Preprint typeset in JHEP style - PAPER VERSION IPPP//56 DCPT// From polygons nd symols to polylogrithmic functions Clude Duhr Institute for Prticle Physics Phenomenology, University of Durhm Durhm, DH 3LE, U.K. nd Institut für theoretische Physik, ETH Zürich, Wolfgng-Pulistr. 7, CH-893, Switzerlnd E-mil: duhrc@itp.phys.ethz.ch Herert Gngl Deprtment of Mthemticl Sciences, University of Durhm Durhm, DH 3LE, U.K. E-mil: herert.gngl@durhm.c.uk John R. Rhodes Deprtment of Mthemticl Sciences, University of Durhm Durhm, DH 3LE, U.K. E-mil: j.r.rhodes@durhm.c.uk Astrct: We present review of the symol mp, mthemticl tool tht cn e useful in simplifying epressions mong multiple polylogrithms, nd recll its min properties. A recipe is given for how to otin the symol of multiple polylogrithm in terms of the comintoril properties of n ssocited rooted decorted polygon. We lso outline systemtic pproch to constructing function corresponding to given symol, nd illustrte it in the prticulr cse of hrmonic polylogrithms up to weight four. Furthermore, prt of the miguity of this process is highlighted y ehiiting fmily of non-trivil elements in the kernel of the symol mp for ritrry weight. Keywords: Multiple polylogrithms, Feynmn integrls, loop computtions, symol mp, iterted integrls, decorted polygons.
. Introduction Polylogrithms nd their multivrile generliztions [, ] ply n eqully importnt role in modern mthemtics nd in physics. In mthemtics they occur for instnce in connection with lgeric K-theory nd mied Tte motives, e.g. [3, 4, 5, 6, 7, 8], with Hilert s third prolem on scissors congruences, e.g. [9,, ], s volume functions for hyperolic spces, e.g. [, 3, 8, 4, 5,, 6], nd re lso relted to chrcteristic clsses, e.g. [7], specil vlues of L-functions in lgeric numer theory, e.g. [8, 5, 7], lgeric cycles, e.g. [9,, ] or, in the form of iterted integrls, in lgeric topology, e.g. [, 3, 4]. In physics, the computtion of higher order corrections to physicl oservles requires the nlyticl evlution of Feynmn integrls tht cn generlly e epressed in terms of specil clsses of multiple polylogrithms, e.g. [5, 6, 7, 8, 9, 3, 3, 3, 33, 34, 35, 36, 37, 38, 39, 4, 4, 4, 43, 44, 45, 46, 47, 48, 49, 5, 5, 5, 53, 54, 55, 56, 57]. While in ll of these pplictions it would e desirle to hve miniml spnning set sis functions in physics prlnce for the polylogrithmic epressions involved in given prolem, it is well known tht these ltter functions stisfy vrious intricte functionl equtions mong themselves, mking the question of how to find miniml spnning set very hrd to nswer in generl. As consequence, seemingly complicted results, sy for Feynmn integrl, my dmit much shorter nlytic representtion, the simplicity of the nswer eing hidden due to the eistence of n undnce of functionl equtions mong these functions. There is thus strong interest for etter understnding of the functionl equtions mong multiple polylogrithms, oth from forml mthemticl stndpoint nd in view of prcticl pplictions in physics. A wy to pproch functionl equtions mong multiple polylogrithms is provided y the so-clled symol mp, liner mp tht ssocites to ech multiple polylogrithm of weight n n element in the n-fold tensor power of some vector spce of one-forms. The virtue of the symol mp is tht it cptures to good etent the min comintoril nd nlyticl properties of certin trnscendentl functions, nd in prticulr it is epected tht ll functionl equtions mong multiple polylogrithms re in the kernel of the symol mp. Loosely speking, this mens tht necessry condition for two epressions written in terms of multiple polylogrithms to e equl modulo functionl equtions is tht they hve the sme symol, condition tht is usully much esier to check thn proving equlity t the level of the functions. The inverse prolem sometimes clled integrtion of symol of finding function whose symol mtches given tensor stisfying certin integrility condition is much hrder nd we know of no generl lgorithm to construct such function. While specil cses of the symol mp hve een profitly used y mthemticins for over two decdes for emple in connection with functionl equtions see e.g. refs. [5, 7, 58, 59], it hs only very recently een introduced into physics in the contet of the N = 4 Super Yng-Mills SYM theory in ref. [6], where it ws pplied to gretly simplify the nlytic epression for the two-loop si-point reminder function otined in ref. [6, 6]. In the wke of tht work, the symol mp hs seen vrious pplictions, mostly in the contet of N = 4 SYM. In prticulr, y now the symols of ll two-loop reminder functions re completely known [63], while t three loops the symols of the reminder functions for the
hegon in generl kinemtics [64] nd for the octgon in specil kinemtics [65] re known up to some free prmeters tht could not e fied from generl considertions. However, only in the ltter octgon cse n integrted form of the symol is lso known. Other pproches, iming t the determintion of the symol of loop mplitudes y eploiting the opertor product epnsion in the colliner limit [66, 67] or the reltionship etween Feynmn integrls nd the volumes of polyhedr in non-eucliden spces [68, 69], hve lso een considered. Furthermore, the symol mp ws recently used to otin compct nlytic epressions for certin one-loop hegon integrls in D = 6 dimensions [7, 7, 7, 73]. More phenomenologicl pplictions, s for emple in ref. [74], hve lso een considered. The im of this pper is twofold: While the symol mp hs lredy een etensively used in the N = 4 SYM community in physics, it seems still rther little known in other res of physics in which the computtion of Feynmn integrls plys n importnt role. On the one hnd, we therefore present concise review on this topic, putting specil emphsis on how to pply the symol mp to otin simpler or shorter nlytic results for functions rising from certin Feynmn integrls. On the other hnd, we elieve tht our work goes eyond the eisting literture on the suject in vrious spects. While so fr the symol of trnscendentl function ws defined recursively y considering iterted differentils, we introduce simple digrmmtic rule tht llows to directly red off the symol from ll possile tringultions of certin decorted polygon ssocited to multiple polylogrithm []. Furthermore, we lso ddress the prolem of how to integrte symol to function y presenting n effective pproch to construct cndidte spnning set of functions in terms of which the symol might e integrted. The structure of the pper is s follows: In section we give short review of multiple polylogrithms nd of their properties. In section 3 we review the min properties of the symol mp nd we show how to otin the symol of multiple polylogrithm s the weighted sum of ll possile miml dissections of certin decorted polygon ssocited to the polylogrithm. In section 4 we give short emple of how to integrte symol following the pproch introduced in refs. [6, 75], efore generlizing this procedure to higher weights in section 5. In order to highlight remining difficulties nd miguities when trying to integrte to function, we lso give fmily of non-trivil elements in the kernel of the symol mp. We illustrte these concepts in section 6 where we pply them to derive spnning set up to weight four for specil clss of multiple polylogrithms, the so-clled hrmonic polylogrithms [5]. The ppendices contin summry of the mthemticl notions used throughout the pper, s well s some technicl detils nd proofs left out in the min tet. We lso include n ppendi with collection of symols for multiple polylogrithms up to weight four. Remrk: The uthors wish the reder to e wre tht this pper contins the work of oth physicists nd mthemticins. The fct tht the topics covered re oth found in, nd indeed re currently of interest to, oth mthemtics nd physics is very eciting. As consequence of this it should e noted tht the pper hs een written to try to ccommodte the widest udience possile nd so the uthors hope the reder understnds when rguments delve deeper thn my e felt necessry.
