A refined machinery to calculate large moments from coupled systems of linear differential equations
aa r X i v : . [ c s . S C ] D ec DESY 19-225, DO-TH 19/13, SAGEX–19-35, PoS(RADCOR19)078
A refined machinery to calculate large momentsfrom coupled systems of linear differentialequations
Johannes Blümlein
Deutsches Elektronen–Synchrotron, DESY,Platanenallee 6, D-15738 Zeuthen, Germany.E-mail: [email protected]
Peter Marquard
Deutsches Elektronen–Synchrotron, DESY,Platanenallee 6, D-15738 Zeuthen, Germany.E-mail: [email protected]
Carsten Schneider ∗ Johannes Kepler University Linz, Research Institute for Symbolic ComputationAltenberger Straße 69, A-4040 Linz, Austria.E-mail: [email protected]
The large moment method can be used to compute a large number of moments of physical quanti-ties that are described by coupled systems of linear differential equations. Besides these systemsthe algorithm requires a certain number of initial values as input, that are often hard to derive ina preprocessing step. Thus a major challenge is to keep the number of initial values as small aspossible. We present the basic ideas of the underlying large moment method and present refinedversions that reduce significantly the number of required initial values. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ refined large moment machinery
Carsten Schneider
1. Introduction
In order to solve open problems at the forefront of elementary particle physics, millions ofcomplicated Feynman integrals in terms of a continuous parameter x and the dimensional parameter ε have to be tackled. In order to crunch these millions of Feynman integrals to a few hundred (orthousand) so-called master integrals, integration-by-parts (IBP) methods [34, 44, 47, 48, 66] areapplied as a preprocessing step. This yields an expression ¯ f ( x , ε ) given as a linear combination ofthese master integrals. Then the main task is to compute the first coefficients of its ε -expansion¯ f ( x , ε ) = ¯ f l ( x ) ε l + ¯ f l + ( x ) ε l + + ¯ f l + ( x ) ε l + + . . . (1.1)for three-loop Feynman integrals the expansion starts usually at l = −
3. More precisely, one com-putes for each master integral in ¯ f ( x , ε ) such an ε -expansion and assembles afterwards the sub-results accordingly to get the coefficients in (1.1). As observed in [24, 36, 43, 53] most of thesemaster integrals, say f i ( x , ε ) , can be determined as solutions of coupled systems of linear differen-tial equations which are of the form D x f ( x , ε ) ... f λ ( x , ε ) ! = A f ( x , ε ) ... f λ ( x , ε ) ! + g ( x , ε ) ... g λ ( x , ε ) ! , (1.2)with A ( x , ε ) being an invertible λ × λ matrix with entries from the polynomial ring K [ x , ε ] inthe variables x and ε with coefficients from a field K . Here the components g i ( x , ε ) are given as a K ( x , ε ) -linear combination of simpler master integrals whose ε -expansions can be computed (eitherby solving again a coupled system or by using, e.g., symbolic summation/integration tools [9, 57,60]).In the last years we have developed general algorithms in [5,8,9,11,35] that enable one to solvesuch systems in terms of indefinite nested sums (and integrals) provided that the inhomogeneouscomponents can be represented in this class. Successful applications of these algorithms to non-trivial physical quantities can be found, e.g., in [8, 12, 14, 27]. For certain instances one may alsoapply methods introduced in [40, 46].In the last years we entered more and more complicated physical problems where the abovetechnologies are too expensive or are not anymore applicable. More precisely, solving these under-lying systems recursively leads to many complicated indefinite nested sums (resp. indefinite nestedintegrals) and the calculation time explodes. Even worse, for more complicated physical problemsthe arising master integrals cannot be expressed anymore in terms of indefinite nested sums (or in-tegrals) but in terms of more complicated sums/integrals, where the simpler cases come, e.g., fromthe class of elliptic or modular functions/forms, F solutions [10, 20, 21, 25, 33, 54] and generaliterative-non-iterative integrals [13]. However, combining all these sub-results to the final expres-sion ¯ f ( x ) most of these sums/integrals vanish and only few sums/functions remain. In particularfor the coefficients ¯ f l ( x , ε ) , . . . , ¯ f − ( x , ε ) in the ε -expansion (1.1) one often expects only indefinitenested sums/integrals, like harmonic sums [30, 68] / harmonic polylogarithms [55]. In other cases,only very few of these more complicated special functions remain and one wants to filter out all We suppose that K is computable and contains the rational numbers Q as a sub–field. refined large moment machinery Carsten Schneider those sub-expression (coming from various color-factors) that are still expressible within the classof indefinite nested sums.Exactly for such types of problems the recently proposed large moment method [32] imple-mented in
