Structural Relations of Harmonic Sums and Mellin Transforms at Weight w=6
aa r X i v : . [ m a t h - ph ] J a n DESY 08–206SFB-CPP/09–002January 2009
Structural Relations of Harmonic Sums andMellin Transforms at Weight w = 6 Johannes Bl¨umlein
Deutsches Elektronen–Synchrotron, DESY,Platanenallee 6, D-15735 Zeuthen, Germany
Abstract
We derive the structural relations between nested harmonic sums and the correspondingMellin transforms of Nielsen integrals and harmonic polylogarithms at weight w = 6 . Theyemerge in the calculations of massless single–scale quantities in QED and QCD, such asanomalous dimensions and Wilson coefficients, to 3– and 4–loop order. We consider theset of the multiple harmonic sums at weight six without index {− } . This restriction issufficient for all known physical cases. The structural relations supplement the algebraicrelations, due to the shuffle product between harmonic sums, studied earlier. The originalamount of 486 possible harmonic sums contributing at weight w = 6 reduces to 99 sumswith no index {− } . Algebraic and structural relations lead to a further reduction to 20basic functions. These functions supplement the set of 15 basic functions up to weight w = 5 derived formerly. We line out an algorithm to obtain the analytic representation ofthe basic sums in the complex plane. Proceedings of the “Motives, Quantum Field Theory, and Pseudodifferential Operators”, held at the ClayMathematics Institute, Boston University, June 2–14, 2008
Introduction
Inclusive and semi-inclusive scattering cross sections in Quantum Field Theories as QuantumElectrodynamics (QED) and Quantum Chromodynamics (QCD) at higher loop order can beexpressed in terms special classes of fundamental numbers and functions. Zero scale quantities,like the loop-expansion coefficients for renormalized couplings and masses in massless filed theo-ries, are given by special numbers, which are the multiple ζ -values [1, 2] in the known orders. Athigher orders and in the massive case other quantities more will contribute [3]. The next class ofinterest are the single scale quantities to which the anomalous dimensions and Wilson coefficientsdo belong [4–6], likewise other hard scattering cross sections being differential in one variable z = ˆ L/L given by the ratio of two Lorentz invariants with support z ∈ [0 , M [ f ( z )] ( N ) = Z dz z N f ( z ) . (1.1)In the light-cone expansion [7] these quantities naturally emerge as moments for physical reasonswith N ∈ N . Their mathematical representation is obtained in terms of nested harmonicsums [8–10] S b,~a ( N ) = N X k =1 (sign b ) k k | b | S ~a ( k ) , S ( k ) = 1 , (1.2)which form a unified language. This is the main reason to adopt this prescription also for otherquantities of this kind. The harmonic sums lead to the multiple ζ -values in the limit N → ∞ for b = 1. In the latter case the harmonic sums diverge. To obtain a representation which isas compact as possible we seek to find all relations between the harmonic sums. There are twoclasses of relations : i ) the algebraic relations, cf. [11]. They are due to the index set of the harmonic sums only andresult from their quasi–shuffle algebra [12]. ii ) the structural relations. These relations depend on the other properties of the harmonicsums. One sub–class refers to relations being obtained considering harmonic sums at N andinteger multiples or fractions of N , which leads to a continuation of N ∈ Q . Harmonic sumscan be represented in terms of Mellin-integrals of harmonic polylogarithms H ~a ( z ) weighted by1 / (1 ± z ) [13], which belong to the Poincar´e–iterated integrals [14]. The Mellin integrals arevalid for N ∈ R , N ≥ N . From these representations integration-by-parts relations can bederived. Furthermore, there is a large number of differentiation relations d l dN l M [ f ( z )] ( N ) = M h ln l ( z ) f ( z ) i ( N ) . (1.3)We analyzed a wide class of physical single scale massless processes and those containing asingle mass scale at two- and three loops [4–6] in the past, which led to the same set of basicharmonic sums and, related to it, basic Mellin transforms . Like in the case of zero scale quantities,this points to a unique representation, which is widely process independent and rather related Due to the algebraic relations [11] of the harmonic sums one may show that this divergence is at most of O (ln m ( N )), where m is the number of indices equal to one at the beginning of the index set. Generalized polylogarithms and Z -sums were considered in [15].
