V and W . The tensor product of V and W isdefined by the following universal property: there is a unique vector space(unique up to isomorphism), denoted by V ⊗ W , together with a bilinearmap τ : V × W → V ⊗ W such that for every vector space E and everybilinear map b : V × W → E there is a unique linear map β : V ⊗ W → E such that b = β ◦ τ .An algebra is a vector space A equipped with a multiplication thatturns it into a ring. In other words, there is a map m : A × A → A such that m ( a, b ) = a · b . The distributivity law implies that m is bilinear.Then, according to the defining property of the tensor product, there isa linear map µ : A ⊗ A → A such that µ ( a ⊗ b ) = m ( a, b ) = ab . Inthe following, we therefore define an algebra as a vector space A with alinear map µ : A ⊗ A → A and a unit element, and the multiplication isassociative, µ (id ⊗ µ ) = µ ( µ ⊗ id) . (169)An algebra homomorphism is a map φ that preserves the algebra structure,i.e., it is linear and φ ( a · b ) = φ ( a ) · φ ( b ).If A and B are algebras, then their tensor product A ⊗ B is also analgebra, and the multiplication is given by( a ⊗ b ) · ( a ⊗ b ) = ( a · a ) ⊗ ( b · b ) . (170)An algebra is called graded if it is a direct sum as a vector space A = ∞ M n =0 A n , (171)and the multiplication preserves the weight A m · A n ⊂ A m + n . (172) Coalgebras and Hopf algebras
Consider two (complex, real or rational) vector spaces V and W and alinear map φ : V → W . Our goal is to understand what this map is interms of the dual spaces. The dual space W ∗ of W is the vector space of alllinear functionals ϕ : W → K (with K = C , R or Q ) and which associatesto w ∈ W an element ϕ ( w ) ≡ h ϕ | w i . Every linear form is determined by ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 53 taking the scalar product with a certain constant vector. The hermitianconjugate φ † : W ∗ → V ∗ is defined by h ϕ | φ ( v ) i ≡ h φ † ( ϕ ) | v i , for v ∈ V and ϕ ∈ W ∗ . (173)In particular, if V = W ⊗ W , we have h ϕ | φ ( a ⊗ b ) i ≡ h φ † ( ϕ ) | a ⊗ b i , for a, b ∈ W and ϕ ∈ W ∗ , (174)where the scalar product in W ⊗ W is defined via h a ⊗ b | c ⊗ d i ≡ h a | c i h b | d i , for a, b, c, d ∈ W . (175)In the special case where W is an algebra, we have a natural map µ : W ⊗ W → W , and so we may ask what the ‘hermitian conjugate’∆ ≡ µ † : W → W ⊗ W of the multiplication µ is. Obviously, ∆ is linear.Writing ∆( ϕ ) = P i ϕ (1) i ⊗ ϕ (2) i , we see that h ϕ | ( a · b ) · c i = h ϕ | µ ( µ ( a ⊗ b ) ⊗ c ) i = h ∆( ϕ ) | µ ( a ⊗ b ) ⊗ c i = X i h ϕ (1) i ⊗ ϕ (2) i | µ ( a ⊗ b ) ⊗ c i = X i h ϕ (1) i | µ ( a ⊗ b ) i h ϕ (2) i | c i = X i h ∆( ϕ (1) i ) | a ⊗ b i h ϕ (2) i | c i = h (∆ ⊗ id)∆( ϕ ) | a ⊗ b ⊗ c i . (176)Similarly, we get h ϕ | a · ( b · c ) i = h (id ⊗ ∆)∆( ϕ ) | a ⊗ b ⊗ c i , and the associativityof µ implies that these two expressions must be equal, and so we must have(∆ ⊗ id)∆ = (id ⊗ ∆)∆ , (177)i.e., we see that ∆ is coassociative. In other words, if W is an algebra then W ∗ is a coalgebra d .A coalgebra W is called graded if it is a direct sum of vector spaces, W = ∞ M n =0 W n , (178)and the coproduct preserves the weight∆( W n ) ⊂ n M k =0 W k ⊗ W n − k . (179) d There is also a similar dual notion of the unit element of W . ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 54 C. Duhr A bialgebra is an algebra that is at the same time a coalgebra, and theproduct and the coproduct are compatible in the sense that∆( a · b ) = ∆( a ) · ∆( b ) , (180)i.e., the coproduct is an algebra homomorphism.A Hopf algebra H is a bialgebra equipped with an antipode, a linearmap S : H → H satisfying certain properties. It turns out that in our casethe antipode does not contain new information because one can show thatif a bialgebra H is graded and H = Q , then there is a unique antipode thatturns H into a Hopf algebra. In other words, the antipode does not carryany information that was not already present at the level of the bialgebra,and we therefore never consider it explicitly in these lectures. Acknowledgements
I would each like the thank the TASI organisers (Lance Dixon and FrankPetriello) for inviting me to lecture at the TASI school 2014 and for creat-ing such a stimulating environment for students and lecturers alike. I amgrateful to Nicolas Deutschmann for comments on the manuscript. Thiswork is supported by the “Fonds National de la Recherche Scientifique”(FNRS), Belgium.