. Short review of multiple polylogrithms Definition. Multiple polylogrithms cn e defined recursively, for n, vi the iterted integrl [, ] dt G,..., n ; = G,..., n ; t,. t with G = G; =, n eception eing when = in which cse we put G = clerly ny epression... should e zero, nd with i C re chosen constnts nd is comple vrile. In the following, we will lso consider G,..., n ; to e functions of,..., n. In the specil cse where ll the i s re zero, we define, using the ovious vector nottion n =,...,, C, }{{} n G n ; = n! logn,. consistent with the cse n = ove. Note tht, while in the Mthemtics literture these functions pper lredy in the erly th century in the works of Poincré nd of Lppo- Dnilevsky [76] s hyperlogrithms, s well s in the 96 s in Chen s work on iterted integrls e.g., [], in the physics literture these functions re often clled Gonchrov polylogrithms, due to the welth of structure tht the ltter hs estlished for them over the lst yers. Throughout this pper, we follow the physics convention for the definition of the iterted integrls, which differs slightly from the mthemticl one; e.g., in ref. [], the function corresponding to G,..., n ; would e denoted I; n,..., ;, i.e., with the reverse order of the i ut keeping the sme vrile. We will refer to the vector =,..., n s the vector of singulrities ttched to the multiple polylogrithm nd the numer of elements n, counted with multiplicities, in tht vector is clled the weight of the multiple polylogrithm. Properties. We collect here numer of useful nd well-known properties cf. e.g. ref. [, 8]. Iterted integrls form shuffle lger [78] see ppendi A for short review of shuffle lgers, which llows one to epress the product of two multiple polylogrithms of weight n nd n s liner comintion with integer coefficients of multiple polylogrithms of weight n + n, vi G,..., n ; G n +,..., n +n ; = G σ,..., σn +n ;,.3 σ Σn,n where Σn, n denotes the set of ll shuffles of n + n elements, i.e., the suset of the symmetric group S n +n defined y cf. ref. [], eq..5.6 Σn, n = {σ S n +n σ <... < σ n nd σ n + <... < σ n +n }..4 The lgeric properties of multiple polylogrithms imply tht not ll the G ; for fied re independent, ut tht there re polynomil reltions mong them. In prticulr, we In sense, they lredy mde n ppernce in Kummer s pioneering work [77] in 84. 3
cn reduce them, modulo products of lower weight functions, to functions whose rightmost inde of ll the vectors of singulrities is non-zero prt from ojects of the form G n ;, e.g., G,, ; = G, ; G; G,, ; G,, ; = G, ; G; G,, ; G, ; G; G,, ; = G, ; G; + G,, ; G, ; G;,.5 where the middle summnd is of the desired form nd the remining summnds re products. If the rightmost inde n of is non-zero, then the function G ; is invrint under rescling of ll its rguments, i.e., for ny k C we hve Gk ; k = G ; n..6 Furthermore, multiple polylogrithms stisfy the Hölder convolution [79], i.e., whenever nd n, we hve, p C, n G,..., n ; = k G k,..., ; G k+,..., n ;..7 p p k= Below in section 5 we will e prticulrly interested in the limiting cse p of this identity, G,..., n ; = n G n,..., ;..8 Whenever they converge, multiple polylogrithms cn eqully well e represented [] s multiple nested sums e.g., for i < Li m,...,m k,..., k = n <n < <n k n n n k k n m nm n m k k = n k = n k k n m k k n k n k = n n... n m. n =.9 Note tht we re using Gonchrov s originl summtion convention []; other uthors define Li m,...,m k,..., k using the reverse summtion convention insted, i.e. n > > n k. The G nd Li functions define in fct the sme clss of functions nd re relted y Li m,...,m k,..., k = k G mk,...,m,...,,. k... k note the reverse order of the indices for G where we used the nottion G m,...,m k t,..., t k = G,...,, t }{{},...,,...,, t }{{} k ;.. m m k It is possile to find closed epressions for very few specil clsses of multiple polylogrithms, for ritrry weight, in terms of clssicl polylogrithm functions, e.g., for we hve G n ; = n! logn, G n, ; = Li n G n ; = n! logn, G n, p ; = p S n,p,,. 4
where S n,p denotes the Nielsen polylogrithm [8]. Moreover, up to weight three, multiple polylogrithms re well-known to e epressile in terms of ordinry logrithms, dilogrithms nd trilogrithms cf. ref. [8], 8.4.3, implicitly, s well s refs. [, 5]. In prticulr, if nd re non-zero nd different, we find G, ; = Li Li + log log..3 Aim. The im of this pper is to present n lgorithmic pproch how to del with in fct rther to circumvent the complicted functionl equtions tht relte multiple polylogrithms, nd how to find, given choice of certin singulrities i, possily miniml spnning set for functions in which to epress multiple polylogrithms with singulrities only in these i, provided such set eists. The pproch we present is rther generic nd cn e pplied to ny epression involving multiple polylogrithms. This is mde possile y using results closely relted to work of Gonchrov, Sprdlin, Vergu nd Volovich [6], which in turn ws inspired y the theory of mied Tte motives, nd in prticulr y using certin tensor clculus ssocited to iterted integrls, which is clled symol clculus in the following the nme symol originting from [6] nd from ref. [89], nd which we will review in the net section. An importnt remrk is tht the construction of symol seems to e rther specil cse of very generl construction y Chen [], where it ppers s the imge of n iterted integrl s -cocycle in the so-clled r construction ttched to, sy, X equl to the projective line minus numer of points more generlly, the construction hs een investigted for hyperplne configurtion [9], 3, nd it lnds in the n-fold tensor product of the vector spce of -forms on the underlying spce X. Moreover, Chen chrcterised the imge s the forml words in these -forms stisfying nturl integrility condition. Therefore, it would seem pproprite to reflect this in the nottion for this oject, e.g. s Chen symol. A polygon ttched to n iterted integrl enjoys the useful property tht it gives very concise wy of eplicitly producing integrle words, i.e. Chen symols, of tht kind. As n ppliction, we restrict ourselves in section 6 to specific suclss of multiple polylogrithms tht re of prticulr