2o the contributing Feynman integrals only. The representation in terms of harmonic sums isusually more compact than a corresponding representation by harmonic polylogarithms, since i ) Mellin convolutions emerge as simple products; ii ) harmonic polylogarithms are multipleintegrals, which are usually not reduced to more compact analytic representations. The latterrequires to solve (part of) these integrals analytically. In the case of harmonic sums the analyticcontinuation of their argument N to complex values has to be performed to apply them in physicsproblems. As lined out in Ref. [16–18] this is possible since harmonic sums can be representedin terms of factorial series [19] up to known algebraic terms. Harmonic sums turn out to bemeromorphic functions with single poles at the non-positive integers. One may derive theirasymptotic representation analytically and they obey recursion relations for complex arguments N . Due to this their unique representation is given in the complex plane.In the present paper we derive the structural relations of the weight w = 6 harmonic sumsextending earlier work on the structural relations of harmonic sums up to weight w = 5 [18]. Thepaper is organized as follows. In Sections 2–6 we derive the structural relations of the harmonicsums of weight w = 6 of depth 2 to 6 for the harmonic sums not containing the index {− } . Therestriction to this class of functions is valid in the massless case at least to three-loop order andthe massive case to two-loop order. In Section 7 we summarize the set of basic functions chosen.The principal method to derive the analytic continuation of the harmonic sums to complex valuesof N is outlined in Section 8 in an example. Section 9 contains the conclusions. Some usefulintegrals are summarized in the appendix. The following w = 6 two–fold sums occur : S ± , ( N ) , S ± , ± ( N ) , S − , ( N ) and S , ( N ) , S − , − ( N ).The latter sums are related to single harmonic sums through Euler’s relation. S a,b ( N ) + S b,a ( N ) = S a ( N ) S b ( N ) + S a ∧ b ( N ) , (2.1)with a ∧ b = sign( a ) · sign( b )( | a | + | b | ). For the former six sums we only consider the algebraicallyirreducible cases. In Ref. [18] the basic functions, which determine the harmonic sums withoutindex {− } through their Mellin transform, up to w = 5 were found : w = : 1 / ( x −
1) (2.2) w = : ln(1 + x ) / ( x + 1) (2.3) w = : Li ( x ) / ( x ±
1) (2.4) w = : Li ( x ) / ( x + 1) , S , ( x ) / ( x ±
1) (2.5) w = : Li ( x ) / ( x ± , S , ( x ) / ( x ± , S , ( x ) / ( x ± , Li ( x ) / ( x ± , [ln( x ) S , ( − x ) − Li ( − x ) / / ( x ±
1) (2.6)In the following we determine the corresponding basic functions for w = 6 .In case of the double sums we show that they all can be related to M " Li ( x )1 + x ( N ) (2.7)3p to derivatives of basic functions of lower degree and polynomials of known harmonic sums.The representation of S ± , ( N ) read : S , ( N ) = M " Li ( x ) x − ! + ( N ) − S ( N ) ζ + S ( N ) ζ − S ( N ) ζ + S ζ (2.8) S − , ( N ) = ( − N M " Li ( x )1 + x ( N ) + 1516 ζ ln(2) − s − S − ( N ) ζ + S − ( N ) ζ − S − ( N ) ζ + S − ζ , (2.9)with Z dxg ( x )[ f ( x )] + = Z dx [ g ( x ) − g (1)] f ( x ) (2.10)and s = 1516 ln(2) ζ + Z dz Li ( z )1 + z (2.11)being one of the basic constants at weight w = 6. For the determination of the constants inthe alternating case we use the tables associated to Ref. [10]. To express one of the sums givenbelow we also give a second representation of S − , ( N ), S − , ( N ) = S − ( N ) S ( N ) + S − ( N )+( − ( N +1) M " Li ( − x ) − ln( x )Li ( − x ) + ln ( x )Li ( − x ) / x + 1 ( N )+( − ( N +1) M " − ln ( x )Li ( − x ) / − ln ( x ) ln(1 + x ) / x + 1 ( N ) − ζ [ S − ( N ) − S ( N )] − ζ + 34 ζ + 238 ζ ln(2) − s . (2.12)The other two-fold sums are S − , − ( N ) = − M " ( − x ) − ln( x )Li ( − x ) x − ! + ( N )+ 12 ζ [ S ( N ) − S − ( N )] − ζ S ( N ) + 218 ζ S ( N ) − ζ S ( N ) (2.13) S − , ( N ) = ( − N M " ( x ) − Li ( x ) ln( x )1 + x ( N )+2 ζ S − ( N ) − ζ S − ( N ) + 4 ζ S − ( N ) + 239840 ζ − ζ − ζ ln(2) + 4 s (2.14) S , − ( N ) = 12 ζ [ S − ( N ) − S ( N )] − ζ S − ( N ) + 218 ζ S − ( N ) − ζ S − ( N )+( − N +1 M " ( − x ) − ln( x )Li ( − x )1 + x ( N ) − ζ + 1516 ζ + 4 ζ ln(2) − s (2.15) S , ( N ) = − M " ( x ) − ln( x )Li ( x ) x − ! + ( N ) + 2 ζ S ( N ) − ζ S ( N ) + 4 ζ S ( N )(2.16)4 − , − ( N ) = M " ( − x ) − x )Li ( − x ) + ln ( x )Li ( − x ) / x − ( N ) − ζ [ S − ( N ) − S ( N )] − ζ S ( N ) + 458 ζ S ( N )= 12 h S − ( N ) + S ( N ) i (2.17) S − , ( N ) = 3 ζ S − ( N ) − ζ S − ( N ) + ( − N +1 M " S , (1 − x ) − ζ x ! + ( N )+( − N +1 M " x ) [ S , (1 − x ) − ζ ] + ln ( x ) [ S , (1 − x ) − ζ ] /
21 + x ! + ( N ) − ζ + 8132 ζ + 458 ζ ln(2) − s (2.18) S , ( N ) = 3 ζ S ( N ) − ζ S ( N ) − M " S , (1 − x ) − ζ x − ! + ( N ) − M " ln ( x ) [ S , (1 − x ) − ζ ] / x ) [ S , (1 − x ) − ζ ] x − ! + ( N )= 12 h S ( N ) + S ( N ) i . (2.19)In the above relations Nielsen integrals, [20], given by S p,n ( x ) = ( − p + n +1 ( p − n ! Z dzz ln p − ( z ) ln n (1 − xz ) (2.20)occur. The corresponding functions S ,k (1 − x ) are given by S , (1 − x ) = − Li ( x ) + log( x )Li ( x ) + 12 log(1 − x ) log ( x ) + ζ (3) S , (1 − x ) = − Li ( x ) + log( x )Li ( x ) −
12 log ( x )Li ( x ) −
16 log ( x ) log(1 − x ) + ζ (4) S , (1 − x ) = − Li ( x ) + ln( x )Li ( x ) −
12 ln ( x )Li ( x ) + 16 ln ( x )Li ( x )+ 124 ln ( x ) ln(1 − x ) + ζ . (2.21)They are used to express the respective sums in terms of the Mellin transforms of basic functionsand their derivatives w.r.t. N .The algebraic relation for S , ( N ) can be used to express M [(Li ( x ) / ( x − + ]( N ). The Mellintransform in S − , − ( N ) allows to express S − , − ( N ) and S − , ( N ) through (2.12). S , ( N ) and S − , ( N ) do not contain new Mellin transforms. Therefore the only non-trivial Mellin transformneeded to express the double sums at w = 6 is M [Li ( x ) / (1 + x )]( N ).In some of the harmonic sums Mellin transforms of the typeLi k ( − x ) x ± . (2.22)contribute. For odd values of k = 2 l + 1 the harmonic sums S , − ( k − ( N ) , S − ( k − , ( N ) and S − l, − l ( N ) allow to substitute the Mellin transforms of these functions in terms of Mellin trans-forms of basic functions and derivatives thereof.5or even values of k this argument applies to M [Li k ( − x ) / (1 + x )]( N ) but not to M [Li k ( − x ) / (1 + x )]( N ). In the latter case one may use the relation12 k − Li k ( x )1 − x = Li k ( x )1 − x + Li k ( x )1 + x + Li k ( − x )1 − x + Li k ( x )1 + x . (2.23)Since in massless quantum field-theoretic calculations both denominators occur, one may applythis decomposition based on the first two cyclotomic polynomials, cf. [21], and the relationbetween Li k ( x ) and Li k ( ± x ), [22]. The corresponding Mellin transforms also require half–integer arguments. In more general situations other cyclotomic polynomials might emerge. Therelation12 k − M " Li k ( x ) x − ! + N − (cid:19) = M " Li k ( x ) x − ! + ( N ) + M " Li k ( x ) x + 1 ! + ( N )+ M " Li k ( − x ) x − ! + ( N ) + M " Li k ( − x ) x + 1 ! + ( N ) − Z dx Li k ( x )1 + x (2.24)determines M [Li k ( − x ) / (1 + x )]( N ). For k = 