Pith. sign in

REVIEW 3 major objections 4 minor 77 references

This paper claims that the epsilon-expansion of any scalar one-loop Feynman integral is governed by a recursion that expresses every coefficient as a multiple polylogarithm, starting from hyperbolic simplex volumes at epsilon=0.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 21:00 UTC pith:UB4CT3BU

load-bearing objection New recursion for one-loop ε-expansions is plausible and useful, but the all-orders step rests on an unproved interchange; k≥2 cases are not actually demonstrated. the 3 major comments →

arxiv 2607.16416 v1 pith:UB4CT3BU submitted 2026-07-17 hep-th

Recursive construction of scalar one-loop integrals in dimensional regularisation

Paul Mork This is my paper
classification hep-th
keywords scalar one-loop integralsdimensional regularisationepsilon expansionmultiple polylogarithmshyperbolic simplex volumesSchlafli differential formulaorthoscheme dissectionhexagon integral
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper claims that every coefficient in the epsilon-expansion of any scalar one-loop Feynman integral—with arbitrary masses, arbitrary number of legs, and off-shell Euclidean kinematics—is a multiple polylogarithm, and it supplies an explicit recursive procedure to compute them all. The recursion moves down one order in the dimensional regulator while moving up one or two in the number of external legs, so every target coefficient is eventually expressed through epsilon=0 integrals, which are known hyperbolic simplex volumes. The engine is Schläfli's differential formula applied to orthoscheme dissections of hyperbolic simplices; a logarithmic subtraction term absorbs the singular part of the higher-point limit. The first genuinely new result is a closed multiple-polylogarithm expression for the O(epsilon) coefficient of the scalar hexagon with arbitrary masses in Euclidean kinematics.

Core claim

The paper's central discovery is an epsilon-recursion: for even N, the k-th Laurent coefficient of the N-point integral equals the q->0 limit of -2 times the (k-1)-th coefficient of an (N+1)-point integral plus a subtraction term I_N^N(0) Li_{k+1}(-1/q), where q is a mass parameter sent to zero. A companion large-mass limit converts even-point coefficients into odd-point ones. Iterating these two operations expresses every coefficient as a limit of an integer-dimensional integral with more external legs—an object already known to be a multiple polylogarithm through its interpretation as a hyperbolic simplex volume. Each step stays inside the class of multiple polylogarithms and raises the tr

What carries the argument

Schläfli's differential formula, which says that the change in volume of a hyperbolic simplex under a deformation equals a signed sum over its codimension-2 faces of their volumes times the changes of their dihedral angles. Combined with the auxiliary-mass representation of the dimensionally regularised integral as a one-fold integral over its integer-dimensional counterpart, the formula is integrated along a one-parameter mass shift in which only one dihedral angle varies. The orthoscheme dissection reduces arbitrary kinematics to nested Gram matrices whose dihedral angles are invariant under that shift, so the recursion's kernel is identified with a higher-point integral. The recursion is

Load-bearing premise

The recursion assumes it is safe to replace a certain integrated kernel by its leading approximation as a higher-point integral and then take the small-mass limit, so that no leftover error term survives.