importnce in pplictions in high-energy physics. These so-clled hrmonic polylogrithms HPL s H ; were first singled out nd thoroughly studied in ref. [5]. HPL s correspond to specil cse of the iterted integrl defined in eq.. where i {,, }. More precisely, they re defined vi H ; = k G ;, i {,, },.4 where k is the numer of elements in equl to +. Mny one nd two-loop Feynmn integrls cn e epressed in terms of HPL s up to weight four nd generliztions Let us point out tht this is fr from eing the first ehiit of direct connection etween mied Tte motives nd mthemticl physics, s such reltionship hs een eplored, e.g., y Kreimer in work with Bloch nd Esnult [8, 83], such connection ws clerly pprent from letter correspondence etween Brodhurst nd Deligne [84] resulting e.g. in ref. [85], work of Belkle Brosnn [86] or more recently y Brown [87] nd others. One should lso mention work of Connes nd Mrcolli [88] in this direction. 5
thereof [6, 7]. As hrmonic polylogrithms re just specil cse of the multiple polylogrithms introduced t the eginning of this section, ll HPL s through weight three cn e epressed through clssicl polylogrithms. By contrst, similr to the generl cse of multiple polylogrithms, it is epected tht HPL s of weight 4 re no longer epressile in terms of clssicl ones lone. In section 6 we illustrte our technique y constructing spnning set of hrmonic polylogrithms in weight 4. 3. Symols nd polygons The differentil structure of multiple polylogrithms cn e cptured very well comintorilly using certin kind of decorted polygons with some dditionl structure, s developed in ref. [], where they were clled R-deco polygons. We note tht there re relted notions tht hd occurred previously in Gonchrov s work, e.g. in refs. [8, 9]. There is n lgeric oject ttched to such polygon, nd hence to the corresponding multiple polylogrithm. This oject, which hs een dued symol in ref. [6], is n element in tensor power of certin vector spce nd contins lot of informtion out the originl function. 3. An emple in nutshell In this susection we give quick ide of how, following ref. [], one cn ssocite to multiple polylogrithm or rther to n ssocited rooted decorted polygon its symol we show in section 3. tht this definition is equivlent to the definition given in ref. [6]. In the following susection we then give more detiled ccount of the construction. A multiple polylogrithm of weight n gives rise to certin n + -gon. As foreshdowing emple, we first give the 4-gon P = P c,,, ttched to some weight 3 multiple polylogrithm G,, c; = Li,, /c, /, / = I; c,, ; : P c,,, = c G,, c; which comes equipped with decortions in this order c,, nd, the ltter decortion eing for the distinguished root side drwn y doule line in the picture, nd lso crries informtion on the orienttion of the polygon in the form of ft verte which should e thought of s the first verte, while the root side djcent to the first verte is the lst side. In first step, one lists ll possile wys to drw the miml numer of non-intersecting rrows n rrow is directed line from verte of P to non-djcent side, which for n n + -gon mounts to n such rrows, nd one formlly dds the resulting ojects the frming polygon eing identicl, ut ech equipped with different miml set of rrows. In our emple n = 3, such miml set contins n = rrows, nd there re precisely different such sets, given y 6
c c c c c c c c c c c c In second step, to ech such miml set A of rrows in P, we ssocite rooted tree s the tree dul to the polygon dissection defined y the rrows whose decortions re decorted nd rooted -gons. As n emple, to the 4-gons in the lst column ove we ttch c c c c c c c c c c Any liner order on the vertices of such rooted tree which is comptile in the sense discussed elow in section 3. with the prtil order on it only the middle tree ove is non-liner hence llows more thn one such liner order now gives term in the symol SP ttched to P. In prctice, this mens tht every rnching in tree contriutes to the symol y the shuffle of the vertices tht pper on ech of its rnches see elow 7
for more detiled description. Third step: Ech of the -gons B in one of the liner orders on the vertices now is mpped vi suitle mp µ to rtionl function in the originl decortions of the polygons in the emple nturl trget spce would e the function field Q,, c, of y rtionl functions in the vriles,, c,. More precisely, if B = where y denotes the root decortion, then we mp B to µb = y/, provided, nd to µb = y otherwise. Lst step: Fiing the signs. We need to invoke sign for the individul elementry tensors, nd this sign is determined y using the numer of ckwrd rrows in dissection. In order to see this quickly, it is convenient to rek up the polygon t its first verte in the pictures it is typiclly indicted y ullet. Then we roll out the sequence of sides nd rrnge it s line from left to right, strting with the first verte nd ending with the root side; dissecting rrows inside the polygon will e stretched out in wy tht they still do not intersect. We give it for the third emple ove: c β α c β α Now ckwrd rrow is one which, in the rolled-out version of the polygon, hs its end point to the left of its strting point i.e. points from right to left, like β ove, while forwrd rrow hs it to its right i.e. points from left to right, like α ove. Here is more forml definition: There is nturl liner order on the sides e,..., e n of n n-gon s ove, strting with the non-root side e incident with the first verte nd ending with the root side e n in the emple ove it is the liner order given y the sides e,..., e 4 decorted y c,, nd, nd the vertices v = e n e the first verte, v = e e, v 3 = e e 3, v 4 = e 3 e 4. This induces liner order on the vertices v j which rise s the intersection of e j nd e j indices tken modulo n, where the first verte is the smllest element in tht order. Then non-trivil rrow cn e encoded y pir v j, e k with k / {j, j}, nd it is ckwrd if nd only if k < j. With these notions, the sign ttched to polygon P with miml rrow set A is given y #{ckwrd rrows of A}. In the three emples discussed ove in more detil we get two ckwrd rrows for the first miml dissection of the squre nd one ckwrd rrow for the remining two dissections. Putting ll of the ove ingredients together nd writing τ A for the tree dul to the miml set of rrows A, nd τ A, for its prtil order, the finl formul for the symol 8