References [1] R. Ap´ery, Irrationali´e de ζ et ζ , Ast´erique . (11-13) (1979).[2] M. Kontsevich and D. Zagier. Periods. In eds. B. Engquis and W. Schmid, Mathematics unlimited – 2001 and beyond , p. 771. Springer (2001).[3] C. Bogner and S. Weinzierl, Periods and Feynman integrals,
J.Math.Phys. , 042302 (2009). doi: 10.1063/1.3106041.[4] A. B. Goncharov, Multiple polylogarithms, cyclotomy and modular com-plexes, Math. Research Letters . (4), 497 (1998).[5] A. Goncharov, Multiple polylogarithms and mixed Tate motives (2001).[6] J. A. Lappo-Danilevsky, M´emoires sur la th´eorie des syst´emes des ´equationsdiff´erentielles lin´eaires. Vol. II, Travaux Inst. Physico-Math. Stekloff . , 5–210 (1935).[7] K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. , 831 (1977).[8] J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, Special val-ues of multiple polylogarithms, Transactions of the American MathematicalSociety . (2001).[9] E. Remiddi and J. A. M. Vermaseren, Harmonic polylogarithms, Int. J. Mod.Phys.
A15 , 725–754 (2000). ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 55 [10] T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polyloga-rithms, Comput. Phys. Commun. , 296–312 (2001).[11] D. Maitre, HPL, a Mathematica implementation of the harmonic polyloga-rithms,
Comput. Phys. Commun. , 222–240 (2006).[12] D. Maitre, Extension of HPL to complex arguments,
Comput.Phys.Commun. , 846 (2012). doi: 10.1016/j.cpc.2011.11.015.[13] J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polyloga-rithms,
Comput. Phys. Commun. , 177 (2005). doi: 10.1016/j.cpc.2004.12.009.[14] S. Buehler and C. Duhr, CHAPLIN - Complex Harmonic Polylogarithms inFortran,
Comput. Phys. Commun. , 2703–2713 (2014).[15] T. Gehrmann and E. Remiddi, Two-loop master integrals for γ ∗ → Nucl. Phys.
B601 , 248–286 (2001).[16] T. Gehrmann and E. Remiddi, Two loop master integrals for γ ∗ → Nucl.Phys.
B601 , 287–317 (2001). doi: 10.1016/S0550-3213(01)00074-8.[17] S. Di Vita, P. Mastrolia, U. Schubert, and V. Yundin, Three-loop masterintegrals for ladder-box diagrams with one massive leg,
JHEP . , 148(2014). doi: 10.1007/JHEP09(2014)148.[18] T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensionalharmonic polylogarithms, Comput. Phys. Commun. , 200–223 (2002).[19] U. Aglietti and R. Bonciani, Master integrals with 2 and 3 massive propaga-tors for the 2 loop electroweak form-factor - planar case,
Nucl.Phys.