2 , Z dx Li ( x )1 + x = ζ ln(2) − ζ (2.25) Z dx Li ( x )1 + x = 25 ln(2) ζ + 3 ζ ζ − ζ . (2.26)The corresponding relations for M [Li k ( − x ) / (1 + x )]( N ) are : M " Li ( − x ) x + 1 ( N ) = − M " Li ( x ) x − ! + N − (cid:19) + M " Li ( x ) x − ! + ( N )+ M " Li ( − x ) x − ! + ( N ) − M " Li ( x ) x + 1 ( N )+ 38 ζ − ζ ln(2) (2.27) M " Li ( − x ) x + 1 ( N ) = − M " Li ( x ) x − ! + N − (cid:19) + M " Li ( x ) x − ! + ( N )+ M " Li ( − x ) x − ! + ( N ) − M " Li ( x ) x + 1 ( N ) − ζ ln(2) − ζ ζ + 2532 ζ . (2.28)In the case of w = 6 these relations do not lead to a further reduction of basic functions but arerequired at lower weights, cf. [18]. 6 Threefold Sums
The triple sums are : S , , ( N ) = − M " S , ( x ) x − ! + ( N ) + S ( N )(2 ζ − ζ ζ ) − ζ S ( N ) + ζ S ( N ) (3.1) S − , , ( N ) = ( − N +1 M " S , ( x )1 + x ( N ) + (2 ζ − ζ ζ ) S − ( N ) − ζ S − ( N )+ ζ S − ( N ) + 71840 ζ + 18 ζ − ζ ln(2) − ζ ζ ln(2) + 32 s (3.2) S − , − , ( N ) = M " H , , − , , ( x ) x − ( N ) + ζ S − , − ( N ) + [ S − ( N ) − S ( N )] (cid:20) ζ ln(2) − ζ (cid:21) + 340 ζ S ( N ) − (cid:18) ζ ζ − ζ (cid:19) S ( N ) (3.3) S − , − , ( N ) = S − ( N ) S − , ( N ) + S , ( N ) + S − , − ( N ) − S − , , − − S − , − , (3.4) S , − , − ( N ) = S − ( N ) S , − ( N ) + S , ( N ) − S ( N ) S − , − ( N ) − S − , − ( N )+ S − , − , ( N ) (3.5) S , − , − ( N ) = S ( N ) S − , − ( N ) + S − , − ( N ) + S − , − ( N ) − S − , , − ( N ) − S − , − , ( N ) (3.6) S − , , − ( N ) = S − , − ( N ) − S − ( N ) S , − ( N ) − S , ( N ) + S ( N ) S − , − ( N ) − S − ( N ) S − , ( N ) − S , ( N ) + S − , − ( N ) + S ( N ) S − , − ( N )+ S − , , − ( N ) (3.7) S − , , − ( N ) = M " A ( − x ) / S , ( − x ) − S , ( − x ) ln( x ) x − ! + ( N ) − ζ [ S − , ( N ) − S − , − ( N )] − (cid:20) ζ − ζ ln(2) (cid:21) [ S − ( N ) − S ( N )]+ 18 ζ S ( N ) + (cid:20) ζ − ζ ζ (cid:21) S ( N ) (3.8) S − , , ( N ) = ( − N M " S , ( x ) − A ( x ) / x + 1 ( N ) + ζ S − , ( N ) − ζ S − ( N ) − (cid:18) ζ − ζ ζ (cid:19) S − ( N ) + 23168 ζ + 5964 ζ + 4132 ζ ln(2) + 12 ζ ln (2)+ 54 ζ ζ ln(2) − ζ ln (2) − ζ Li (cid:18) (cid:19) − s (3.9) S , − , ( N ) = ( − N M " H , − , , , ( x )1 + x ( N ) − (cid:18) ζ − ζ ζ (cid:19) S − ( N )+ ζ S , − ( N ) − ζ S , − ( N )+ (cid:20) − ζ + 2Li (cid:18) (cid:19) + 34 ζ ln(2) − ζ ln (2) + 112 ln (2) (cid:21) [ S ( N ) − S − ( N )] − ζ ζ ln(2) − ζ ln (2) − ζ Li (cid:18) (cid:19) + ζ ln (2) + 1112 ζ + 8764 ζ − ζ ln(2) − s (3.10) S , , − ( N ) = S − ( N ) S , ( N ) + S , − ( N ) − S ( N ) S − , ( N ) − S − , ( N ) + S − , , ( N ) (3.11) S , − , ( N ) = − S ( N ) S − , ( N ) − S − , ( N ) + S , − , ( N ) + S ( N ) S − , ( N )+ S − , ( N ) (3.12)7 , , − ( N ) = S , − ( N ) − S − ( N ) S , ( N ) − S , − ( N ) + S ( N ) S , − ( N ) − S , − , ( N )+ S , − ( N ) + S ( N ) S − , ( N ) + S − , ( N ) − S − , , ( N ) (3.13) S − , , ( N ) = S ( N ) S − , ( N ) + S − , ( N ) + S − , ( N ) − S , − , ( N ) − S − , , ( N ) (3.14) S − , , ( N ) = ( − N M " (3 / A ( x ) − Li ( x )Li ( x ) x + 1 ( N ) + ζ S − , ( N ) − ζ S − , ( N )+ (cid:18) ζ − ζ ζ (cid:19) S − ( N ) − ζ − ζ + 5132 ζ ln(2) − ζ ln (2)+ 32 ζ ζ ln(2) + 16 ζ ln (2) + 4 ζ Li (cid:18) (cid:19) + 32 s (3.15) S , − , ( N ) = S , − ( N ) − S , , − ( N ) − S − , , ( N ) + S − ( N ) S , ( N ) + S − , ( N ) (3.16) S , − , ( N ) = S − , ( N ) − S ( N ) S − , ( N ) − S − , ( N ) + S − ( N ) S , ( N ) + S , − ( N )+2 S − ( N ) S , ( N ) + S , − ( N ) − S ( N ) S , − ( N ) − S , − ( N ) + S − , ( N ) − S , , − ( N ) − S − , , ( N ) (3.17) S , , − ( N ) = − S − ( N ) S , ( N ) − S − , ( N ) + S − , , ( N ) + S ( N ) S , − ( N )+ S , − ( N ) (3.18) S − , , ( N ) = S ( N ) S − , ( N ) + S − , ( N ) − S − ( N ) S , ( N ) − S , − ( N ) + S , , − ( N ) (3.19) S , , − ( N ) = ( − N M " A ( − x ) / S , ( − x ) − ln( x ) S , ( − x )1 + x ( N ) − ζ [ S , ( N ) − S , − ( N )] − (cid:20) ζ − ζ ln(2) (cid:21) [ S ( N ) − S − ( N )]+ 18 ζ S − ( N ) + (cid:20) ζ − ζ ζ (cid:21) S − ( N )+ 113560 ζ − ζ − ζ ln(2) − ζ ζ ln(2) + s (3.20) S , , ( N ) = M " S , ( x ) − A ( x ) / x − ! + ( N )+ ζ S , ( N ) − ζ S ( N ) − (cid:18) ζ − ζ ζ (cid:19) S ( N ) (3.21) S , , ( N ) = M " (3 / A ( x ) − Li ( x )Li ( x ) x − ! + ( N )+ ζ S , ( N ) − ζ S , ( N ) + (cid:18) ζ − ζ ζ (cid:19) S ( N ) (3.22) S , , ( N ) = S ( N ) S , ( N ) + S , ( N ) − S ( N ) S , ( N ) − S , ( N ) + S , , ( N ) (3.23) S , , ( N ) = 2 S , ( N ) − S ( N ) S , ( N ) − S , ( N ) + S ( N ) S , ( N ) − S , , ( N )+ S , ( N ) + S ( N ) S , ( N ) − S , , ( N ) (3.24) S , , ( N ) = − S ( N ) S , ( N ) − S , ( N ) + S , , ( N ) + S ( N ) S , ( N ) + S , ( N ) (3.25) S , , ( N ) = S ( N ) S , ( N ) + S , ( N ) + S , ( N ) − S , , ( N ) − S , , ( N ) (3.26) S , , ( N ) = − M " A ( x ) + Li ( x ) ln( x ) / − S , ( x ) ln( x ) x − ! + ( N ) − M " S , ( x ) − ( x )Li ( x ) x − ! + ( N )+2 ζ S , ( N ) + 2 ( ζ − ζ ζ ) S ( N )8 16 S ( N ) + 12 S ( N ) S ( N ) + 13 S ( N ) (3.27) S − , , ( N ) = ( − ( N +1) M " A ( x ) + Li ( x ) ln( x ) / − S , ( x ) ln( x ) x + 1 ( N )+( − ( N +1) M " S , ( x ) − ( x )Li ( x ) x + 1 ( N )+2 ζ S − , ( N ) + 2 ( ζ − ζ ζ ) S − ( N ) − ζ ζ ln(2) + 112 ζ ln (2) + 2 ζ Li (cid:18) (cid:19) − ζ ln (2) + 3780 ζ − ζ − ζ ln(2) + 4 s (3.28) S , − , ( N ) = − S − , , ( N ) + S ( N ) S − , ( N ) + S − , ( N ) + S − , ( N ) (3.29) S , , − ( N ) = S − , , ( N ) −
12 [ S ( N ) S − , ( N ) + S − , ( N ) + S − , ( N ) − S ( N ) S , − ( N ) − S , − ( N ) − S , − ( N )] (3.30) S − , , − ( N ) = M " − S , ( − x ) − ln( x )Li ( − x ) / x ) S , ( − x ) x − ! + ( N )+ M " ( − x )Li ( − x ) − A ( − x ) x − ! + ( N ) − ζ S − , − ( N ) + 12 ζ [ S − , − ( N ) − S − , ( N )]+ (cid:20) − ζ + 4Li (cid:18) (cid:19) + 2 ζ ln(2) − ζ ln (2) + 16 ln (2) (cid:21) [ S − ( N ) − S ( N )]+ (cid:18) ζ ζ − ζ (cid:19) S ( N ) (3.31) S , − , − ( N ) = 12 [ − S − , , − ( N ) + S − ( N ) S , − ( N ) + S − , − ( N ) + S , ( N )] (3.32) S − , − , ( N ) = 12 [ − S − , , − ( N ) + S − ( N ) S − , ( N ) + S , ( N ) + S − , − ( N )] (3.33) S − , − , − ( N ) = 16 S − ( N ) + 12 S − ( N ) S ( N ) + 13 S − ( N ) . (3.34)There emerge numerator functions, which do not belong to the class of Nielsen integrals , A ( x ) = Z x dyy Li ( y ) A ( x ) = Z x dyy ln(1 − y ) S , ( y ) A ( x ) = Z x dyy [Li (1 − y ) − ζ ] . (3.35)As seen in Eqs. (3.27), ( A ( x ) / ( x − + is not a basic function since its Mellin transformreduces to single harmonic sums and known Mellin transforms algebraically. Furthermore, somenumerator functions are given by harmonic polylogarithms H a ,... ,a k ( x ) , a i ∈ {− , , +1 } , which cannot be reduced significantly further. Harmonic polylogarithms are Poincar´e–iterated Note a misprint in Eq. (14), [17]. Li ( y ) should read Li (1 − y ). f , f , f − ] = [1 /x, / ( x − , / ( x + 1)], [13], with H ( x ) = ln( x ) (3.36) H ( x ) = − ln(1 − x ) (3.37) H − ( x ) = ln(1 + x ) (3.38)and H a,~b ( x ) = Z x dy f a ( y ) H ~b ( y ) . (3.39) The quadruple–index sums are : S − , , , ( N ) = ( − N M " S , ( x ) x + 1 ( N ) + ζ S − ( N ) − (2 ζ − ζ ζ ) S − ( N )+ 18 ζ ζ ln(2) − ζ ln (2) − ζ Li (cid:18) (cid:19) + 14 ζ ln (2) − ζ + 724 ζ ln (2)+ 4164 ζ − ζ ln(2) + 2 ln(2)Li (cid:18) (cid:19) + ln (2)Li (cid:18) (cid:19) + 136 ln (2) + 2Li (cid:18) (cid:19) − s S , , , ( N ) = − M " S , ( x ) x − ! + ( N ) + ζ S ( N ) − (2 ζ − ζ ζ ) S ( N ) (4.2) S − , , , ( N ) = ( − N +1 M " S , ( x ) + A ( x ) x + 1 ( N ) + ζ S − , ( N ) + (cid:18) ζ − ζ ζ (cid:19) S − ( N ) − ζ ζ ln(2) + 13 ζ ln (2) + 2 ζ Li (cid:18) (cid:19) − ζ ln (2) + 411560 ζ − ζ ln (2) − ζ + 7364 ζ ln(2) − (cid:18) (cid:19) − (2)Li (cid:18) (cid:19) −