SP of n n-gon P is SP = m sets A of rrows in P #{ckwrd rrows of A} liner orders λ comptile with τ A, n µ µ As n emple, the first nd third of the three miml sets of rrows ove give n. 3. +µ c c µ c µ = c c c nd µ c c µ µ = c, c respectively, while the middle term corresponding to non-liner dul tree, i.e. dul tree with rnchings contriutes vi the shuffle product of the two rnches µ c = µ c µ c = c µ c µ c µ µ c c where we introduced the symol for the shuffle product µ c µ c, = +. 3. Motivtion nd justifiction of this ssignment hs een given to n etent in ref. [], where it forms prt of n epression rising from the well-known r construction in lgeric topology pplied to differentil grded lger on the polygons ove which in turn is motivted y certin lgeric cycles originlly studied y Bloch [9] nd Bloch Kriz [9]. For n erlier ppernce of very similr structure clled the m invrint there, see ref. [8], 4.4. To summrize: n importnt prt of the differentil structure of weight n multiple polylogrithm is cptured y certin decorted n + -gon. More precisely, if the rguments of the multiple polylogrithm re epressed in terms of vriles/constnts,..., m for some m, the polygon is n n + -gon with decortion y simple epressions in,..., m ; now to this rooted decorted oriented polygon there is ttched in nturl wy n epression its symol in V n where V is finite rnk sumodule it might e convenient for the reder to think of V s finite dimensionl vector spce of the spce Q,..., m of infinite rnk, i.e., the invertile rtionl functions in the vriles,..., m. 9
3. Rules of symol clculus Roughly, symol is forml sum of elementry n-fold tensors n, nd one works in ech tensor fctor s with refined form of d log terms. In other words, ech fctor i in tensor product is tcitly understood s d log i d i i. 3.3 Furthermore, we use shuffle products nd the following rules essentilly oiling down to multilinerity, ut in n unusul form, s we pss from multiplictive to dditive nottion: Distriutivity. nd consequently C D = C D + C D 3.4 C n D = n C D, n Z, 3.5 where C nd D denote fied elementry tensors. Note tht n here is coefficient rther thn prt of the first tensor fctor; in prticulr, putting n = we see tht C D =. Neglecting torsion. We will work up to torsion, which mens tht we will put for ρ n n n-th root of unity. C ρ n D =, n Z, 3.6 Shuffle product. An importnt property of the symol is tht it preserves products: more precisely, it mps the product of two multiple polylogrithms to the shuffle product of their respective symols, i.e. S G,..., r ; G,..., s ; y = S G,..., r ; S G,..., s ; y 3.7 where is the symol used for the shuffle product of two tensors, defined on elementry tensors y... n n +... n +n = σ... σ n +n, σ Σn,n 3.8 where Σn, n ws defined in eq..4. For more detils on shuffle lgers, we refer to ppendi A. We note tht, on the left hnd side of eq. 3.7, the shuffle permuttions re pplied to the rguments of the two functions cf. e.g. eq..3, while on right hnd side one shuffles the tensor fctors insted, in completely nlogous fshion.
Note tht eq. 3.7 is rther non-trivil fct, s one cn lredy see in the first non-ovious cse: S G; G; = S G, ; + G, ; = = + + +, which grees with S G; S G; = +, due to cncelltions of terms. We will encounter epressions which involve oth tensor nd shuffle products in order to void writing mny prentheses, our convention is tht shuffle tkes precedence over tensor, i.e. c c. 3.9 Furthermore, we revite elementry tensors with the sme fctors s follows: n = } {{ }. 3. n times Refined d log terms. We emphsise here lredy, though, tht we will not tret d log c for rtionl constnt c s zero s opposed to ref. [6] since we would lose lot of importnt informtion this wy. Insted we etend the ove clculus to rtionl numers in complete nlogy with the ove; so we hve, e.g., C m 3 n 5 D = mc D + nc 3 D 5 C D. 3. Root decortion nnihiltes: Since G... ; =, we lso need to put SG... ; =, nd this indictes tht we cn nd will ignore polygons whose root side is decorted y. Liner orders of tree. For rooted tree T, which we view without fied emedding into the plne, hence e.g. we consider s equl the two trees v v v nd v v v There is nturl prtil order on its vertices v j j J, given s follows: the root verte v v j for ny j J, nd v j v k for v j v if nd only if there is direct pth from root to lef pssing first through v j nd then through v k. A liner order on the vertices of T which is comptile with the order is sequence v, v j,..., v jr of ll the vertices v j j J such tht v ji v jk implies j i j k. This mens tht if two vertices re in reltion with respect to the prtil order, then they
should e relted in ny comptile liner order in the sme wy, while if they re not relted in the prtil order, there is no condition for how they should e relted in tht liner order. In the emple, there re precisely two liner orders which re comptile with the prtil order, s the root verte lwys comes first: v, v, v nd v, v, v. Definition of symol. Now we re redy to give complete definition of the symol ttched to rooted decorted oriented n + -gon P with decortions t,..., t n,, nd then etend it y linerity nd shuffle product to ny sum of products of polygons, hence lso for multiple polylogrithms: SP = m sets A of rrows in P #{ckwrd rrows of A} liner orders λ comptile with τ A, µ π A,λ µ π A,λ, 3. where the -gons πν A,λ re determined y the miml dissection A together with the liner order λ which is comptile with the prtil order on τ A, the dul tree of the dissection A, in the mnner given ove in the second step of section 3. i.e. for ech -gon rising from the dissection of A there is verte of τ A decorted y tht -gon, nd for ny two -gons tht re djcent there is n edge in τ A connecting the corresponding vertices. Integrility condition. A very useful property of the rooted decorted polygons, found y the second uthor in collortion with F. Brown nd A. Levin, is tht ech polygon or rther its symol stisfies certin integrility condition. Indeed, n ritrry sum of elementry tensors does not necessrily lie in the imge of the symol mp. Insted, it ws pointed out in ref. [9], mking eplicit in specil cse the very generl pproch of Chen [], tht necessry nd sufficient condition for symol S = c I ω i ω im c I Q, 3.3 I=i,...,i m to e integrle to function is tht c I ω i ω ij ω ij+ ω im = for ll j m, 3.4 I=i,...,i m where ω ij ω ij+ denotes the usul eterior product of two differentil forms. We rewrite this for our purposes s [ ] c I d log ωij d log ω ij+ ωi ω ij ω ij+ ω im =, 3.5 I=i,...,i m where the hts indicte tht we omit the corresponding fctors in the tensor product. As n emple, we indicte the sttement for G, ;, whose symol S G, ; = + 3.6 stisfies d log d log d log d log +d log d log =, 3.7 n