B698 ,277–318 (2004). doi: 10.1016/j.nuclphysb.2004.07.018.[20] R. Bonciani, G. Degrassi, and A. Vicini, On the Generalized Harmonic Poly-logarithms of One Complex Variable,
Comput.Phys.Commun. , 1253–1264 (2011). doi: 10.1016/j.cpc.2011.02.011.[21] J. Ablinger, J. Bl¨umlein, and C. Schneider, Harmonic Sums and Polylog-arithms Generated by Cyclotomic Polynomials,
J.Math.Phys. , 102301(2011). doi: 10.1063/1.3629472.[22] J. Bl¨umlein, D. Broadhurst, and J. Vermaseren, The Multiple Zeta ValueData Mine, Comput.Phys.Commun. , 582–625 (2010). doi: 10.1016/j.cpc.2009.11.007.[23] M. E. Hoffman, The Algebra of Multiple Harmonic Sums,
Jounal of Algebra . , 477–495 (1997).[24] F. C. Brown, Mixed Tate Motives over Spec( Z ), Annals of Mathematics . (1) (2012).[25] L. Lewin, Polylogarithms and Associated Functions . North Holland (1981).[26] L. Lewin, ed.,
Structural Properties of Polylogarithms . Amer. Math. Soc.(1991).[27] J. Ablinger, A Computer Algebra Toolbox for Harmonic Sums Related toParticle Physics (2010).[28] J. Ablinger, J. Bl¨umlein, and C. Schneider, Analytic and Algorithmic As-pects of Generalized Harmonic Sums and Polylogarithms,
J.Math.Phys. ,082301 (2013). doi: 10.1063/1.4811117.[29] A. Goncharov, Galois symmetries of fundamental groupoids and non- ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 56 C. Duhr commutative geometry,
Duke Math.J. , 209 (2005). doi: 10.1215/S0012-7094-04-12822-2.[30] C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgsboson amplitudes,
JHEP . , 043 (2012). doi: 10.1007/JHEP08(2012)043.[31] F. Brown, On the decomposition of motivic multiple zeta values (2011).[32] A. B. Goncharov, M. Spradlin, C. Vergu, and A. Volovich, Classical Poly-logarithms for Amplitudes and Wilson Loops, Phys.Rev.Lett. , 151605(2010). doi: 10.1103/PhysRevLett.105.151605.[33] F. C. Brown, Multiple zeta values and periods of moduli spaces M ,n ( R ), Annales Sci.Ecole Norm.Sup. , 371 (2009).[34] C. Duhr, H. Gangl, and J. R. Rhodes, From polygons and symbols to poly-logarithmic functions, JHEP . , 075 (2012). doi: 10.1007/JHEP10(2012)075.[35] L. Landau, On analytic properties of vertex parts in quantum field theory, Nucl.Phys. , 181–192 (1959).[36] D. Gaiotto, J. Maldacena, A. Sever, and P. Vieira, Pulling the straps ofpolygons, JHEP . , 011 (2011). doi: 10.1007/JHEP12(2011)011.[37] F. Brown and K. Yeats, Spanning forest polynomials and the transcendentalweight of Feynman graphs, Commun.Math.Phys. , 357–382 (2011). doi:10.1007/s00220-010-1145-1.[38] F. Brown, O. Schnetz, and K. Yeats, Properties of c invariants of Feynmangraphs (2012).[39] K. Chetyrkin and F. Tkachov, Integration by Parts: The Algorithm to Cal-culate beta Functions in 4 Loops, Nucl.Phys.
B192 , 159–204 (1981). doi:10.1016/0550-3213(81)90199-1.[40] S. Laporta and E. Remiddi, The Analytical value of the electron ( g − α in QED, Phys.Lett.
B379 , 283–291 (1996). doi: 10.1016/0370-2693(96)00439-X.[41] S. Laporta, High precision calculation of multiloop Feynman integrals bydifference equations,
Int.J.Mod.Phys.
A15 , 5087–5159 (2000). doi: 10.1016/S0217-751X(00)00215-7.[42] C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higherorder perturbative calculations,
JHEP . , 046 (2004). doi: 10.1088/1126-6708/2004/07/046.[43] A. Smirnov, Algorithm FIRE – Feynman Integral REduction, JHEP . ,107 (2008). doi: 10.1088/1126-6708/2008/10/107.[44] A. Smirnov and V. Smirnov, FIRE4, LiteRed and accompanying tools tosolve integration by parts relations, Comput.Phys.Commun. , 2820–2827(2013). doi: 10.1016/j.cpc.2013.06.016.[45] A. V. Smirnov, FIRE5: a C++ implementation of Feynman Integral RE-duction (2014).[46] A. von Manteuffel and C. Studerus, Reduze 2 - Distributed Feynman IntegralReduction (2012).[47] R. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction(2012). ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 57 [48] A. Smirnov and A. Petukhov, The Number of Master Integrals is Finite, Lett.Math.Phys. , 37–44 (2011). doi: 10.1007/s11005-010-0450-0.[49] R. N. Lee and A. A. Pomeransky, Critical points and number of masterintegrals, JHEP . , 165 (2013). doi: 10.1007/JHEP11(2013)165.[50] J. M. Henn, Multiloop integrals in dimensional regularization made sim-ple, Phys.Rev.Lett. (25), 251601 (2013). doi: 10.1103/PhysRevLett.110.251601.[51] T. Gehrmann, A. von Manteuffel, L. Tancredi, and E. Weihs, The two-loopmaster integrals for qq → V V , JHEP . , 032 (2014). doi: 10.1007/JHEP06(2014)032.[52] S. Caron-Huot and J. M. Henn, Iterative structure of finite loop integrals, JHEP . , 114 (2014). doi: 10.1007/JHEP06(2014)114.[53] M. Argeri, S. Di Vita, P. Mastrolia, E. Mirabella, J. Schlenk, et al., Magnusand Dyson Series for Master Integrals, JHEP . , 082 (2014). doi: 10.1007/JHEP03(2014)082.[54] A. Kotikov, Differential equations method: The Calculation of vertextype Feynman diagrams, Phys.Lett.