What would settle it

Compute, for a small nonzero q, the exact primitive on the left of Eq. (3.25) and compare it with the (N+1)-point approximation on the right; if the difference, integrated against the logarithmic kernels, does not vanish as q->0 for a concrete case such as the bubble O(epsilon^2) coefficient, the recursion fails. Alternatively, evaluate the paper's hexagon O(epsilon) expression at generic Euclidean points using independent high-precision numerical integration; any mismatch beyond numerical error would falsify the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Every Laurent coefficient of any scalar one-loop integral—any number of legs, arbitrary masses, off-shell Euclidean kinematics—is a multiple polylogarithm of uniform transcendental weight d/2+1+k.
  • Higher orders in epsilon can be generated algorithmically from epsilon=0 data without rationalising square roots; the epsilon=0 seeds are known hyperbolic simplex volumes.
  • The O(epsilon) coefficient of the scalar hexagon with arbitrary masses and off-shell Euclidean kinematics is now available in closed multiple-polylogarithm form.
  • The O(epsilon^2) contributions to lower-point integrals such as the box and triangle become accessible by applying the recursion to the hexagon result.
  • The resulting O(epsilon) differential equation matches the known diagrammatic-coaction differential equation for one-loop integrals, providing a direct consistency check.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the recursion is sound, the epsilon-expansion problem reduces to computing one hyperbolic volume: the k-th coefficient of an N-point integral comes from an integer-dimensional (N+1+2k)-point orthoscheme, so the bottleneck becomes geometric rather than analytic.
  • The alternating limits in the recursion suggest a systematic geometric signature ladder—hyperbolic, then AdS-like, then further transitions—that the paper only illustrates for the bubble; one could test whether this picture organises every order in epsilon.
  • A practical testable extension is to push the recursion to the massless on-shell hexagon at O(epsilon^2) and compare with existing analytic results, checking that the multiple-polylogarithm class survives singular kinematic limits.
  • The method may have an algorithmic payoff: automating the limiting procedures would let one generate epsilon expansions for arbitrary numbers of legs on demand, provided simplification identities among multiple polylogarithms keep expressions manageable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper claims a recursive construction of all Laurent coefficients in the dimensional regulator ε for scalar one-loop integrals. For even N, Eq. (3.30) expresses the coefficient I_N^k(Q) as a q→0 limit of the (k−1)-coefficient of an (N+1)-point integral plus a polylogarithmic subtraction; odd N are covered by the large-mass limit (3.31). The recursion is seeded by ε=0 integrals, which are known in terms of hyperbolic simplex volumes and multiple polylogarithms (MPLs). The authors derive the recursion from Schläfli's differential formula, prove the even-to-odd conversion in Appendix A, verify several O(ε) and one O(ε²) cases against known results and pySecDec, and present an explicit MPL expression for the hexagon O(ε) coefficient.

Significance. If the all-orders statement is correct, the result is significant: it would establish that all orders of the ε-expansion of arbitrary one-loop scalar integrals remain in the MPL class and provide a constructive algorithm that avoids simultaneous square-root rationalisation. The hexagon O(ε) expression is a concrete new result, and the numerical checks and the comparison with the O(ε) differential equation of ref. [49] lend support. The recursion is not circular: it uses the ε=0 volume results as seeds, and the even-to-odd proof in Appendix A is a genuine derivation. The main weakness is the lack of a rigorous justification of the q→0 interchange in §3.3 and the absence of any test of the recursion at orders k≥2.

major comments (3)
  1. [§3.3, Eq. (3.25)] The replacement of the primitive by 2 I_{N+1}^{N+2}(u) is only stated 'to leading order in q', and this replacement is then used inside the remaining u-integration before q→0. The exact kernel obtained from Eq. (3.24) after the change v=η/(1+η) is 1/[(1−v)(q+(1−q)v)], not 1/(v+q)+1/(1−v); the difference is −q²/[(q+(1−q)v)(v+q)]. Integrated against I_N^N(v), this remainder is O(q), but no uniform error bound is given after convolution with [1/(u+δ)+1/(1−u)] log((u+δ)/(1−u))^{k−1} and integration over u. Without such a bound, the limit q→0 in Eq. (3.30) is not established.
  2. [§3.3, Eqs. (3.23)–(3.30)] The derivation removes the δ-regulator before taking q→0. Even after correcting the integration-by-parts sign (see minor comments), the exact primitive is F_q(u)=2 I_{N+1}^{N+2}(u)+R_q(u) with R_q(u)=O(q). The δ→0 limit of the R_q terms is not controlled: R_q(0) is O(q), and the combined plus-distribution and 1/(1−u) integrals of R_q must be shown to vanish as q→0. This is particularly relevant for k≥1, where the subtraction term Li_{k+1}(−1/q) has finite parts and any surviving O(q) contribution would shift the finite limit. The manuscript does not supply the needed dominated-convergence or remainder estimate.
  3. [§5 and §6] The explicit applications do not exercise the all-orders claim beyond the k=0 case and one k=1 case whose input is taken from ref. [36] rather than generated by the full recursion. The bubble O(ε²) computation uses I_3^0 from ref. [36] instead of deriving it from the pentagon via Eq. (3.30). No k≥2 coefficient is computed. Therefore the central assertion in §4.1 that all Laurent coefficients are MPLs of uniform weight rests on an unverified recursion step. At minimum, the author should provide a k=1 computation that uses the complete recursion (e.g., triangle O(ε²) from pentagon O(ε)) and a k=2 example, or give the missing analytic estimate that makes such checks unnecessary.
minor comments (4)
  1. [§3.3, Eq. (3.23)] The integration-by-parts sign is incorrect as displayed. With F(u)=∫_u^1 dv K(v) I(v) and L(u)=(u+δ)/(1−u), one has F'(u)=−K(u)I(u), so I1 = log^k(δ) F(0) + k∫_0^1 du [1/(u+δ)+1/(1−u)] L(u)^{k−1} F(u), not minus k∫. The narrative after Eq. (3.26) uses the plus sign; the displayed minus must be corrected.
  2. [§2.2.1, Eq. (2.19)] The notation I^k_N(Q) and I^k_{N,d}(Q) is used almost interchangeably after Eq. (2.19). A short remark stating that the explicit d is kept only when the integer dimension matters would improve readability.
  3. [§6, text after Eq. (6.2)] The statement that 'setting log(Q_{7,7})→0' is a simplification specific to the O(ε) coefficient is correct, but it would be helpful to state explicitly that at order O(ε²) and higher the finite parts of Li_{k+1}(−1/q) must be subtracted before taking the limit; otherwise a reader may assume the shortcut is generally valid.
  4. [§3.2, Eq. (3.18)] The derivation of the k=0 case is carried out separately, and the statement that Eq. (3.30) 'also applies for k=0' is true only after using I^{-1}_{N,d}=−I^d_N. This is clear from the text but could be stated as a check rather than as a consequence of the k≥1 derivation.