118 ln (2) − (cid:18) (cid:19) + 94 s (4.3) S , − , , ( N ) = ( − N +1 M " H , − , , , ( x )1 + x ( N ) + ζ S , − ( N ) + (cid:20) ζ ζ − ζ (cid:21) S − ( N )+ (cid:20) − Li (cid:18) (cid:19) + 18 ζ + 18 ζ ln(2) + 14 ζ ln (2) −
124 ln (2) (cid:21) [ S ( N ) − S − ( N )] − ζ ζ ln(2) − ζ ln (2) − ζ Li (cid:18) (cid:19) + 12 ζ ln (2) − ζ + 712 ζ ln (2)+ 105128 ζ − ζ ln(2) + 4 ln(2)Li (cid:18) (cid:19) + 2 ln (2)Li (cid:18) (cid:19) + 118 ln (2)+4Li (cid:18) (cid:19) + s (4.4) S − , , , ( N ) = ( − N M " A ( x )1 + x ( N ) − ( ζ ζ − ζ ) S − ( N ) − ζ S − , ( N ) + ζ S − , , ( N )+ 516 ζ ζ ln(2) + 116 ζ ln (2) + 32 ζ Li (cid:18) (cid:19) − ζ ln (2) + 11120 ζ − ζ ζ ln(2) + 52 s (4.5) S − , − , , ( N ) = − M " H , − , , , ( x ) x − ! + ( N ) + ζ S − , − ( N ) + (cid:18) ζ ζ − ζ (cid:19) S ( N )+ (cid:18) − Li (cid:18) (cid:19) + 18 ζ + 18 ζ ln(2) + 14 ζ ln (2) −
124 ln (2) (cid:19) [ S − ( N ) − S ( N )](4.6) S , , , ( N ) = − M " S , ( x ) + A ( x ) x − ! + ( N ) + ζ S , ( N ) + (cid:18) ζ − ζ ζ (cid:19) S ( N ) . (4.7)Here, the harmonic polylogarithm H , − , , , ( x ) is given by H , − , , , ( x ) = Z x dyy Z y dz S , ( z )1 + z . (4.8)We tested the above sum-relations containing harmonic polylogarithms in the Mellin transformsnumerically using the code of Ref. [23]. Two 5–fold sums contribute : S , , , , ( N ) = − M " S , ( x ) x − ! + ( N ) + ζ S ( N ) (5.1) S − , , , , ( N ) = ( − N +1 M " S , ( x )1 + x ( N ) + ζ S − ( N )+ 716 ζ ζ ln(2) + 112 ζ ln (2) + 12 ζ Li (cid:18) (cid:19) − ζ ln (2) − ζ ln (2) − ζ − ln(2)Li (cid:18) (cid:19) −
12 ln (2)Li (cid:18) (cid:19) −
172 ln (2) − Li (cid:18) (cid:19) + ln(2) ζ . (5.2)All other sums can be traced back to these sums using algebraic relations [11]. The other Mellintransforms emerging in their representation were all calculated in Refs. [9, 18] before. Only one sixfold sum contributes at w = 6 , S , , , , , ( N ). This sum is completely reducible intoa polynomial of single harmonic sums, cf. [9], S . . . ,1 | {z } = 1720 S + 148 S S + 118 S S + 18 S S + 15 S S + 116 S S + 16 S S S + 148 S + 18 S S + 118 S + 16 S (6.1)11 The Basic Functions
In the following we summarize the basic functions the Mellin transforms of which represents theharmonic sums up to weight w = 6 without those carrying an index {− } . The correspondingsums of lower weight were determined in Refs. [8, 18, 24]. The new 20 functions are given by w = : Li ( x ) / ( x + 1) , S , ( x ) / ( x ± , S , ( x ) / ( x ± ,S , ( x ) / ( x ± , Li ( x )Li ( x ) / ( x ± ,A ( x ) / ( x + 1) , A ( x ) / ( x ± , A ( x ) / ( x + 1) (7.1) H , − , , , ( x ) / ( x ± , H , , − , , ( x ) / ( x ± A ( − x ) + 2 S , ( − x ) − S , ( − x ) ln( x )] / ( x ± A ( − x ) + 2 S , ( − x ) − S , ( − x ) ln( x ) + Li ( − x ) ln( x ) / − Li ( − x )Li ( − x )] / ( x − The anomalous dimensions and Wilson coefficients expressed in Mellin space allow simple rep-resentations of the scale evolution of single-scale observables, which are given by ordinary differ-ential equations. The experimental measurement of the observables requires the representationin z –space. Therefore, one has to perform the analytic continuation of harmonic sums to com-plex values of N . Precise