where we recll tht d log = d/. Indeed, writing d log α = β dy dβ y β α y= 3.8 the left-hnd side of eq. 3.7 ecomes dy d y y= dy d y y= dy d y y= dy d y y= nd we find, e.g., tht the coefficient of d d is given y + The coefficients of d d nd d d vnish in similr wy. + dy d y y= dy d y y= 3.9 =. 3. Reltionship to the symol of ref. [6]. In ref. [6], the Gonchrov, Sprdlin, Vergu nd Volovich use the differentil eqution for multiple polylogrithms recursively to rrive t the definition of symol. More precisely, if F : C n C denotes comple vlued function depending on n comple vriles k, k n, the uthors of ref. [6] define the symol of the trnscendentl function F in the following recursive wy: if the totl differentil of F cn e epressed in the form df = i F i d log R i, 3. where F i nd R i re functions of the vriles k, nd R i re moreover rtionl functions, then the symol of F is defined recursively vi SF = i SF i R i. 3. In the cse where F is multiple polylogrithm, we cn write down the differentil of F in n eplicit form. For emple, in the specil cse where ll the rguments of the multiple polylogrithm re generic i.e., they re mutully different nd do not tke prticulr vlues, we otin [] n i i+ dg n,..., ; n = G n,..., â i,..., ; n d log. 3.3 i i i= The symol of G,..., n ; n is then defined in the form n i i+ SG n,..., ; n = SG n,..., â i,..., ; n. 3.4 i i i= The symol we just otined looks seemingly different from the definition we gve in eq. 3., which consists in summing over ll possile miml sets of rrows of the polygon P,..., n, n ssocited to G n,..., ; n. In the following we show 3
tht the two definitions re equivlent up to rerrngement of the terms in the sum, nd hence give rise to the sme tensor. Let us consider the n-gon P = P,..., n i.e. with sides decorted y i, the lst one n decorting the root side. We will show tht the symol of P stisfies the recursion 3.4. For simplicity, we will concentrte here on the cse of generic decortions. Let Λ P e the set of ll liner orders on the dul tree ttched to ny of the miml sets of rrows of P. Then there is n ovious ijection etween the terms in the doule sum in eq. 3. nd the elements in Λ P. We cn prtition Λ P y collecting ll those liner orders into suset which shre the sme lst -gon tht decortes the lst verte of this liner order. This prtitions Λ P into priori n susets, s those lst vertices correspond precisely to the -gons tht we cn cut off from P. Note tht cutting off the lst -gon in liner order on miml dissection corresponds to contrcting the ssocited edge in the dul tree. Note lso tht, clerly, the lst verte must e lef of the rooted dul tree, nd hence ech lst -gon necessrily cuts off two successive sides of P. Remrk: For ech n-gon P with n >, the three -gons involving the root side of P cn never ecome the lst one in ny liner order in Λ P. More eplicitly, these re the two -gons n n nd n. The former cn rise only y cutting off the root side, while the ltter cn rise oth y cutting off the root side nd y cutting off the first side nd the first verte. As consequence, Λ P prtitions into only n 3 non-empty susets of sme crdinlity of the ove type. In view of the ove, it is cler tht ny such suset indees ectly the liner orders Λ P on the dul trees of the miml dissections of the supolygon P of P which is otined from P y cutting off fied -gon, followed y contrcting the dissecting rrow to point. There re typiclly two wys of cutting off -gon in which such supolygon P cn i± occur: cutting off the -gons i =,..., n leves complementry supolygons P ± i i which re identified with P i upon contrction of the dissecting rrow. Note tht the two -gons will give terms of opposite prity, s precisely one of them will e forwrd rrow. The only eception rises from cutting off, which corresponds to forwrd rrow, for which only one complementry supolygon P + In summry, we get: Clim: There is ijection of sets n { : i+ } n Λ P i= Λ P + i i i= Λ P i cn occur. { i i }. 3.5 Moreover, the sign of miml set of rrows of Λ P grees with the sign of the corresponding miml set of rrows in Λ P + nd is opposite to the sign of the corresponding miml i set of rrows of Λ P. i 4
All we need to note here is tht the ove remrk ensures tht oth Λ P ± n nd Λ P re empty, so re left out t the right hnd side, nd tht cutting off -gon of the form i+ corresponds to forwrd rrow, hence contriutes sign + to the miml i i dissection, while corresponds to ckwrd rrow, hence contriutes sign. i Due to the ijection 3.5, we cn rewrite eq. 3. y first summing over ll supolygons P i ±, followed y sum over ll possile elements in Λ ± P. The inner sum then i evlutes to the symol of the supolygon P i ±, nd we re left with n SP = S P i + µ i= i+ i n S P i µ where the reltive minus sign rises ecuse, s discussed ove, Λ P ± i opposite signs. After identifiction of P + i i= i i, 3.6 contriute with nd P i, eq. 3.6 grees with the recursion 3.4 modulo the dditivity of the symol. In order to finish the proof, we need to show tht lso the ses of the recursions re the sme. It is indeed esy to check y eplicit computtion tht, e.g., the symol of G, ; 3 otined from the recursive definition 3.4 grees with the symol otined from our polygon construction, eq. 3.. We note tht in ref. [6] the d log of constnt is put to. Although this seems rther nturl nd turns out to e sufficient in severl cses, we dvocte to use refined version