Circularity Check

0 steps flagged

No construction-level circularity; minor same-author reliance on orthoscheme dissection from ref. [36].

full rationale

The central recursion (3.30) is not circular: it is derived from Schläfli's differential formula, the auxiliary-mass representation (2.7)/(2.19), and explicit distributional identities. It expresses I_N^k(Q) as a q→0 limit of a combination of I_{N+1,N+2}^{k-1}(Q̌) and an explicit Li_{k+1}(-1/q) subtraction — i.e., a different object with more legs and lower ε-order, not the same fitted or input quantity. The seeds are ε=0 hyperbolic volumes from Rudenko [19] and Ren et al. [20], which are external to this paper. The new hexagon O(ε) result is checked numerically against direct Feynman-parameter integration with pySecDec, giving independent external support for the main application. The main same-group reliance is the orthoscheme-dissection compatibility taken from ref. [36] (Duhr–Mork), used via Eq. (2.37) to extend results from nested Gram matrices to arbitrary kinematics; this is load-bearing for the all-N claim but is a prior independent lemma, not a definition of the predicted quantity, and its use is partially validated by the numerical hexagon check. The unproved q→0 interchange in deriving Eq. (3.25) is a genuine technical correctness risk, especially for k≥2 where the Li_{k+1}(-1/q) subtraction contributes to the finite part, but it is an analytic-limit assumption rather than a circular reduction. No equation in the paper reduces to its own input by construction, and no fitted parameter is renamed as a prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

There are no fitted free parameters; the construction is a pure derivation from geometric theorems. Its load-bearing assumptions are the volume representation of one-loop integrals, the orthoscheme dissection of ref. [36], the analytic continuation of Eq. (3.13), and an unguarded interchange in the all-orders limit.

axioms (7)
  • standard math Schlafli's differential formula dVol = -(1/N) * sum(Vol(codim-2 face) d(angle)) for hyperbolic simplices
    Invoked in Sec. 3.1, Eq. (3.1), as the basis of the recursion.
  • domain assumption Integer-dimensional one-loop integrals equal hyperbolic simplex volumes (Eq. 2.30), and those volumes are MPLs via Rudenko's formula
    Seed of recursion; taken from refs [13,19,20,36]. Not re-derived.
  • domain assumption Orthoscheme dissection BS(Q) commutes with the auxiliary-mass deformation and with dimensional regularisation (Eq. 2.37)
    From ref. [36]; needed to reduce arbitrary Q to nested Gram matrices.
  • ad hoc to paper Eq. (3.13), derived in the hyperbolic region, continues analytically along q=Q_{N+1,N+1} to q->0+ and the Feynman-parameter integrals remain well-defined
    Asserted after Eq. (3.13); no proof of the continuation is given.
  • standard math Distributional identities (3.16) and (3.28) for 1/(u+q) and log(u+delta)^n/(u+delta)
    Used to isolate singularities in the q->0 and delta->0 limits.
  • ad hoc to paper In Eq. (3.25), the primitive of the auxiliary-mass integrand equals 2 I_{N+1}^{N+2}(u) to leading order in q, and this replacement is safe inside the remaining u-integration before q->0
    Not uniformly justified; this is the weakest technical link in the all-orders recursion.
  • standard math For N>=4 and generic Euclidean kinematics, F and U satisfy bounds allowing dominated convergence in the Q_{N,N}->infinity limit
    Proved in Appendix A; root of the even-to-odd conversion Eq. (3.31).