SolveCoupledSystem [5, 9, 11, 35] can support the user. It allows to compute for avery large µ , say µ = F k ( ) , . . . , ¯ F k ( µ ) with l ≤ k ≤ r of its power series¯ f k ( x ) = ∞ ∑ n = ¯ F k ( n ) x n . (1.3)We remark that standard procedures like Mincer [38, 45] or
MATAD [65] allow the calculationof a comparable small number of Mellin moments, e.g., µ =
20. Given such a large amount ofmoments, one can utilize guessing packages like [42] in order to compute recurrences for thephysical quantity ¯ F k ( n ) . Note that the found recurrences describe precisely the desired solution¯ F k ( n ) (i.e., the recurrence order is minimal). Finally, one can apply our recurrence solving toolsfrom Section 2 to hunt for representations in terms of indefinite nested sums. For instance, asdemonstrated in [28], we could calculate from about µ = µ = µ = ε − and big parts of the constantterms could be expressed in terms of indefinite nested sums. This also applies to the first ordercontributions for the massive three-loop operator matrix element A ( ) Qg , see [10].The article is organized as follows. First, we will recall our recurrence solving machinery inSection 2. Using this technology we will elaborate in Section 3 how one can solve coupled systemsof the form (1.2) in terms indefinite nested sums. Finally we will deviate these ideas in Section 4to extract our large moment machinery [32]. In particular, we will present key ideas from [31] toobtain refined versions that allowed us to perform the recent calculations in [22, 31]. A conclusionis given in Section 5.
2. Solving linear recurrence relations in terms of indefinite nested sums
In the following we recall the basic ideas to solve linear recurrence relations within the classof indefinite nested sums defined over hypergeometric products . Definition 2.1.
A product ∏ kj = l f ( j ) , l ∈ N , is called hypergeometric in k over K if f ( j ) is anelement from the rational function field K ( j ) where the numerator and denominator of f ( j ) are Note that the class of indefinite nested sums contains as special cases harmonic sums [30,68], cyclotomic harmonicsums [16], generalized harmonic sums [17, 49], and nested binomials sums [15]. refined large moment machinery Carsten Schneider nonzero if one replaces j by an integer λ ∈ N with λ ≥ l. An expression in terms of indefinite nestedsums over hypergeometric products in k over K is composed recursively by the three operations( + , − , · ) with • elements from the rational function field K ( k ) , • hypergeometric products in k over K , • and sums of the form ∑ kj = l f ( j ) with l ∈ N where f ( j ) is an expression in terms of indefinitenested sums over hypergeometric products in j over K ; here it is assumed that the evaluationof f ( j ) | j λ for all λ ∈ Z with λ ≥ l does not introduce any poles.If K and k are clear from the context, we call an expression in terms of indefinite nested sumsdefined over hypergeometric products in k over K also an expression in terms of indefinite nestedsums . Basic recurrence solving.
Suppose that we are given polynomials a ( x ) , . . . , d d ( x ) ∈ K [ x ] where a d ( n ) = n ≥ δ for some δ ∈ N and we are given an expression b ( n ) in terms of indefinitenested sums. Consider the sequence ( F ( n )) n ≥ ∈ K N that is defined by the initial values F ( i ) = c i ∈ K for 0 ≤ i ≤ max ( δ , d ) − a ( n ) F ( n ) + a ( n ) F ( n + ) + · · · + a d ( n ) F ( n + d ) = b ( n ) . (2.1)Then using the summation package Sigma [57, 60, 61] one can decide constructively if F ( n ) canbe calculated, up to finitely many start values, by an expression in terms of indefinite nested sums.Internally, this problem can be rephrased in the setting of difference rings [51, 62–64], and theproblem can be decided afterwards in this setting using the algorithms from [19, 39, 52, 56, 58, 59,67]. More precisely, if these algorithms fail to find a solution, one obtains a proof that the sequence F ( n ) cannot be represented within the class of indefinite nested sums. Otherwise, the solutionin the setting of difference rings can be translated back yielding an explicit solution in terms ofindefinite nested sums. Remark 2.2.
