Forcer, a FORM program for the parametric reduction of four-loop massless propagator diagrams

1 Introduction

• Four-loop splitting functions are needed to determine proton parton distributions, while complete four-loop calculations are currently out of reach, making Mellin moments a next-best approach.Those moments can be obtained by expanding in parton momentum and converting to massless propagator integrals; Mincer handled the resulting three-loop integrals.Source: §1 · p. 2
• Generic Laporta reductions substitute parameter values and reduce many subsystems, while spurious poles can require exact rational-polynomial arithmetic that is very slow.The poles cancel only after contributions to a master-integral coefficient are combined, making fixed-depth epsilon expansions unsafe when their number is unknown.Source: §1 · p. 2
• Forcer is a FORM program with a precomputed graph for reducing topologies and mapping those with missing propagators; 21 difficult cases required hand-guided computer-assisted solutions.It can use rational polynomials in epsilon or a fixed expansion whose depth must be selected carefully.Source: §1 · pp. 2–3
• Forcer has supported large calculations including four-loop splitting-function results [29] [30], four-loop propagator and vertex computations [31] [32] [33], and the five-loop beta function [34].The paper also previews benchmarks for complicated cases.Source: §1 · p. 3

2 Reducing known substructures

• The one-loop subgraph rule integrates out a loop when both propagators are massless, replacing it with a single line of non-integer power.For the rule to apply simply, numerator dot products must depend on only one of the two loop lines.Source: §2 · p. 3, Figure 1 · p. 4
• The carpet rule integrates out the loop momentum associated with a propagator connecting the two external lines.The rule was generalized to the four-loop case.Source: §2 · p. 3
• The diamond rule generalizes the triangle rule, reduces a green triple line or blue external line, and can be explicitly summed to avoid spurious poles.Its numerator dot products must not depend on red dashed lines.Source: §2 · p. 3, Figure 1 · p. 4
• The rules reduce 417 of 438 classified topologies; among the 21 without these substructures, 17 contain master integrals and four need hand-guided IBP solutions.The four include one four-loop diagram, two three-loop diagrams with a non-integer-power line, and one two-loop diagram with two such lines.Source: §2 · p. 4

3 IBP identities

• Forcer represents integrals with signed denominator and numerator powers and defines complexity by deviations from unit denominators and zero numerators.The basic integral has complexity zero; each extra denominator power or numerator power raises complexity by one. The authors choose dot products as numerator variables.Source: §3 · pp. 4–5

3.1 Solving parametric IBP identities

• For four-loop diagrams, 20 initial IBP relations form S0; Gaussian elimination and boundary-sensitive rules yield reductions, while some identities can eliminate a line when indices are integer.Simultaneous reductions may fail at boundary values and therefore only speed up reduction; equations with an ϵ coefficient can still reduce an index to zero.Source: §3.1 · pp. 6–7

3.2 Reduction rules beyond S0

• A scheme using a different variable order and more sophisticated equation combinations reduced the same benchmark from 53000 seconds to 555 seconds.The method expands S0 to S1, then eliminates complexity-3 and -2 objects to construct a variable-by-variable reduction scheme.Source: §3.2 · pp. 7–8
• Reduction schemes must control recursion and coefficient growth, since yoyo equations can cycle and large rational coefficients can make reductions impractical.Breaking a yoyo by combining equations can restore a regular condition; oversized coefficients can strain FORM and slow reductions across millions of integrals.Source: §3.2 · pp. 10–11

3.3 Identities for topologies with insertions

• For topologies with insertions, IBPs are generated by shifting the affected index by ϵ, and separate complexity conventions reduce positive and negative index shifts.The two-loop t1star05 integrals reduce to its master integral with all indices one and to integrals with simpler topologies.Source: §3.3 · pp. 11–12