pith-pipeline@v1.3.0-alltime-deepseek · 30105 in / 18532 out tokens · 151300 ms · 2026-08-01T21:00:35.358603+00:00 · methodology

0 comments
read the original abstract

We derive a novel recursive structure for dimensionally regularised scalar one-loop Feynman integrals based on Schl\"afli's differential formula for hyperbolic simplices. The recursion relates the Laurent coefficients in the dimensional regulator $\varepsilon$ of an $N$-point integral to lower-order coefficients of integrals with additional external legs. The construction is seeded by the $\varepsilon=0$ contributions, which admit a geometric interpretation as volumes of simplices in hyperbolic space and are known in terms of multiple polylogarithms (MPLs). Iterating the recursion therefore provides a constructive algorithm for computing arbitrary orders in the $\varepsilon$-expansion of scalar one-loop integrals with arbitrary masses and kinematics, while remaining entirely within the class of MPLs. In particular, this establishes that all coefficients in the Laurent expansion of dimensionally regularised scalar one-loop integrals can be expressed in terms of MPLs. As a first application beyond existing results, we obtain an explicit closed MPL expression for the $\mathcal{O}(\varepsilon)$ coefficient of the scalar hexagon with arbitrary masses and off-shell Euclidean kinematics.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

77 extracted references · 49 linked inside Pith

  1. [1]

    Heinrich,Collider Physics at the Precision Frontier,Phys

    G. Heinrich,Collider Physics at the Precision Frontier,Phys. Rept.922(2021) 1 [2009.00516]

    Pith/arXiv arXiv 2021

  2. [2]

    Caola, W

    F. Caola, W. Chen, C. Duhr, X. Liu, B. Mistlberger, F. Petriello et al.,The Path forward to N3LO, inSnowmass 2021, 2022 [2203.06730]. – 34 –

    Pith/arXiv arXiv 2021

  3. [3]

    Henn,Multiloop integrals in dimensional regularization made simple,Phys

    J.M. Henn,Multiloop integrals in dimensional regularization made simple,Phys. Rev. Lett. 110(2013) 251601 [1304.1806]

    Pith/arXiv arXiv 2013

  4. [4]

    C. Duhr,Mathematical aspects of scattering amplitudes, inTheoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders, pp. 419–476, 2015, DOI [1411.7538]

    Pith/arXiv arXiv 2015

  5. [5]

    Dixon, J.M

    L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington,Hexagon functions and the three-loop remainder function,JHEP12(2013) 049 [1308.2276]

    Pith/arXiv arXiv 2013

  6. [6]

    Chetyrkin and F.V

    K.G. Chetyrkin and F.V. Tkachov,Integration by parts: The algorithm to calculate β-functions in 4 loops,Nucl. Phys. B192(1981) 159

    1981

  7. [7]

    Laporta,High-precision calculation of multiloop Feynman integrals by difference equations, Int

    S. Laporta,High-precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A15(2000) 5087 [hep-ph/0102033]

    Pith/arXiv arXiv 2000

  8. [8]

    Goncharov, M

    A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich,Classical Polylogarithms for Amplitudes and Wilson Loops,Phys. Rev. Lett.105(2010) 151605 [1006.5703]

    Pith/arXiv arXiv 2010

  9. [9]

    Brown,Feynman amplitudes, coaction principle, and cosmic Galois group,Commun

    F. Brown,Feynman amplitudes, coaction principle, and cosmic Galois group,Commun. Num. Theor. Phys.11(2017) 453 [1512.06409]

    Pith/arXiv arXiv 2017

  10. [10]

    Schnetz,Graphical functions and single-valued multiple polylogarithms,Commun

    O. Schnetz,Graphical functions and single-valued multiple polylogarithms,Commun. Num. Theor. Phys.08(2014) 589 [1302.6445]