Suppose that b ( n ) is not given as an expression in terms of indefinite nested sums, butonly by a large number of moments, say, b ( ) , b ( ) , . . . , b ( µ ) . Then the recurrence (2.1) togetherwith the initial values F ( i ) = c i ∈ K with ≤ i ≤ max ( δ , d ) − enable one to compute in lineartime the moments F ( ) , F ( ) , . . . , F ( µ ) . ε -recurrence solving. More generally, suppose that one is given a recurrence of the form a ( n , ε ) F ( n , ε ) + a ( n , ε ) F ( n + , ε ) + · · · + a d ( n , ε ) F ( n + d , ε ) = b ( n , ε ) (2.2)with multivariate polynomials a i ( x , ε ) ∈ K [ x , ε ] for 0 ≤ i ≤ d (not all zero) and where the inhomo-geneous part can be given by a formal Laurent series expansion b ( n , ε ) = b l ( n ) ε l + b l + ( n ) ε l + + b l + ( n ) ε l + + . . . (2.3)for some l ∈ Z where at least the coefficients b l ( n ) , . . . , b r ( n ) ∈ K can be computed for n ∈ N byexpressions in terms of indefinite nested sums. We may suppose that not all a ( x , ) , . . . , a d ( x , ) are zero. Otherwise we find a common factor ε u with u ≥ ε u . Let d ′ ∈ N refined large moment machinery Carsten Schneider be maximal such that a d ′ ( x , ) = δ ∈ N such that a d ′ ( n , ) = n ≥ δ .Then following the construction from [29] implemented in Sigma one can decide constructivelyif there is a solution F ( n , ε ) = F l ( n ) ε l + F l + ( n ) ε l + + F l + ( n ) ε l + + . . . (2.4)of (2.2) with the given initial values F j ( i ) = c j , i ∈ K for l ≤ j ≤ r and 0 ≤ i ≤ max ( d ′ , δ ) − F j ( n ) can be calculated by an expression in terms of indefinite nested sums.Internally, one proceeds as follows. Plugging in (2.4) into (2.2) and comparing coefficients at ε l yield the constraint a ( n , ) F l ( n ) + a ( n , ) F l ( n + ) + · · · + a d ′ ( n , ) F l ( n + d ′ ) = b l ( n ) . (2.5)Together with the initial values F l ( i ) = c l , i for 0 ≤ i ≤ max ( d ′ , δ ) − Sigma (seeabove) to decide algorithmically if F l ( n ) can be represented in terms of indefinite nested sums. Ifthis is not possible, the algorithm stops. Otherwise, one obtains such an expression for F l ( n ) andone gets a ( n , ε ) F ′ ( n , ε ) + a ( n , ε ) F ′ ( n + , ε ) + · · · + a d ( n , ε ) F ′ ( n + d , ε ) = b ′ ( n , ε ) (2.6)with the updated right hand side b ′ ( n , ε ) = b ( n , ε ) − (cid:16) a ( n , ε ) F l ( n ) + a ( n , ε ) F l ( n + ) + · · · + a d ( n , ε ) F l ( n + d ) (cid:17) (2.7)and the unknown ε -expansion F ′ ( n , ε ) = F l + ( n ) ε l + + F l + ( n ) ε l + + . . . (2.8)Note that the first r − l coefficients in b ′ ( n , ε ) = b ′ l + ( n ) ε l + + b ′ l + ( n ) ε l + + . . . , (2.9)i.e., in the ε -expansion of (2.7) can be given in terms of indefinite nested sums. Thus we can repeatthe above strategy and can decide algorithmically if the remaining coefficients F l + ( n ) , . . . , F r ( n ) can be represented in terms of indefinite nested sums. Remark 2.3.
Note that the above algorithm computes the maximal λ ∈ N with l − ≤ λ ≤ rsuch that F l ( n ) , . . . , F r ( n ) can be represented in terms of indefinite nested sums and returns thecoefficients F l ( n ) , . . . , F λ ( n ) in such a representation. Remark 2.4.