numerical representations for the analytic continuation of the basicfunctions up to weight w = 5 were derived in [25] based on the MINIMAX -method [26]. Onemay even obtain corresponding representations for quite general functions Φ( z ) , z ∈ [0 , For other effectiveparameterizations see [28].Here we aim on an exact representations. The inverse Mellin transforms are obtained by acontour integral around the singularities of the respective functions in the complex plane.We traced back all the harmonic sums to Mellin transforms of basic functions f i ( z ), F − i ( N ) = Z dzf i ( z ) z N − z − , F + i ( N ) = Z dzf i ( z ) ( − z ) N − z + 1 . (8.1)Eqs. (8.1) imply the recursion relations F − i ( N + 1) = − F − i ( N ) + Z dzz N f i ( z ) , (8.2) F + i ( N + 1) = F + i ( N ) + ( − N +1 Z dzz N f i ( z ) . (8.3)The remaining integrals are simpler Mellin transforms, which correspond to harmonic sums oflower weight. For another proposal for the analytic continuation of harmonic sums to N ∈ R , for which some simpleexamples were presented, cf. [29].
12f the functions f i ( z ) / ( z − , f i ( z ) / ( z + 1) are analytic at z = 1 the Mellin transforms (8.1)can be represented in terms of factorial series [19]. Not all basic functions chosen above havethis property. A corresponding analytic relation replacing f i ( z ) → f i (1 − z ) (8.4)always exists. The additional terms are lower weight functions in N or are related to these bydifferentiation. We use this representation and consider the factorial series. Due to this both thepole–structure and the asymptotic relation for | N | → ∞ are known. . The poles are locatedat the integers below a fixed value N . The recursion relations (8.1) are used to express therespective harmonic sums at any value N ∈ C except the poles.Let us illustrate this representation in an example for the harmonic sum S , , , , ( N ). Thecorresponding basic function is S , ( z ) z − ! + . (8.5)The recursion relation is given by M " S , ( z ) z − ( N + 1) = M " S , ( z ) z − ! + ( N ) + M [ S , ( z )] ( N ) , (8.6)with M [ S , ( z )] ( N ) = 1 N + 1 (cid:20) ζ − N + 1 S , , , ( N ) (cid:21) , (8.7)cf. [33].The numerator function possesses a branch–point at z = 1. The contributions related toterms ln k (1 − z ) / ( z ±
1) contained have to be subtracted explicitely due to there logarithmicgrowth (to a power) for | N | → ∞ . This is either possible using the relation S , ( z ) to Li (1 − z ) S , ( z ) = − Li (1 − z ) + ln(1 − z )Li (1 − z ) −
12 ln (1 − z )Li (1 − z ) + 16 ln (1 − z )Li (1 − z )+ 124 ln (1 − z ) ln( z ) + ζ (8.8)or considering harmonic sums, which are algebraic equivalent to the above and are related to abasic function which is regular at z →
1. We will follow the latter way and use the algebraicrelations [11] to express S , , , , ( N ) afterwards, S , , , , = S , , , , + 14 S S , , , + S , , , , + S , , , + S , , , + S , , , − S S , , , + S , , , + S , , , + S , , , − S S , , , − S , , , − S , , , − S , , , − S S , , , + S , , , + S , , , + S , , , + S , , , (8.9) In [30] asymptotic relations for non-alternating harmonic sums to low orders in 1 /N k were derived. Ouralgorithm given below is free of these restrictions. The main ideas were presented in January 2004 [31], seealso [32]. S , , , , ( N ) = − M " Li (1 − x )1 − x ( N ) + ζ S , , , ( N ) − ζ S , , ( N )+ ζ S , ( N ) − ζ S ( N ) + ζ . (8.10)The function in the remaining Mellin transform is regular at z = 1 and can be represented interms of a factorial series. The remainder terms in (8.10) are polynomials of single harmonicsums. Therefore