    Pith/arXiv arXiv 2014

  11. [11]

    ’t Hooft and M.J.G

    G. ’t Hooft and M.J.G. Veltman,Scalar One Loop Integrals,Nucl. Phys. B153(1979) 365

    1979

  12. [12]

    Passarino and M.J.G

    G. Passarino and M.J.G. Veltman,One Loop Corrections fore +e− Annihilation Intoµ +µ− in the Weinberg Model,Nucl. Phys. B160(1979) 151

    1979

  13. [13]

    Davydychev and R

    A.I. Davydychev and R. Delbourgo,A Geometrical angle on Feynman integrals,J. Math. Phys.39(1998) 4299 [hep-th/9709216]

    Pith/arXiv arXiv 1998

  14. [14]

    Schnetz,The geometry of one-loop amplitudes,1010.5334

    O. Schnetz,The geometry of one-loop amplitudes,1010.5334

    Pith/arXiv arXiv

  15. [15]

    Mason and D

    L. Mason and D. Skinner,Amplitudes at Weak Coupling as Polytopes in AdS 5,J. Phys. A44 (2011) 135401 [1004.3498]

    Pith/arXiv arXiv 2011

  16. [16]

    Nandan, M.F

    D. Nandan, M.F. Paulos, M. Spradlin and A. Volovich,Star Integrals, Convolutions and Simplices,JHEP05(2013) 105 [1301.2500]

    Pith/arXiv arXiv 2013

  17. [17]

    Goncharov,Multiple polylogarithms, cyclotomy and modular complexes,Math

    A.B. Goncharov,Multiple polylogarithms, cyclotomy and modular complexes,Math. Res. Lett.5(1998) 497 [1105.2076]

    Pith/arXiv arXiv 1998

  18. [18]

    Goncharov,Volumes of hyperbolic manifolds and mixed Tate motives,J

    A. Goncharov,Volumes of hyperbolic manifolds and mixed Tate motives,J. Am. Math. Soc. 12(1999) 569 [alg-geom/9601021]

    Pith/arXiv arXiv 1999

  19. [19]

    Rudenko,On the Goncharov depth conjecture and a formula for volumes of orthoschemes, J

    D. Rudenko,On the Goncharov depth conjecture and a formula for volumes of orthoschemes, J. Am. Math. Soc.36(2023) 1079 [2012.05599]

    Pith/arXiv arXiv 2023

  20. [20]

    L. Ren, M. Spradlin, C. Vergu and A. Volovich,One-loop integrals from volumes of orthoschemes,JHEP05(2024) 104 [2306.04630]

    Pith/arXiv arXiv 2024

  21. [21]

    ’t Hooft and M.J.G

    G. ’t Hooft and M.J.G. Veltman,Regularization and Renormalization of Gauge Fields,Nucl. Phys. B44(1972) 189

    1972

  22. [22]

    Cicuta and E

    G.M. Cicuta and E. Montaldi,Analytic renormalization via continuous space dimension, Lett. Nuovo Cim.4(1972) 329

    1972

  23. [23]

    Bollini and J.J

    C.G. Bollini and J.J. Giambiagi,Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter,Nuovo Cim. B12(1972) 20. – 35 –

    1972

  24. [24]

    Del Duca, C

    V. Del Duca, C. Duhr, E.W.N. Glover and V.A. Smirnov,The One-loop pentagon to higher orders in epsilon,JHEP01(2010) 042 [0905.0097]

    Pith/arXiv arXiv 2010

  25. [25]

    Dixon, J.M

    L.J. Dixon, J.M. Drummond and J.M. Henn,The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM,JHEP06(2011) 100 [1104.2787]

    Pith/arXiv arXiv 2011

  26. [26]

    Del Duca, C

    V. Del Duca, C. Duhr and V.A. Smirnov,The massless hexagon integral inD= 6 dimensions,Phys. Lett. B703(2011) 363 [1104.2781]

    Pith/arXiv arXiv 2011

  27. [27]

    Del Duca, C

    V. Del Duca, C. Duhr and V.A. Smirnov,The One-Loop One-Mass Hexagon Integral in D= 6Dimensions,JHEP07(2011) 064 [1105.1333]

    Pith/arXiv arXiv 2011

  28. [28]