Suppose that the coefficients b i ( n ) in (2.3) are not given as expressions in terms ofindefinite nested sums, but only by a large number of moments, say, b j ( ) , b j ( ) , . . . , b j ( µ ) . Thenthe recurrence (2.1) together with the initial values F j ( i ) = c j , i ∈ K with ≤ i ≤ max ( δ , d ′ ) − enables one to compute in linear time the moments F j ( ) , F j ( ) , . . . , F j ( µ ) . More precisely, onestarts with j = l, obtains the recurrence (2.5) with the moment b l ( ) , b l ( ) , . . . , b l ( µ ) and computesthe moments F l ( ) , F l ( ) , . . . , F l ( µ ) ; see Remark 2.2. Then one computes the moments for b ′ j ( n ) in (2.9) using the formula (2.7) . This yields (2.6) with (2.8) and we are in the position to repeatthis process. Namely, we can calculate iteratively the moments F j ( ) , F j ( ) , . . . , F j ( µ ) for j = l + , l + , . . . , r. refined large moment machinery Carsten Schneider
3. Solving coupled systems of first-order linear differential equations
In the following we present our main tools to solve coupled systems in terms of indefinitenested sums.
Coupled system solving.
Suppose that we are given a coupled system of the form (1.2) with A ( x , ε ) being an invertible λ × λ matrix with entries from the polynomial ring K [ x , ε ] . Furthermore,suppose that the inhomogeneous parts can be given in form of a power series g i ( x , ε ) = ∞ ∑ n = G i ( n , ε ) x n (3.1)where the coefficients itself can be given in form of the ε -expansions G i ( n , ε ) = G i , l ( n ) ε l + G i , l + ( n ) ε l + + G i , l + ( n ) ε l + . . . (3.2)If the coefficients G i , k ( n ) , free of x and ε , for l ≤ k ≤ r ′ i ( r ′ i sufficiently high) can be representedin terms of indefinite nested sums, one can decide algorithmically if also the unknown functions f ( x , ε ) , . . . , f λ ( x , ε ) can be given in such a form where the highest ε orders are r , . . . , r λ , respec-tively.Here one proceeds as follows.1. By uncoupling algorithms, like, e.g., Zürcher’s algorithm [71] implemented in the package OreSys [37], one obtains a scalar linear differential equation of the form α ( x , ε ) f ( x , ε ) + α ( x , ε ) D x f ( x , ε ) + · · · + α λ ( x , ε ) D λ x f ( x , ε ) = β ( x , ε ) (3.3)with β ( x , ε ) = ∑ i , j ≥ β i , j ( x , ε ) D ix g j ( x , ε ) for explicitly given β i , j ∈ K ( x , ε ) ; note that only finitely many β i , j are non-zero. In addition,one gets f k ( x , ε ) = ∑ k , i ≥ φ k , i ( x , ε ) D ix f ( x , ε ) , ≤ k ≤ λ , (3.4)for explicitly given φ k , i ∈ K ( x , ε ) ; note that only finitely many φ k , i are non-zero2. Plugging in the ansatz f ( x , ε ) = ∞ ∑ n = F ( n , ε ) x n (3.5)into (3.3) and performing coefficient comparison w.r.t. x n yield a recurrence of the form (2.2)(with F ( n , ε ) replaced by F ( n , ε ) ) for some explicitly given a ( x , ε ) , . . . , a d ( x , ε ) ∈ K [ x , ε ] and (2.3) where the first coefficients b i ( n , ε ) can be represented in terms of indefinite nestedsums; note that the orders r ′ i of the ε -expansions in (3.2) must be set high enough to obtainthe correct expressions for b i ( n , ε ) up to the order r .Note further that the recurrence order d ∈ N of (2.2) is bounded by d ≤ λ + max ≤ i ≤ λ deg x ( α i ) . (3.6)5 refined large moment machinery Carsten Schneider
3. Applying the tools from Section 2 together with the corresponding initial values one candecide if the first coefficients F , i ( n ) of the ε -expansion F ( n , ε ) = F , l ( n ) ε l + F , l + ( n ) ε l + + F , l + ( n ) ε l + + . . . (3.7)can be given in terms of indefinite nested sums. If this fails, stop.4. Plugging the ε -expansion (3.7) (where the first coefficients are given explicitly) into (3.4)and extracting the coefficient of x n yield F k ( n , ε ) = F k , l ( n ) ε l + F k , l + ( n ) ε l + + F k , l + ( n ) ε l + + . . . , ≤ k ≤ λ (3.8)where the first coefficients F k , i ( n ) can be represented in terms of indefinite nested sums; notethat the orders r , r ′ , . . . , r ′ λ of the ε -expansions F ( n , ε ) , G ( n , ε ) , . . . , G λ ( n , ε ) must be setsufficiently high to get the desired ε -expansions of F k ( n , ε ) up to the orders r k .We note that this machinery has been implemented efficiently in the package SolveCoupled-System [5, 9, 11, 35] that relies on the following sub-packages:1.
