Determining the closed forms of the $O(a_s^3)$ anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra

Single-scale structure

• Mellin transformation yields structural simplifications for single-scale processes, which are represented at lower orders by harmonic sums and rational functions of N.Less is known about function spaces for processes involving two or more scales.Source: §1 · p. 2

Method

• Given sufficiently many moments, the method automatically finds a polynomial-coefficient recurrence that uniquely propagates the sequence from its initial values.The recurrence is then solved with Sigma in generalized harmonic sums and reduced to an algebraically independent basis.Source: §1 · p. 3
• The recurrence solver also handles d’Alembertian solutions expressible through indefinite nested sums and products, extending beyond ordinary harmonic-sum recurrences.The authors note that more general functions may arise in higher-order or massive calculations.Source: §1 · p. 3

Motivation

• Single-scale anomalous dimensions and Wilson coefficients are more easily calculated through Mellin moments than through complete expressions in the Mellin variable N.The study targets unpolarized three-loop splitting functions and deep-inelastic-scattering Wilson coefficients.Source: §1 · pp. 2–3

Application

• The paper applies the approach separately to color contributions of three-loop anomalous dimensions and Wilson coefficients, using moments from exact calculations.It compares this exact reconstruction with earlier approximate methods based on special ansatzes [7] [25].Source: §1 · p. 3

Validation and scope

• Finite data cannot prove that a recurrence continues beyond the known moments, so recurrence detection requires additional bounds or validation against the supplied sequence.A failed ansatz excludes only the tested order and degree, not all possible recurrences.Source: §2 · pp. 4–5

Recurrence ansatz

• A finite moment array is tested against an ansatz of order l and degree d, producing a homogeneous linear system for (l + 1)(d + 1) unknown coefficients.Overdetermined systems can identify recurrences valid beyond the supplied terms, whereas underdetermined solutions typically fit only the observed range.Source: §2 · p. 4

Scale of the computation

• An ansatz for a recurrence of order 3 with polynomial coefficients of degree 3 produced an overdetermined homogeneous system with 16 unknowns and 17 equations, yet two independent solutions.Source: §2 · p. 4

Example

• Two independent solutions in an overdetermined order-3, degree-3 system for Pgg,0(N) correspond to valid recurrences rather than numerical coincidence.The supplied example gives explicit third-order recurrence equations valid for all N ≥ 1.Source: §2 · pp. 4–5

Modular computation

• The implementation solves systems modulo word-size primes, combines results by Chinese remaindering, and reconstructs exact rational coefficients.This avoids explicit rational systems that can exceed memory limits, including an estimated 11 Tb system for a large problem.Source: §2 · p. 5

Operator reduction

• The procedure computes modular greatest common right divisors of candidate operators to combine small linear systems with compact minimal-order output recurrences.For the large problem, this reduces the direct-system burden from 33804 unknowns to 5022.Source: §2 · p. 6

Algorithm

• The workflow chooses a prime and order-and-degree bounds, solves recurrence systems modulo each prime, computes greatest common right divisors by the Euclidean algorithm, repeats across primes, applies Chinese remaindering and rational reconstruction, and checks the resulting recurrence against the given data.Source: §2 · p. 6

3 Solving the Recurrence Equations

• The overall workflow first computes d’Alembertian solutions and then converts their nested sums into closed forms such as harmonic sums for further problem solving.The general solver operates in ΠΣ-difference fields, allowing polynomial expressions in indefinite nested sums and products, while the section’s simplified setting assumes polynomial coefficients and zero inhomogeneity.Source: §3 · p. 7

3.1 Finding all d’Alembertian solutions

• The method factors recurrences into linear right factors to construct all d’Alembertian solutions, which are linearly independent and span every solution expressible through indefinite nested sums and products.Recursion reduces the operator order, while product-solution improvements can remove several factors at once.Source: §3.1 · p. 9
• For the hardest Wilson-coefficient recurrence, order 35, the solver derived 35 linearly independent solutions in 25 hours using 3 GB of memory.The representation required 478 summation quantifiers instead of 595; summand denominators had irreducible factors up to degree 1000 and polynomial coefficients up to 700 decimal digits.Source: §3.1 · p. 12