    Del Duca, L.J

    V. Del Duca, L.J. Dixon, J.M. Drummond, C. Duhr, J.M. Henn and V.A. Smirnov,The one-loop six-dimensional hexagon integral with three massive corners,Phys. Rev. D84 (2011) 045017 [1105.2011]

    Pith/arXiv arXiv 2011

  29. [29]

    Chavez and C

    F. Chavez and C. Duhr,Three-mass triangle integrals and single-valued polylogarithms, JHEP11(2012) 114 [1209.2722]

    Pith/arXiv arXiv 2012

  30. [30]

    Kozlov and R.N

    M.G. Kozlov and R.N. Lee,One-loop pentagon integral inddimensions from differential equations inϵ-form,JHEP02(2016) 021 [1512.01165]

    Pith/arXiv arXiv 2016

  31. [31]

    Bourjaily, E

    J.L. Bourjaily, E. Gardi, A.J. McLeod and C. Vergu,All-massn-gon integrals inn dimensions,JHEP08(2020) 029 [1912.11067]

    Pith/arXiv arXiv 2020

  32. [32]

    Haug and F

    J. Haug and F. Wunder,The massless single off-shell scalar box integral — branch cut structure and all-order epsilon expansion,JHEP02(2023) 177 [2211.14110]

    Pith/arXiv arXiv 2023

  33. [33]

    Haug and F

    J. Haug and F. Wunder,The massless non-adjacent double off-shell scalar box integral — branch cut structure and all-order epsilon expansion,JHEP05(2023) 059 [2302.01956]

    Pith/arXiv arXiv 2023

  34. [34]

    Buccioni, P.A

    F. Buccioni, P.A. Kreer, X. Liu and L. Tancredi,One loop QCD corrections togg→t ¯tHat O(ϵ2),JHEP03(2024) 093 [2312.10015]

    Pith/arXiv arXiv 2024

  35. [35]

    Becchetti, M

    M. Becchetti, M. Delto, S. Ditsch, P.A. Kreer, M. Pozzoli and L. Tancredi,One-loop QCD corrections to¯ud→t ¯tWatO(ε 2),JHEP09(2025) 126 [2502.14952]

    arXiv 2025

  36. [36]

    Duhr and P

    C. Duhr and P. Mork,Analytic results for one-loop integrals in dimensional regularisation, 2512.14363

    arXiv

  37. [37]

    Brown,The Massless higher-loop two-point function,Commun

    F. Brown,The Massless higher-loop two-point function,Commun. Math. Phys.287(2009) 925 [0804.1660]

    Pith/arXiv arXiv 2009

  38. [38]

    Anastasiou, C

    C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger,Soft triple-real radiation for Higgs production at N3LO,JHEP07(2013) 003 [1302.4379]

    Pith/arXiv arXiv 2013

  39. [39]

    Ablinger, J

    J. Ablinger, J. Bl¨ umlein, C. Raab, C. Schneider and F. Wißbrock,Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms,Nucl. Phys. B885 (2014) 409 [1403.1137]

    Pith/arXiv arXiv 2014

  40. [40]

    Panzer,Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals,Comput

    E. Panzer,Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals,Comput. Phys. Commun.188(2015) 148 [1403.3385]

    Pith/arXiv arXiv 2015

  41. [41]

    Schnetz,HyperlogProcedures,Maple package, version 0.8(2025)

    O. Schnetz,HyperlogProcedures,Maple package, version 0.8(2025)

    2025

  42. [42]

    Kardos, S.-O

    A. Kardos, S.-O. Moch and O. Schnetz,HyperFORM: a FORM package for parametric integration with hyperlogarithms,Eur. Phys. J. C86(2026) 659 [2511.19992]

    arXiv 2026

  43. [43]

    Besier, D

    M. Besier, D. Van Straten and S. Weinzierl,Rationalizing roots: an algorithmic approach, Commun. Num. Theor. Phys.13(2019) 253 [1809.10983]. – 36 –

    Pith/arXiv arXiv 2019

  44. [44]

    Besier, P

    M. Besier, P. Wasser and S. Weinzierl,RationalizeRoots: Software Package for the Rationalization of Square Roots,Comput. Phys. Commun.253(2020) 107197 [1910.13251]

    Pith/arXiv arXiv 2020

  45. [45]