3.4 Solving strategy

• The solving strategy begins by selecting a numerator basis, whose quality affects the IBPs, and constructing the IBP identities.Source: §3.4 · p. 12
• Gaussian elimination simplifies S0, removes complexity-2 and -3 terms from raised-index options in S1, and uses S0 again to reduce complexity-1 terms before combining reductions.This can simplify final formulas and break up difficult yoyos; useful double-reduction equations are also included in the topology's reduction procedure.Source: §3.4 · pp. 12–13
• The elimination order can change program efficiency by orders of magnitude; if the last variable lacks a good equation, the nearly complete scheme yields alternatives from which the shortest is selected.Form procedures execute the steps, but choosing the elimination order and equations is laborious; for the bebe topology, complexity-raising use of two S0 equations helped eliminate two variables and reduced the remaining equations to 18.Source: §3.4 · pp. 12–13
• The custom scheme is checked numerically for irreducible master integrals and loops, while relations among equations containing only missing-line integrals are left unaccounted for.The completed reductions are combined with useful double-reduction equations based on S0 or Gaussian-elimination substitutions.Source: §3.4 · p. 13
• For schemes still too slow, automated combinations using S2 and S−1 can produce shorter reductions, but may take substantial time because of complicated parameter-dependent rational polynomials.Source: §3.4 · p. 13
• Rules can run slowly when subleading terms force returns to earlier rules or complicated coefficients multiply later terms; expanding in ϵ can help if carried deep enough to avoid spurious poles.Source: §3.4 · p. 13
• Monte Carlo Tree Search for elimination-order selection is under investigation, with proposed evaluation factors including term counts, spectators, and rational-polynomial size; this use of AI was also hinted at in ref..The proposal cites MCTS as an example.Source: §3.4 · pp. 13–14

4 The topologies that need custom reductions

• The master-topology classification excludes factorized diagrams and integrates out one-loop two-point functions, which can change the master-integral values.The authors give master-integral values in this convention in appendix C.Source: §4 · p. 14
• 21 topologies require custom reduction schemes: eight four-loop master integrals and four non-master topologies join the master cases in this category.The four non-master cases comprise one four-loop diagram, two three-loop diagrams with a non-integer edge, and one two-loop diagram with two non-integer edges.Source: §4 · p. 14
• Performance is tested on complete four-loop reductions at complexity 14 and 28, confirming that simpler-topology terms do not create bottlenecks.The two tests raise all indices by 1 and by 2, respectively.Source: §4 · p. 17
• For nono, the authors avoid a circular reduction path by processing a looping term separately, yielding a compact 39-term formula for n4.The alternative formula would exceed 1,000 lines.Source: §4 · p. 17
• The bebe scheme requires hand guidance because reducing n1 and n3 with available equations raises complexity and consumes equations needed for other variables.The remaining equations still suffice for the next 11 variables, after which n2 must be derived again from S0.Source: §4 · p. 17
• For bubu, eliminating the last of five numerators requires case-split, lengthy formulas, making variable order critical to prevent term growth.Finding a first suitable scheme took more than two months.Source: §4 · p. 18
• The reduction outcome depends on how and in what order compact Gaussian-elimination equations are applied; some faster schemes incur more spectators or higher powers of ϵ.The ordering must also avoid loops, and the authors cannot always identify an optimal choice.Source: §4 · p. 18

5 The Forcer framework

• For each topology, Forcer precomputes its reduction action and momentum rewrites, avoiding runtime cost for costly optimizations such as choosing an optimal basis.Source: §5 · p. 18

5.2 Reduction graph execution

• The graph links topologies when a propagator is removed; at runtime, Forcer symmetrizes each topology, applies its action, and routes resulting integrals to masters or topologies with fewer lines.Source: §5.1 · p. 18, §5.2 · p. 21

5.1 Reduction graph generation

• The generated reduction graph contains 438 topologies and requires 40,000 lines of Form code.Source: §5.1 · p. 21

5.3 Example

• For d366, the insertion action integrates the left bubble using dot products in p1, then rewrites the two remaining dot products to topology d118.Source: §5.3 · p. 23