the poles of S , , , , ( N ), resp. S , , , , ( N ), are located at the non-positiveintegers. Finally we need the asymptotic representations of M [Li (1 − x ) / (1 − x )] ( N ), M " Li (1 − x )1 − x ( z ) ∼ z + 132 z − z + 51541472 1 z − z − z + 4644828197653456160000 1 z + 15337530749787136000 1 z − z + 959290541160030080000 1 z + 57513437734302116913534146740000 1 z − z − z + 225456132288901603788601079506240000 1 z + 2635677027013005586811053965342760089760000 1 z − z − z + 19214070284092333591693913028192458306945920000 1 z − z + O (cid:18) z (cid:19) (8.11)The corresponding representations for all other harmonic sums of weight w = 6 will be given ina forthcoming paper. We derived the basic functions spanning the nested harmonic (alternating) sums up to weight w = 6 with no index {− } . This sub-class governs the functions contributing to the masslesssingle-scale quantities, like the anomalous dimensions and Wilson coefficients to 3-loop order inQED and QCD. There are first indications, that in the massive case, even in the limit Q ≫ m this class needs to be extended at 3-loop order, cf. [34]. Up to weight w = 5 all basic functionswere given by polynomials of Nielsen integrals, Eq. (2.20), of argument x or − x weighted by1 / ( x ± w = 6 share this property, some contain1-dimensional integrals over polynomials of Nielsen integrals A i ( ± x ) | ... and more dimensionalintegrals, which are not reducible. This is generally expected and the cases up to w = 5 form anexception.We lined out how the exact the representation of the Mellin transforms of the basic functionscan be obtained, generalizing effective numerical high-precision representations [25, 27]. Upto terms which can be determined algebraically the Mellin transforms of the basic functionsare factorial series. The singularities of the Mellin transforms are located at the non-positiveintegers. They obey recursion relations for N → N + 1. The asymptotic representation of14he Mellin transforms can be determined analytically. The basic Mellin transforms are thusgeneralizations of Euler’s ψ -function and their derivatives, which describe the single harmonicsums. Acknowledgment.
I would like to thank Steve Rosenberg for his invitation and the organization of a very interestingand stimulating workshop. For useful conversations I would like to thank J. Ablinger, Y. Andre,F. Brown, and C. Schneider. This paper was supported in part by the Clay MathematicsInstitute, DFG Sonderforschungsbereich Transregio 9, Computergest¨utzte Theoretische Physik,and by the European Commission MRTN HEPTOOLS under Contract No. MRTN-CT-2006-035505. 15
In this appendix we list useful constants and integrals.Li k (1) = ζ k (10.1) S ,k (1) = ζ k +1 (10.2) S , (1) = 110 ζ (10.3) S , (1) = 2 ζ − ζ ζ (10.4) S , ( −
1) = − ζ + 12 ζ ζ (10.5) S , (1) = 2 ζ − ζ ζ (10.6) A (1) = − ζ + 2 ζ ζ (10.7) A ( −
1) = − ζ + 34 ζ ζ (10.8) A (1) = − ζ (10.9) A (1) = − ζ + ζ ζ (10.10) Z x dy Li ( − y )1 + y = ln(1 + x )Li ( − x ) + 12 Li ( − x ) (10.11) Z x dy ln( y )Li ( − y )1 + y = ln(1 + x ) ln( x )Li ( − x ) + 12 Li ( − x ) − S , ( − x ) + 2 ln( x ) S , ( − x ) (10.12) Z x dy S , ( y ) y − − x ) S , ( x ) + 3 S , ( x ) (10.13) Z x dyy [Li (1 − y ) − ζ ] = − ( x ) + ln( x )Li ( x ) (10.14) Z x dy Li ( y ) y − ( x ) + ln(1 − x )Li ( x ) (10.15) Z x dy ln( y ) y − ( y ) = 12 Li ( x ) + ln( x ) ln(1 − x )Li ( x ) − S , ( x ) + 2 ln( x ) S , ( x ) (10.16) Z x dyy ln(1 − y )Li ( y ) = − Li ( x )Li ( x ) + A ( x ) (10.17) Z x dyy Li ( y ) ln( y ) ln(1 − y ) = −
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