    Schl¨ afli,Theorie der vielfachen Kontinuit¨ at,Denkschr

    L. Schl¨ afli,Theorie der vielfachen Kontinuit¨ at,Denkschr. Schweiz. Naturforsch. Ges.38 (1901) 1

    1901

  46. [46]

    J.M. Henn, S. Moch and S.G. Naculich,Form factors and scattering amplitudes in N=4 SYM in dimensional and massive regularizations,JHEP12(2011) 024 [1109.5057]

    Pith/arXiv arXiv 2011

  47. [47]

    Loebbert, J

    F. Loebbert, J. Miczajka, D. M¨ uller and H. M¨ unkler,Massive Conformal Symmetry and Integrability for Feynman Integrals,Phys. Rev. Lett.125(2020) 091602 [2005.01735]

    Pith/arXiv arXiv 2020

  48. [48]

    Loebbert, J

    F. Loebbert, J. Miczajka, D. M¨ uller and H. M¨ unkler,Yangian Bootstrap for Massive Feynman Integrals,SciPost Phys.11(2021) 010 [2010.08552]

    Pith/arXiv arXiv 2021

  49. [49]

    Abreu, R

    S. Abreu, R. Britto, C. Duhr and E. Gardi,Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case,JHEP12(2017) 090 [1704.07931]

    Pith/arXiv arXiv 2017

  50. [50]

    Tkachov,A theorem on analytical calculability of 4-loop renormalization group functions,Phys

    F.V. Tkachov,A theorem on analytical calculability of 4-loop renormalization group functions,Phys. Lett. B100(1981) 65

    1981

  51. [51]

    Tarasov,A New approach to the momentum expansion of multiloop Feynman diagrams, Nucl

    O.V. Tarasov,A New approach to the momentum expansion of multiloop Feynman diagrams, Nucl. Phys. B480(1996) 397 [hep-ph/9606238]

    Pith/arXiv arXiv 1996

  52. [52]

    Anastasiou,Two loop integrals and QCD scattering, Ph.D

    C. Anastasiou,Two loop integrals and QCD scattering, Ph.D. thesis, Durham University, 2001

    2001

  53. [53]

    Lee,Space-time dimensionality D as complex variable: Calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D,Nucl

    R.N. Lee,Space-time dimensionality D as complex variable: Calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D,Nucl. Phys. B 830(2010) 474 [0911.0252]

    Pith/arXiv arXiv 2010

  54. [54]

    Melrose,Reduction of Feynman diagrams,Nuovo Cim.40(1965) 181

    D.B. Melrose,Reduction of Feynman diagrams,Nuovo Cim.40(1965) 181

    1965

  55. [55]

    Huang, D.-S

    R.-J. Huang, D.-S. Jian, Y.-Q. Ma, D.-M. Mu and W.-H. Wu,Efficient computation of one-loop Feynman integrals and fixed-branch integrals to high orders inε,Phys. Rev. D111 (2025) 094028 [2412.21054]

    Pith/arXiv arXiv 2025

  56. [56]

    Lobatschewsky,G´ eom´ etrie imaginaire,J

    N. Lobatschewsky,G´ eom´ etrie imaginaire,J. Reine Angew. Math.17(1837) 295

  57. [57]

    Coxeter,The Functions of Schl¨ afli and Lobatschefsky,Quart

    H.S.M. Coxeter,The Functions of Schl¨ afli and Lobatschefsky,Quart. J. Math.6(1935) 13

    1935

  58. [58]

    B¨ ohm,Zu Coxeters Integrationsmethode in gekr¨ ummten R¨ aumen,Math

    J. B¨ ohm,Zu Coxeters Integrationsmethode in gekr¨ ummten R¨ aumen,Math. Nachr.27(1964) 179

    1964

  59. [59]

    Wagner,A Volume Formula for Asymptotic Hyperbolic Tetrahedra with an Application to Quantum Field Theory,Indag

    P. Wagner,A Volume Formula for Asymptotic Hyperbolic Tetrahedra with an Application to Quantum Field Theory,Indag. Math.7(1996) 527

    1996

  60. [60]

    Kellerhals,Volumes in Hyperbolic 5-Space,Math

    R. Kellerhals,Volumes in Hyperbolic 5-Space,Math. Proc. Cambridge Philos. Soc.118 (1995) 447

    1995

  61. [61]