• Users specify vertices with vx, call the external momentum Q, and can mark propagator insertions with ex; Forcer matches input topologies and rewrites dot products automatically.Source: §6 · p. 24
• The default output is exact in master integrals and rational coefficients, while the optional msbarexpand argument returns the unrenormalized answer expanded in MS.Source: §6 · p. 24
• Topology matching, momentum substitution, and reduction can also be executed as separate steps.Source: §6 · p. 24

7 From physical diagrams to Forcer

• Physical diagrams must be mapped to Forcer topology and momentum notation; notation.h supports topology matching, relabelling, and symmetry operations.Source: §7 · pp. 24–26

7.1 Self-energy filtering

• The self-energy filter tests possible vertex combinations and eliminates a diagram when a super-vertex is reduced to two lines; the complete procedure uses 30 Form statements.Source: §7.1 · pp. 27–28

7.2 Colour split-off

• Splitting colour from Lorentz structure lets each distinct colour object be evaluated only once.Source: §7.2 · p. 29

7.3 Diagram database

• Grouping diagrams with the same topology and colour factor into superdiagrams supports distribution across computers and typically speeds calculation by O(3).Source: §7.3 · p. 29

7.4 Momentum substitutions

• Momentum substitutions use a layered, order-dependent rewrite to limit term growth; merging terms during rewrites makes this approach faster.Source: §7.4 · p. 30

Examples and performance

• The four-loop beta function agreed with refs.; its background-field calculation took under three minutes in Feynman gauge and about four hours across gauge-parameter powers on 32 cores.Source: §8 · pp. 30, 33
• Forcer computed QCD propagators and nullified-momentum three-point vertices; the triple-gluon vertex took one week on 24 cores.Source: §8 · p. 31
• OPE moments reached N = 16 (N = 19 at large Nc), enough to reconstruct and validate the non-singlet all-N form.Simpler leading and sub-leading nf diagrams reached N = 40; the large-Nc cusp anomalous dimension agreed with the quark-form-factor calculation [49].Source: §8 · p. 31
• High-complexity integrals exceeded current Laporta-style methods by a large margin, and some Forcer computations still took weeks.Source: §8 · p. 31
• The five-loop Yang–Mills beta function computation confirmed the SU(3) result [50] and took three days on a cluster.Source: §8 · p. 31
• Ten times more sampled configurations increased runtime only about 20%, while each complexity step multiplied runtime by about 2.5.Shared configurations deeper in the reduction merge through symmetries and grouping.Source: Figure 5 · p. 32, §8 · p. 32
• The background-field propagator yields the beta function without another propagator and vertex, while its benchmark timing nearly matches the gluon’s.Source: §8 · p. 32, Figure 6 · p. 33

Conclusions and outlook

• About 30% of gluon-propagator diagrams can be computed by the five-loop Forcer equivalent, despite over 200 missing topology actions and undetermined master integrals.The authors state that automatically working out the master topologies could allow a five-loop program to be constructed shortly afterward.Source: §9 · p. 33
• The five-loop extension is challenging because the number of parameters to reduce grows from 14 at four loops to 20 at five loops.The authors condition a near-term construction on the master topologies being worked out automatically.Source: §9 · pp. 33–34
• For Yang-Mills theory with fermions, propagators and one-vanishing-momentum three-point functions were determined exactly in master integrals with rational coefficients in ϵ.The five-loop beta function with generic colour group was computed in three days on a cluster.Source: §9 · p. 34
• Operator vertices reduce resource demands for higher DIS moments; non-singlet splitting-function moments up to N ≥16 have been obtained.Using the methods of ref. for three-loop moments becomes demanding at moment six or higher; the authors hope to extend the operator-vertex approach to singlet splitting functions.Source: §9 · p. 34