3.2 Simplification of d’Alembertian solutions

• Sigma automatically simplifies d’Alembertian expressions into representations with algebraically independent sums, minimal nested depth and denominator degree, and algorithmically efficient tuning.This automation extends Karr’s ΠΣ-theory because Karr’s algorithm alone requires the target sum elements to be known explicitly.Source: §3.2 · p. 12
• The harmonic sums are algebraically independent, obtained using HarmonicSums with relations also available in Sigma.HarmonicSums was chosen for convenience and efficiency.Source: §3.2 · pp. 13–14
• A depth-35 expression was simplified in around four days and 20 hours to 65 sums, including 47 sums of depth two.The resulting sums satisfy properties 1–3 stated in the paper.Source: §3.2 · p. 14
• Less-nested sum representations make the difference-field algorithms more efficient, whereas harmonic-sum forms with depths up to five can exceed time and memory limits.The construction therefore prioritizes minimal nesting depth for subsequent algorithmic processing.Source: §3.2 · p. 14

Method and scope

• The method derives recurrence equations from Mellin moments and solves them into algebraically reduced harmonic-sum representations for 3-loop QCD quantities.The calculation treats individual color- and ζ-value contributions and compares recurrence parameters and harmonic-sum counts across the results.Source: §4 · pp. 15–16

Largest example

• The most demanding example reaches N = 5114, while its recursive computation requires roughly 3 GB and 270 minutes for one C_F contribution.That contribution produces a fraction with 13,888 numerator and 13,881 denominator digits, and the moment set occupies 69 MB.Source: §4 · p. 15

Anomalous dimensions

• Anomalous-dimension results use at most 26 harmonic sums, reducing the non-singlet case P^(k),± from 68 to 26 and another case from 130 to 21.The reported reductions are relative to the representations in the cited earlier results.Source: §4 · p. 15

Wilson coefficients

• The largest Wilson-coefficient calculation uses 60 harmonic sums instead of 290 for C^(3)2,q,C_F^2 C_A after algebraic reduction.The 60 sums are the possible weight-w = 6 sums without index {-1} after reduction.Source: §4 · p. 15

Structural results

• After reduction, harmonic-sum counts cluster around 58–60, 26–29, 11–15, or below 10, with the pattern possibly related to diagram topology or color and field structure.This clustering is not observed when counting sums in the earlier representations.Source: §4 · pp. 15–16

Computational requirements

• 110.3 CPU days were required overall, with the largest calculations using up to 10 GB on a 2 GHz processor.For the largest Wilson-coefficient contributions, each case required about four weeks of computation and no more than 10 GB.Source: §4 · pp. 15–16

Harmonic-sum basis

• The 3-loop Wilson coefficients require functions through weight w = 6, represented by 38 functions and 35 basic Mellin transforms.An algebraically equivalent basis can replace two of the listed functions without changing the representation.Source: §4 · pp. 16–17

Limitations

• The method cannot yet be applied to whole 3-loop color-factor contributions because mincer and MATAD become prohibitively expensive at the high moments required.For 3-loop quantities, reaching N = 16 can require about 0.5 CPU year, motivating cheaper fixed-moment methods.Source: §4 · p. 17

• The method computes exact single-scale quantities from finite moment sets and applies to anomalous dimensions and Wilson coefficients through three-loop order.The results are compactified using algebraic relations among harmonic sums.Source: §5 · p. 17
• The full computation required about four months on 2 GHz processors and less than 10 GB of memory, with fivefold naive parallelization possible.Source: §5 · p. 17
• Whole color-factor contributions cannot yet be solved because available methods require too many moments, motivating cheaper high-order fixed-moment methods.Source: §5 · p. 17
• Large difference equations of order 35 and degree approximately 1000 were established and solved reliably and unconditionally.Source: §5 · pp. 17–18

• The non-singlet anomalous dimensions are expressed using nested harmonic sums such as S−2,1, S−3,1, S−2,1,1, and S−2,1,−2.Source: §6 · pp. 26, 28, 31, 32
• The expressions combine harmonic sums with rational functions of n and high-degree polynomial coefficient factors.Source: §6 · pp. 28, 29, 32, 33
• The appendix defines the color-factor conventions used in the formulae, including CA, CF, NF, Nc, and TR.Source: §6 · p. 33
• Additional polynomial functions P1(n) through P8(n) provide coefficient data for the displayed anomalous-dimension expressions.Source: §6 · p. 33