    Cho and H

    Y. Cho and H. Kim,On the Volume Formula for Hyperbolic Tetrahedra,Discrete Comput. Geom.22(1999) 347

    1999

  62. [62]

    Murakami and M

    J. Murakami and M. Yano,On the Volume of a Hyperbolic and Spherical Tetrahedron, Commun. Anal. Geom.13(2005) 379

    2005

  63. [63]

    Kneser,Der Simplexinhalt in der nichteuklidischen Geometrie,Deutsche Mathematik1 (1936) 337

    H. Kneser,Der Simplexinhalt in der nichteuklidischen Geometrie,Deutsche Mathematik1 (1936) 337

    1936

  64. [64]

    Kellerhals,On the Volume of Hyperbolic Polyhedra,Math

    R. Kellerhals,On the Volume of Hyperbolic Polyhedra,Math. Ann.285(1989) 541. – 37 –

    1989

  65. [65]

    Kellerhals,On Schl¨ afli’s Reduction Formula,Math

    R. Kellerhals,On Schl¨ afli’s Reduction Formula,Math. Z.206(1991) 193

    1991

  66. [66]

    Zhao,Analytic continuation of multiple polylogarithms,Anal

    J. Zhao,Analytic continuation of multiple polylogarithms,Anal. Math.33(2007) 301 [math/0302054]

    Pith/arXiv arXiv 2007

  67. [67]

    Danciger,A Geometric Transition from Hyperbolic to Anti-de Sitter Geometry,Geom

    J. Danciger,A Geometric Transition from Hyperbolic to Anti-de Sitter Geometry,Geom. Topol.17(2013) 3077 [1302.5687]

    Pith/arXiv arXiv 2013

  68. [68]

    Hodgson,Degeneration and Regeneration of Geometric Structures on Three-Manifolds, Ph.D

    C.D. Hodgson,Degeneration and Regeneration of Geometric Structures on Three-Manifolds, Ph.D. thesis, Princeton University, 1986

    1986

  69. [69]

    Porti,Regenerating Hyperbolic and Spherical Cone Structures from Euclidean Ones, Topology37(1998) 365

    J. Porti,Regenerating Hyperbolic and Spherical Cone Structures from Euclidean Ones, Topology37(1998) 365

    1998

  70. [70]

    Danciger,Geometric Transitions: From Hyperbolic to AdS Geometry, Ph.D

    J. Danciger,Geometric Transitions: From Hyperbolic to AdS Geometry, Ph.D. thesis, Stanford University, 2011

    2011

  71. [71]

    Riolo and A

    S. Riolo and A. Seppi,Geometric Transition from Hyperbolic to Anti-de Sitter Structures in Dimension Four,Ann. Sc. Norm. Super. Pisa Cl. Sci.23(2022) 115

    2022

  72. [72]

    J.M. Henn, A. Matijaˇ si´ c and J. Miczajka,One-loop hexagon integral to higher orders in the dimensional regulator,JHEP01(2023) 096 [2210.13505]

    Pith/arXiv arXiv 2023

  73. [73]

    Duhr and F

    C. Duhr and F. Dulat,PolyLogTools — polylogs for the masses,JHEP08(2019) 135 [1904.07279]

    Pith/arXiv arXiv 2019

  74. [74]

    Borowka, G

    S. Borowka, G. Heinrich, S. Jahn, S.P. Jones, M. Kerner and J. Schlenk,A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec,Comput. Phys. Commun.240(2019) 120 [1811.11720]

    Pith/arXiv arXiv 2019

  75. [75]

    Heinrich, S

    G. Heinrich, S. Jahn, S.P. Jones, M. Kerner, F. Langer, V. Magerya et al.,Expansion by regions with pySecDec,Comput. Phys. Commun.273(2022) 108267 [2108.10807]

    Pith/arXiv arXiv 2022

  76. [76]

    Heinrich, S.P

    G. Heinrich, S.P. Jones, M. Kerner, V. Magerya, A. Olsson and J. Schlenk,Numerical scattering amplitudes with pySecDec,Comput. Phys. Commun.295(2024) 108956 [2305.19768]

    Pith/arXiv arXiv 2024

  77. [77]

    Arkani-Hamed and E.Y

    N. Arkani-Hamed and E.Y. Yuan,One-Loop Integrals from Spherical Projections of Planes and Quadrics,1712.09991. – 38 –

    Pith/arXiv arXiv