One-loop sub integrals

• The one-loop subintegral formula can be written directly for loop-momentum numerators, avoiding further recursions, and FORM evaluates d’Alembertians efficiently with each term generated once.The implementation uses the combinatoric functions distrib and dd.Source: A One-loop sub integrals · p. 35
• The FORM example generated 1,133,616 terms in 3.09 seconds.The reported output contains 1,133,616 terms.Source: A One-loop sub integrals · p. 35
• The d’Alembertians are used in both one-loop integrals and harmonic projections for structure-function moments, making the implementation relevant to higher Mellin moments.The source states that the code is essential when higher Mellin moments are needed.Source: A One-loop sub integrals · p. 35
• The method has three preprocessing costs: rewriting numerator dot products, converting resulting dot products to a lower-loop basis, and generating lower-loop reductions for non-integer denominator powers.The authors describe these costs as small compared with the computer resources saved.Source: A One-loop sub integrals · p. 35
• The function G is normalized to a version with unit denominator powers plus possible ϵ multiples and no numerators; Pochhammer factors carry the difference, while the remaining G is substituted with master integrals at the program’s end.Pochhammer factors may be kept as rational polynomials or expanded in ϵ.Source: A One-loop sub integrals · pp. 35–36

Generalised carpet rule

• For a subgraph embedded in an outer one-loop graph, scaling and Lorentz invariance allow the outer loop to be integrated first.The rule applies to an embedded subgraph with loop count Lsub.Source: B Generalised carpet rule · p. 36
• The subgraph integrand comprises propagator factors and a degree-n polynomial Pn({pi}, q) in the loop momenta and external momentum, expressed using dot products in Forcer.The squared-momentum factors also cover quadratic Lorentz scalars produced by subgraph integration, such as pi·pj and pi·p.Source: B Generalised carpet rule · p. 36
• When Pn({pi}, q) = 1 and n = 0, the subgraph’s dimension alone suffices to write the outer-loop integral result.The formula also uses Pochhammer symbols and the function G defined by equation (25).Source: B Generalised carpet rule · p. 36
• The formula’s d’Alembertian can be implemented efficiently with FORM’s distrib and dd functions.Source: B Generalised carpet rule · p. 36

The master integrals

• The four-loop master integrals are copied from refs. [38] [39]; ref. [38] gives sufficient ϵ precision for four-loop calculations, while ref. [39] contains more terms.For inserted master integrals, the paper sets the insertion line power to 1 + mϵ; thus a one-loop insertion means propagator power ϵ, compared with 1 + ϵ in its convention.Source: C The master integrals · pp. 36–37
• Forcer normalizes the master integrals to powers of the fundamental one-loop integral multiplied by ϵ, a convention called the G-scheme.The displayed expansions are truncated at a precision sufficient for a five-loop program, although the program contains the full precision of ref. [39].Source: C The master integrals · p. 37
• Conversion to MS requires multiplying by the appropriate number of powers of the conversion factor, with ζ2 dropped because it never appears in physical results of massless propagators.Source: C The master integrals · pp. 38–39

D Expansions

• Rational-polynomial arithmetic can become costly, while spurious poles may require extra powers of ϵ that cancel only after terms for a master integral are summed.The authors encountered numerator and denominator powers above 700, even in an incomplete reduction.Source: D Expansions · p. 39
• A trial run estimates the required expansion depth; the program uses the worst depth across diagrams plus one guard power to generate expansion tables.The trial run keeps only the minimum power of ϵ with coefficient one to prevent such terms from canceling.Source: D Expansions · pp. 39–40
• Running the calculation twice with different expansion depths and checking where coefficients change is a simpler, safer alternative.The authors compare this check to testing numerical instability at different floating-point precisions.Source: D Expansions · p. 40
• Form’s PolyRatFun rat(divergence,variable) option keeps only the lowest power of ϵ, while rat(expand,variable,maxpow) expands through a specified maximum power.The program is generally quite fast in divergence mode.Source: D Expansions · p. 40