Sigma [57, 60, 61] to find the solutions of the recurrence (2.2).2.
HarmonicSums [1–4, 15–17, 30, 68] to support
Sigma for the elimination of algebraicrelations among the arising sums using quasi-shuffle algebras [18, 26, 41].3.
SumProduction [61] to obtain from (3.4) the coefficients in (3.8) in terms of indefinitenested sums.Using all these technologies we succeeded in calculating, e.g., various non-trivial physical quanti-ties in [6,23]. We remark further, that one can apply also differential equation solvers (available e.g.in
HarmonicSums , see [4]) to (3.3) in order to find closed form solutions in terms of indefinitenested integrals defined over hyperexponential functions. For more details we refer to [8, 14].
4. The large moment method and refined variants
Suppose that we utilized IBP methods [34, 44, 47, 48, 66] and obtained a physical expression¯ f ( x , ε ) in terms of master integrals that are solutions of coupled systems of linear differential equa-tions of the form (1.2). A natural tactic is then to solve these coupled systems, e.g., in terms ofindefinite nested sums as described in Section 3 and to combine the solutions to obtain the first co-efficients of the ε -expansion (1.1); here the coefficients ¯ f k ( x ) itself are considered in power seriesexpansions (1.3) and one seeks for an all- n solution of the corresponding coefficients ¯ F k ( n ) .As mentioned in the introduction, master integrals may pop up that cannot be expressed interms of indefinite nested sums. In particular, the inhomogeneous components in (1.2) and thusthe coefficients in (2.3) of the underlying recurrence (2.2) cannot be expressed in terms of indef-inite nested sums, or the recurrence cannot be solved within this class. In order to overcome thissituation, we have introduced the large moment machinery in [32] that enables one to compute themoments ¯ F k ( n ) with n = , , . . . , µ for a large number µ ∈ N of the power series expansion (1.3).This enables one to analyze this physical quantity further by numerical methods. In addition, one6 refined large moment machinery Carsten Schneider can try to compute linear recurrences by guessing methods [42] and to solve them as described inSection 2; for concrete calculations see also the end of Section 1.Internally, the large moment machinery works as follows.
The original large moment method.
Suppose that we are given a coupled system of the form (1.2)where the inhomogeneous part g i ( x , ε ) has the coefficients G i , k ( n ) in (3.1). But the coefficientsare not given as an all- n representation (e.g., in terms of indefinite nested sums as assumed inSection 3), but by a finite number of moments G i , k ( ) , G i , k ( ) , . . . , G i , k ( µ ) where µ is large (e.g., µ = • other coupled systems to which the large moment method is applied recursively; • symbolic summation or integration methods [9, 57, 60] that yield representations in terms ofindefinite nested sums or integrals from which one can produce a large number of moments; • by standard procedures like Mincer [38, 45] or
MATAD [65] if only a small number ofmoments contributes. Also analytically solvable (multiple) Mellin-Barnes representationscan be used in some cases.Given this input, one follows the calculation steps given in Section 3. But instead of dealingwith expressions in terms of indefinite nested sums, one performs the calculation steps for listswith entries from K that encode the moments of the underlying expressions. Adapting this pro-cedure, one obtains a recurrence (2.2) (with F ( n , ε ) replaced by F ( n , ε ) given in (3.7)) wherethe polynomials a i ( x , ε ) ∈ K [ x , ε ] are given explicitly but where coefficients b i ( n ) in (2.3) arenot represented as expressions in terms of indefinite nested sums, but are given explicitly by themoments b i ( ) , b i ( ) , . . . , b i ( µ ) . Thus using the first initial values we can calculate the moments F , k ( ) , F , k ( ) , . . . , F , k ( µ ) for k = l , l + , . . . , r in (3.7) by Remark 2.4.Finally, we can calculate the moments F j , k ( ) , F j , k ( ) , . . . , F j , k ( µ ) of (3.8) for j = , . . . , λ and k = l , . . . , r j using the formulas (3.4) and the moments F , k ( ) , . . . , F , k ( µ ) for k = l , . . . , r .This general machinery introduced in [32] worked successfully, e.g., for the calculation of thesplitting functions [7]. But to carry out larger problems such as [22, 31] we faced the following Bottleneck.