of this which is wht is typiclly used when working with symols in numer field: for ech element of set of multiplictively independent elements in given numer field one cn choose logrithm independently ut then the logrithm of ny product formed from those elements is determined. For emple, we will see in section 6 tht in the contet of hrmonic polylogrithms the only constnts tht need to e treted in this fshion re powers of, nd hence it is sufficient to think of s n irreducile element. The reson for considering this refined version is tht it is very helpful for recognising functions from which given symol might originte. In prticulr, it hs proved to e very useful, e.g., when recognizing HPL s for keeping trck of terms which come from shuffle product of polylogrithms, see section 6. While the symol of multiple polylogrithms otined y considering the miml set of rrows of the ssocited decorted polygon is equivlent to the symol otined from the recursive definition 3.4, we elieve tht oth pproches hve their virtues. While the ltter might e esier to implement into computer progrm in generl, it is strictly speking only vlid in the cse of generic rguments of the polylogrithms. Indeed, if the rguments re non-generic, we otin divergences in the right-hnd side of eq. 3.3, e.g. when i = i+ for some i. It is then in principle necessry to resort to creful regulriztion to del with the degenerte cses []. The definition of the symol sed on the decorted polygons, eing comintoril in nture, voids this prolem y construction nd llows to identify very esily the degenerte cses they correspond, e.g., to rrows ending on side whose decortion is zero, nd to discrd them from the strt, voiding in this wy the need of regulriztion. Furthermore, s we will see in the net section, 5
the polygon pproch hs nturlly uilt in the refined d log-prescription, ecuse the comintoril nture of the construction does not mke distinction etween constnts s e.g. for which one might e typiclly tempted to define d log =. Symols for clssicl polylogrithms. The polygons ttched to clssicl polylogrithms Li m = G,...,, ;, re given y decortions for the first side nd }{{} m terms for the remining non-root sides s well s for the root side. Their ttched symol consists of the negtive of single elementry tensor, in fct we hve S Li m = } {{ }, 3.7 m fctors where we hve m fctors weight m on the right hnd side. Note the prentheses which seprte the coefficient, here, from the ctul tensor, to void misinterprettion s. Such tensors hve long een considered in connection with functionl equtions of polylogrithms in fct, Zgier [5, 58] hs given criterion for such equtions uilt on those tensors, which hs een used cf. ref. [59] to find the first non-trivil equtions for Li 6 nd Li 7 eyond weight 7 none re known, nd the corresponding epressions for multiple polylogrithms re importnt lredy in Gonchrov s erly work e.g. [7] where he generlises the underlying tensor lger considerly. 4. A simple emple The symol ttched to G, ;. In this section we illustrte the fct tht the symol clculus provides convenient tool to detect functionl equtions mong multiple polylogrithms, on the emple of G, ; which hppens to coincide with the HPL H, ;. Even though we could of course immeditely pply eq..3 to epress G, ; in terms of clssicl polylogrithms, we will derive similr functionl eqution using the tensor clculus introduced in the previous section. The multiple polylogrithm G, ; is ssocited to trigon, G, ; P,, =. 4. The dissection of the trigon cn esily e trnslted into the tensor ssocited to the polylogrithm, µ µ + + µ µ µ µ 6
The lst line llows to red off the symol of G, ;, SG, ; = + = + + +. 4. Before turning to the question of how to ttch function to this symol, let us riefly comment on how this symol could hve een otined y using the recursive definition 3.4. Using eq. 3.3, we otin dg, ; = G ; d log + G; d log 4.3 = G ; d log + G; d log + G; d log. The three terms in the lst line of this eqution re in one-to-one correspondence with the three terms in the symol in eq. 4.. Note, however, tht we hd to tret ll the rguments of the three-vrile function G, ; s generic, nd to use the refined d logprescription, i.e. d log. Putting to zero ll the d log terms is equivlent to putting to zero ll elementry tensors in the symol where fctor inside tensor product is constnt [6]. As we will see elow, we prefer to keep these terms s they provide us with vlule informtion out the function tht should e ssocited to the symol. Attching function to symol. Since every multiple polylogrithm of weight two cn e epressed s comintion of clssicl polylogrithms, we mke the nstz tht G, ; cn e written, up to n dditive constnt, in the form c i Li f i + c jk logg j logh k, 4.4 i j,k such tht the tensor ssocited to this epression corresponds to the tensor in eq. 4., where c i nd c jk re rtionl numers nd f i, g j nd h k Q re rtionl functions. We sudivide this prolem into smller ones y postulting tht we cn distinguish etween the three different contriutions in eq. 4.. By using procedure suggested in ref. [6, 75] we cn distinguish the first sum from the second y projecting the respective symols onto their symmetric or lternting prts: ech term in the second sum will give zero contriution for the ltter one, while ech summnd in the first sum will give non-zero contriution. Indeed the tensor ssocited to product of logrithms is totlly symmetric, nd hence its contriution to the tensor vnishes if we project onto the ntisymmetric component of the tensor in eq. 4.. Preprtory steps: decomposing tensors into symmetric nd ntisymmetric prts. We recll tht, for vector spce V over C, sy, or more generlly over field of chrcteristic there is direct sum decomposition V V = V V V V other nottions, s used e.g. in refs. [5] or [7], re V V = Sym V, nd V V = V, nd V V is generted y for some, V, while V V is generted y where 7