B259 , 314–322 (1991). doi: 10.1016/0370-2693(91)90834-D.[55] A. Kotikov, Differential equation method: The Calculation of N pointFeynman diagrams,
Phys.Lett.
B267 , 123–127 (1991). doi: 10.1016/0370-2693(91)90536-Y.[56] T. Gehrmann and E. Remiddi, Differential equations for two loop four pointfunctions,
Nucl.Phys.
B580 , 485–518 (2000). doi: 10.1016/S0550-3213(00)00223-6.[57] V. Del Duca, C. Duhr, and V. A. Smirnov, An Analytic Result for the Two-Loop Hexagon Wilson Loop in N = 4 SYM,
JHEP . , 099 (2010). doi:10.1007/JHEP03(2010)099.[58] V. Del Duca, C. Duhr, and V. A. Smirnov, The Two-Loop Hexagon WilsonLoop in N = 4 SYM, JHEP . , 084 (2010). doi: 10.1007/JHEP05(2010)084.[59] T. Gehrmann, M. Jaquier, E. Glover, and A. Koukoutsakis, Two-Loop QCDCorrections to the Helicity Amplitudes for H → JHEP . ,056 (2012). doi: 10.1007/JHEP02(2012)056.[60] A. Brandhuber, G. Travaglini, and G. Yang, Analytic two-loop form factorsin N=4 SYM, JHEP . , 082 (2012). doi: 10.1007/JHEP05(2012)082.[61] R. Bonciani, A. Ferroglia, T. Gehrmann, A. von Manteuffel, and C. Studerus,Light-quark two-loop corrections to heavy-quark pair production in thegluon fusion channel, JHEP . , 038 (2013). doi: 10.1007/JHEP12(2013)038.[62] T. Gehrmann, L. Tancredi, and E. Weihs, Two-loop master integrals for q ¯ q → V V : the planar topologies,
JHEP . , 070 (2013). doi: 10.1007/JHEP08(2013)070.[63] A. B. Goncharov, Volumes of hyperbolic manifolds and mixed Tate motives, J. Amer. Math. Soc. , 569–618 (1999).[64] R. Kellerhals, Volumes in hyperbolic 5–space, GAFA . , 640–667 (1995).[65] A. B. Goncharov, Polylogarithms in arithmetic and geometry, Proc. of the ecember 1, 2014 1:42 World Scientific Review Volume - 9in x 6in Duhr˙TASI page 58 C. Duhr
International Congress of Mathematicians . (1995).[66] J. Golden, M. F. Paulos, M. Spradlin, and A. Volovich, CLuster Polyloga-rithms for Scattering Amplitudes (2014).[67] R. Ree, Lie Elements and an Algebra Associated With Shuffles, Annals ofMathematics . (2), 210–220 (1958).[68] G. Griffing, Dual Lie Elements and a Derivation for the Cofree Coassocia-tive Coalgebra, Proceeding of the American Mathematical Society . (11),3269–3277 (1995).[69] F. Brown,The Massless higher-loop two-point function, Commun.Math.Phys. ,925–958 (2009). doi: 10.1007/s00220-009-0740-5.[70] F. C. Brown, On the periods of some Feynman integrals (2009).[71] E. Panzer, On the analytic computation of massless propagators in dimen-sional regularization (2013). doi: 10.1016/j.nuclphysb.2013.05.025.[72] E. Panzer, On hyperlogarithms and Feynman integrals with divergences andmany scales,
JHEP . , 071 (2014). doi: 10.1007/JHEP03(2014)071.[73] E. Panzer, Algorithms for the symbolic integration of hyperlogarithms withapplications to Feynman integrals (2014).[74] C. Anastasiou, C. Duhr, F. Dulat, and B. Mistlberger, Soft triple-realradiation for Higgs production at N3LO, JHEP . , 003 (2013). doi:10.1007/JHEP07(2013)003.[75] J. Ablinger, J. Bl¨umlein, C. Raab, C. Schneider, and F. Wißbrock, Calcu-lating Massive 3-loop Graphs for Operator Matrix Elements by the Methodof Hyperlogarithms, Nucl.Phys.
B885 , 409–447 (2014). doi: 10.1016/j.nuclphysb.2014.04.007.[76] C. Bogner and F. Brown, Feynman integrals and iterated integrals on modulispaces of curves of genus zero (2014).[77] M. Caffo, H. Czyz, S. Laporta, and E. Remiddi, The Master differentialequations for the two loop sunrise selfmass amplitudes,
Nuovo Cim.
A111 ,365–389 (1998).[78] S. Laporta and E. Remiddi, Analytic treatment of the two loop equal masssunrise graph,
Nucl.Phys.