In order to execute the large moment machinery, sufficiently many initial values mustbe provided in a preprocessing step. In general, the number of initial values equals the order d of the recurrence (2.6) (sometimes it might be reduced if a d ( x , ) = a d ( x , ) = a d ( x , d ) = α ( x , ε ) , . . . , α λ ( x , ε ) have rather high degrees (e.g., up to 50) and thus also the recurrence orderis of similar magnitude. Unfortunately, the calculation of this amount of initial values is often toohard and thus the above large moment machinery is out of scope.Based on these observations we succeeded in [31] in reducing the required initial values sig-nificantly to a number that could be computed in reasonable time (or that have been calculatedalready in earlier projects). We remark that this improvement has its price: the calculation of themoments gets more involved. Nevertheless, a slower method is better than a method that cannot beapplied due to missing initial values. 7 refined large moment machinery Carsten Schneider
Refinement 1.
Suppose that the greatest common divisor of the coefficients α ( x , ε ) , . . . , α λ ( x , ε ) in (3.3) is 1, i.e., there is no common polynomial factor that depends on x or ε . This impliesthat there is at least one i such that α i ( x , ) =
0. By coefficient comparisons w.r.t. ε l and x n (andappropriate expansions) one gets directly a recurrence of the form (2.1) (with F ( n ) replaced by F , l ( n ) ). Analogously to the method in Section 2 one can now repeat the game: Given sufficientlymany initial values, one can calculate the moments F , l ( ) , F , l ( ) , . . . , F , l ( µ ) . Afterwards, oneplugs (3.5) with (3.7) into (3.3), updates the inhomogeneous side β ( x , ε ) and obtains the differentialequation (3.3) (with modified β ( x , ε ) ) where f ( x , ε ) takes over the role of f ′ ( x , ε ) = f , l + ( x ) ε l + + f , l + ( x ) ε l + + . . . . Thus we can loop up to get F , k ( ) , F , k ( ) , . . . , F , k ( µ ) with k = l + , . . . , r . Afterwards, one com-putes the remaining coefficients F j , k ( ) , . . . , F j , k ( µ ) with j = , . . . , λ and 1 ≤ k ≤ r j as describedabove.A benefit of this modified procedure is that the uncoupling method does not have to deal with ε ,i.e., the rational function arithmetic is reduced from K ( x , ε ) to K ( x ) . We note that the obtainedrecurrence contains only the information for one coefficient F , k ( n ) , while the recurrence of theoriginal method (depending on ε ) contains the information to describe the full ε -expansion withthe coefficients F , l ( n ) , F , l + ( n ) , . . . – this extra information often requires a recurrence of higherorder. Another major improvement is obtained by refining this approach further with Refinement 2.
Let p ( x ) ∈ K [ x ] \ { } be the greatest common divisor of α ( x , ) , . . . , α λ ( x , ) , i.e., p ( x ) contains all common polynomial factors in x of the α i ( x , ) . Dividing (3.3) by p ( x ) yields λ ∑ i = α i ( x , ε ) p ( x ) D ix f ( x , ε ) = β ( x , ε ) p ( x ) . (4.1)Performing the expansions w.r.t. x and ε on the right-hand side and doing coefficient comparisonw.r.t. ε l and x j afterwards yield a linear recurrence of the form (2.1) (with F ( n ) replaced by F , l ( n ) )where the order d is bounded by d ≤ λ + max ≤ i ≤ λ deg x ( α i ) − deg x ( p ) . (4.2)After the computation of the moments F , l ( ) , F , l ( ) , . . . , F , l ( µ ) with that recurrence we proceedas in Refinement 1 incorporated the ideas of Refinement 2 in each step.As it turns out, the degree of the common factor p ( x ) of the coefficients α i ( x , ) with 0 ≤ i ≤ λ isoften rather large and thus also the upper bound of the recurrence order can be reduced significantly;see (4.2). Even better, the recurrence order d is reduced by the value deg x ( p ) and therefore thenumber of required initial values can be reduced substantially; see Example 4.1 below.Using, e.g., Zürcher’s algorithm [71], one obtains a fully decoupled system with usually onescalar linear differential equation (3.3) for the unknown f ( x , ε ) and further formulas (3.4) thatenable one to express the remaining f k ( x , ε ) in terms of f ( x , ε ) . Alternatively, one may take thesimple Gauss method (available in OreSys [37]) that yields the following8 refined large moment machinery
Carsten Schneider
Refinement 3.