we introduce the following rther stndrd nottions for symmetric nd ntisymmetric tensors, +,. 4.5 Bck to the emple. With this nottion, the decomposition of generic elementry rnk two tensor i.e., for some, V into its symmetric nd ntisymmetric components cn e epressed s = +. 4.6 Concentrting solely on the ntisymmetric component of eq. 4., nd using the ntisymmetry of the wedge product, which, e.g., induces =, we otin + + + = + = + +. 4.7 As the tensor ssocited to product of logrithms does not hve n ntisymmetric component, eq. 4.7 suggests tht it is the ntisymmetric prt of the tensor ssocited to some sum of dilogrithms, nd from eq. 3.7, it is esily identified s the ntisymmetric prt of S Li +. Hving identified the dilogrithm contriutions to G, ;, we cn proceed vi ootstrp procedure nd sutrct off this contriution, leving only totlly symmetric tensor S + G, ; + Li Fiing the constnt. the tensor ssocited to the comintion Li + = + = S log log + log. 4.8 We hve shown tht the tensor ssocited to G, ; equls + log log + log. It would e premture, however, to conclude tht oth epressions re equl, ut they re equl only up to n dditive constnt independent of. Indeed, specilizing to =, nd using the fct tht G, ; = = nd Li / = π log, we see tht { G, ; Thus, we otin [ + Li + log log + log G, ; = Li + ] } = = π. 4.9 + log log + log + π. 4. Note tht this dditive constnt is not detected y the symol, ecuse Sπ = Slog =. 4. 8
5. Integrting symols: n lgorithmic pproch In the previous section we illustrted how the symol clculus cn e used to derive functionl eqution mong polylogrithms. While in tht weight two emple ll the steps were esily crried out y hnd, n lgorithmic pproch is desirle in more complicted cses. In this section, we present our pproch tht often llows to integrte symol of trnscendentl function, i.e., to find function F, written s liner comintion of products of multiple polylogrithms, whose symol mtches given tensor S, which in the following we lwys ssume to stisfy the integrility condition 3.5. From the emple of the previous section it should e cler tht the min chllenges to ddress when going to higher weight w re. choosing pproprite rguments of the desired function types s few emples of function types of weight four, we list Li 4, Li, or Li log log such tht their symols spn the vector spce in which the tensor S lies;. finding the generliztion of the decomposition of weight two tensors into symmetric nd ntisymmetric prts indicted in the simple emple of weight two in section 4 to higher weights. This prolem ws ddressed up to weight four in refs. [6, 75]. Let us ssume tht we hve liner comintion S with rtionl coefficients of elementry tensors where the fctors in ech elementry tensor re rtionl functions of some vriles,..., n. In the following we ssume the tensor to e of pure weight w, i.e., ech elementry tensor is ssumed to hve the sme numer of fctors. Without loss of generlity we cn then ssume tht S tkes the form ll sums re ssumed to e finite S = R i,..., n... R iw,..., n, 5. i,...,i w c i,...,i w where R il,..., n i l ms for some ms determined y the initil tensor S re rtionl functions in the vriles i nd the c i,...,i w re rtionl numers. Distriutivity cf. eqs. 3.4, 3.5 then implies tht, without loss of generlity, S cn e rewritten with new constnts c j,...,j w Q s S = c j,...,j w π j... π jw, 5. j,...,j w where π j = π j,..., n j K for some K re suitly chosen rtionl functions which re multiplictively independent i.e., there is no non-trivil reltion of the form K j= πr j j = ±, for r j Z. In prctice, we chieve this y simply fctorizing the rtionl functions R i,..., n in eq. 5. into irreducile polynomils over Q, sy i.e., polynomils in Q[,..., n ] tht cnnot e written s the product of two non-constnt polynomils in Q[,..., n ]. Let us denote the set of these polynomils y P S = {π,..., π K }, which will constitute our uilding locks in the following. The symol cn then e seen s n element of the tensor lger over the Q-vector spce generted y the forml sis vectors in the set P S more precisely, we should consider it s n element of the wth grding of 9
Weight Bsic function types of pure weight log Li 3 Li 3 4 Li 4, Li,, y 5 Li 5, Li,3, y 6 Li 6, Li,4, y, Li 3,3, y, Li,,, y, z Tle : Indecomposle multiple polylogrithms of pure weight 6. the tensor lger over the Z-module ± K j= πr j j r j Z. Our gol is now to find function, sy F, tht is rtionl liner comintion of multiple polylogrithms nd products thereof whose rguments re rtionl functions in the i such tht SF = S. The procedure to chieve this proceeds in two steps: first we hve to decide on the types of functions tht should pper in F, nd then we hve to concoct suitle rtionl functions in the i s rguments of these functions such tht for some liner comintion of these functions the resulting epression fulfills the condition SF = S. Note tht this ltter step is not lgorithmic in generl, s it my involve finding rguments for the functions tht hve singulrities outside P S. 5. Choosing the types of functions Our first gol is to construct set of function types our sic types out of which we cn construct our cndidte function F. This involves two steps, nd we wnt oth the functions nd their rguments to e s simple s possile, ut we need to tke into ccount tht ll the possile function types one cn write down for given weight re relted y n undnce of functionl equtions. The min criterion we will use in the following is tht function type tht cn e written s product of lower weight function types is simpler thn function type of pure weight/trnscendentlity i.e., function tht cnnot e written s sum of terms, ech of which eing product of lower weights. Furthermore, we re guided y the conjecture which the second uthor lerned mny yers go from Gonchrov tht multiple polylogrithm Li m,...,m k with m j = for some j cn e epressed in terms of multiple polylogrithms where no inde is equl to. This conjecture suggests to put restriction on the function types of pure trnscendentlity tht cn pper for given weight. Furthermore, the shuffle nd stuffle reltions provide us with further constrints. As n emple, we cn deduce from Li m,m, y + Li m,m y, = Li m Li m y Li m +m y. 5.3 tht we cn hence ignore Li m,m with m < m. For low weights, the corresponding sets of presumly independent functions which re indecomposle, i.e. cnnot e written in terms of products of lower order functions, re given in Tle.
5. Finding the rguments Hving t hnd suitle set to construct the sic function types from, we still need to find the rguments of these function types. In the contet of prticle physics it hs proved helpful to use guidnce from educted guesses, motivted y physicl constrints cf. refs. [64, 65], to construct the symol nd/or the functions epressing the desired physicl quntities. To see how one might proceed even without ny such guidnce, let us concentrte first on clssicl polylogrithms only. We strt y defining, for P S = {π j } j s ove, P S = P S P S, 5.4 where P S is the set of ll prime fctors tht pper in π i ± π j nd ± π i, π i, π j P S. Let us denote the elements of P S y π i. Since S is constructed out of the irreducile polynomils π i P S P S, it is perhps nturl to hope tht ll rguments ppering in the polylogrithmic epressions needed for S cn e written in the form R ± n,...,n k,..., n = ±π n,..., n... π n k k,..., n, 5.5 where n,..., n k re integers. Let us denote the set of these functions R ± n,...,n k,..., n y R S, i.e. this is, up to sign, the multiplictive spn of the π j P S. Note tht in prctice it is often enough to consider R S to e the spn of only suset of the polynomils in P. Finlly, note tht the set R S crries group structure, given y the multipliction of rtionl functions. Choosing rguments for clssicl polylogrithms. However, not ll of these functions re good cndidtes for rguments of polylogrithms. Indeed, if for emple such function ppers s n rgument of clssicl polylogrithm, then y eq. 3.7 for R = R n ±,...,n k,..., n we cn write SLi n R = R } R {{ R }. n times 5.6 It is now esy to see tht if we wnt this tensor to e n element of the tensor lger of the vector spce generted y the set P S, then we need to impose the constrint Let us introduce the nottion R R S. 5.7 R S = {R R S R R S } R S. 5.8 It follows tht, for R R S, the symol of Li nr n is liner comintion of tensors of the form π l... π ln. Hence ll the rtionl functions in the set R S re good cndidtes for rguments of the clssicl polylogrithms tht cn pper in our function F. Note tht R S is no longer group in generl. It cn however e given some more structure
y considering the following ction of the symmetric group S 3 on rtionl functions, defined for rtionl function R nd rtionl functions σ i of one vrile y σ R = R, σ R = R, σ 3 R = /R, σ 4 R = / R, σ 5 R = /R, σ 6 R = R/R. 5.9 Note tht the σ j form group under composition of functions isomorphic to the permuttion group S 3 on three letters. It is esy to check tht R S is closed under this ction of S 3. As S 3 is generted y the two elements σ nd σ 3, it is enough to check tht R S is closed under these two mps. Closure under σ is trivil y definition of R S. To see tht it is lso closed under σ 3 we hve to check tht R R S hve ecuse of the group structure of R S., σ 3R R S. Indeed, we σ 3 R = /R = R R R S, 5. Choosing rguments for polylogrithms of depth >. So fr we hve only considered clssicl polylogrithms, ut in generl we should lso e le to mke sensile nstz for the rguments of multiple polylogrithms of depths greter thn one. In the following, we find it more convenient to work with the functions G m,...,m k defined in eq.. rther thn with the functions Li m,...,m k. As the two function types re relted vi eq.., one cn esily convert from one representtion to the other. Let us consider multiple polylogrithm of depth two, sy G,. We re hence looking for pir of rtionl functions R, R R S R S tht re good cndidtes for the rguments of G,. The symol of G, R, R is given y [ SG, R, R = R R R R ] R R [ R R R R ] R R R R R R R + R R R R R R R R R ] R R [ R R, reclling our nottion for the shuffle products see eq. 3.9, 5. A B C D = A B C D + A B D C, A B C D = A B C D + A C B D + A C D B. 5. Using the sme resoning s for clssicl polylogrithms, we see tht the cndidte rguments for multiple polylogrithms of depth two re pirs of rtionl functions from the set R S = {R, R R S R S R R R S }. 5.3
An importnt consequence is tht no new rtionl functions re needed to construct the set R S, ut ll the informtion is lredy contined in R S. The new set R S then consists of pirs of elements of R S, suject to the dditionl constrint tht their difference must gin e fctorizle in terms of the sme prime elements. Moreover, we sw tht R S is endowed with nturl ction of the group S 3, defined in eq. 5.9. It is hence nturl to sk for non-trivil symmetry groups tht leve the set R S invrint. First, it is esy to see tht the defining condition for R S is invrint up to n overll sign under swpping the two entries of ny given pir. Second, the ction of S 3 defined in eq. 5.9 induces simultneous on oth fctors ction on R S R S, defined for σ i S 3 y It is now esy to check tht R S R, R σ i σ i R, σ i R. 5.4 is closed under this ction. To see this, it is enough to check tht σ i R σ i R R S for i =, nd whenever R, R R S hve. Indeed, we σ R σ R = R R = R R R S, σ 3 R σ 3 R = /R /R = R R R R R S, 5.5 where we used the fct tht R R R S nd tht R S is multiplictive group. Comining this S 3 symmetry with the invrince under n echnge of rguments, here R nd R, we see tht R S is closed under the ction of the group S 3 S, defined for σ, ρ S 3 S y R, R σ,ρ σr ρ, σr ρ, 5.6 i.e. the fctor S simply cts s permuttion of the entries. The previous discussions for depths one nd two redily generlize to higher depth. Our cndidte rguments for the multiple polylogrithms of depth k re k-tuples of rtionl functions from the set R k S = {R,..., R k R S... R S R i R j R S, i < j k}, 5.7 nd using ectly the sme rgument s in the depth two cse, we see tht R k S equipped with n ction of the group S 3 S k, cting on R,..., R k R k S vi cn e R,..., R k σ,ρ σr ρ,..., σr ρk, 5.8 i.e. the fctor S k simply cts s permuttion of the entries. 5.3 Integrting the symol We now turn to the prolem of integrting the tensor S tht stisfies the integrility condition 3.5. In prctice, such tensors could come from computing Feynmn integrl in terms of multiple polylogrithms, or y computing its symol y other mens [63, 64, 65, 67, 7, 73]. Our gol is to find function F, more precisely liner comintion of multiple polylogrithms, such tht SF = S. The considertions of the previous section 3