One obtains a linear differential equation of f λ ( x , ε ) of order o λ where the inhomo-geneous part depends only on the inhomogeneous components g i ( x , ε ) of (1.2) together with theapplication of D ux with u ∈ N on these components. Next, one obtains a linear differential equationfor f λ − ( x , ε ) of order o λ − whose inhomogeneous part depends additionally on D ux f λ − ( x , ε ) with u ∈ N . More generally, one gets a linear differential equation for f j ( x , ε ) of order o j with 1 ≤ j ≤ λ whose inhomogeneous part depends on the components g ( x , ε ) , . . . , g λ ( x , ε ) and additionally onthe unknowns f j + ( x , ε ) , . . . , f λ ( x , ε ) and the application of D ux with u ∈ N to them. Thus one cancompute iteratively the unknowns f j ( x , ε ) for j = λ , λ − , . . . , o , . . . , o λ is usually much smaller than λ which implies that for each F j , k ( n ) the number of initial values is reduced. In particular, we get theimproved upper bound d ≤ o i + max ≤ i ≤ λ deg x ( α i ) − deg x ( p ) . Note that the total number of initial values needed for all components F j , k ( n ) might be even larger.However, it is usually much harder to calculate initial values F , k ( i ) for large values i (as proposedin the earlier strategies) than computing initial values for more components F j , k ( i ) with 1 ≤ j ≤ λ where i is small. Example 4.1.
Consider a typical system (1.2) with λ = coming from [31]. For the uncoupledsystem we obtain 4 linear differential equations of orders o = , o = , o = and o = .Here the inhomogeneous part of the linear differential equation in f k ( x , ε ) of order o k dependson the components D ux g ( x , ε ) , D ux g ( x , ε ) , D ux g ( x , ε ) , D ux g ( x , ε ) and D ux f k + ( x , ε ) , . . . , D ux f ( x , ε ) with u ∈ N . Thus we compute the moments stepwise for f ( n , ε ) , f ( n , ε ) , f ( n , ε ) , and finally forf ( n , ε ) . More precisely, we compute the moments of the coefficients F j , k ( n ) in (3.7) and (3.8) with ≤ j ≤ and l = − ≤ k. E.g., for F , − ( n ) we obtain a recurrence of the form (2.1) (F ( n ) replacedby F , − ( n ) ) of order d = . We note that this small recurrence order was possible by Refinement 2:we sneaked in a polynomial p ( x ) ∈ Q [ x ] of degree within the linear differential equation (4.1) ;setting p ( x ) = would have delivered a recurrence of order + = . Finally using 4 initialvalues enabled us to compute the moments F , − ( ) , F , − ( ) , . . . , F , − ( ) . Similarly all theother coefficients can be calculated.
5. Conclusion
We presented our general large moment method introduced in [32]. In order to activate it, suf-ficiently many initial values have to be provided as a preprocessing step. However, the calculationof these starting points is often the show-stopper and a major challenge is to reduce the requirednumber of initial values as much as possible. In particular, the maximum of the number of initialvalues among all components F j , k ( n ) should be kept as small as possible. In this article we haveelaborated three refinements from [31] that improved this problem significantly. This new largemoment engine enabled us to recalculate the polarized three-loop anomalous dimensions in [22]and to tackle big parts of the heavy fermion contributions of the massive three loop form factorsin [31]. We remark that additional improvements have been introduced in [31] that require furtherinvestigations. 9 refined large moment machinery Carsten Schneider
Acknowledgment.
This work was supported in part by the Austrian Science Fund (FWF) grantSFB F50 (F5009-N15), by the EU TMR network SAGEX Marie Skłodowska-Curie grant agree-ment No. 764850 and COST action CA16201: Unraveling new physics at the LHC through theprecision frontier.
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