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Off-shell form factor: factorization is violated

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The off-shell Sudakov form factor in $\mathcal{N}=4$ sYM cannot be written as a product of hard, collinear, and ultrasoft factors; an extra $R(\varepsilon,t)$ factor is required.

desk verdict A careful three-loop computation showing the simple product ansatz fails, but the title overstates the claim because the authors concede a convolution could restore factorization. read the letter →

arxiv 2505.22595 v1 pith:S3DP5WA2 submitted 2025-05-28 hep-th hep-ph

A.V. Belitsky , V.A. Smirnov This is my paper
classification hep-thhep-ph
keywords Sudakovformfactorfactorizationviolationhard-collinear-ultrasoftN=4supersymmetricYang-MillsCoulombbranchmethodofregionsoff-shellinfraredpowercounting
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that the off-shell Sudakov form factor in $\mathcal{N}=4$ super-Yang-Mills on the Coulomb branch cannot be split into a product of hard, collinear, and ultrasoft factors. Working at three loops in the near mass-shell limit, the authors separate Feynman integrals into momentum regions defined by infrared power counting. They find that the hard region factorizes cleanly, but the collinear and ultrasoft regions stay entangled: a mismatch first appears at two loops at subleading order in the dimensional regulator and cannot be repaired at three loops by redefining the jet and soft functions. The conclusion is that a strictly multiplicative factorization $F_2 = h\,J\,S$ fails, and an additional factor $R(\varepsilon,t)$ is required.

What carries the argument

The load-bearing tool is the Method of Regions, which assigns to each loop-momentum configuration one of three scaling patterns—hard, $P_i$-collinear, or ultrasoft—and decomposes every Feynman integral into region contributions with powers $t^{-j\varepsilon}$. Within this decomposition, the obstruction is a single propagator in the two-loop double-ladder graph, $D(k_1+k_2-P_1) = [(k_2-p_1)^2 + 2k_1\cdot(k_2-p_1)]^{-1}$, whose second term entangles the collinear momentum $k_2$ with the ultrasoft momentum $k_1$ and cannot be dropped on power-counting grounds. The factorization-checking identities are the mismatch factors $z_1(\varepsilon)=\Gamma(1+2\varepsilon)/\Gamma^2(1+\varepsilon)$ and $z'_{21}(\varepsilon)\neq z'_{22}(\varepsilon)$, which quantify how far the mixed regions deviate from products of lower-loop factors.

What would settle it

Compute the four-loop mixed region $T^{c\text{-}c\text{-}us}$ to higher order in $\varepsilon$ and check whether the analogues of $z'_{21}$ and $z'_{22}$ become equal; if they do, the irreparability claim would collapse. Alternatively, explicitly constructing a convolution ansatz that reproduces Eqs. (6.25)-(6.28) would show the 'violation' is an artifact of the multiplicative restriction.

Watch

Extended reading notes

Core claim

The central claim is that the near-mass-shell Sudakov form factor in $\mathcal{N}=4$ sYM on the Coulomb branch violates the standard hard-collinear-ultrasoft factorization. Explicit evaluation of all three-loop momentum regions shows that mixed regions involving both collinear and ultrasoft modes cannot be expressed as products of the one- and two-loop collinear and ultrasoft integrals. The two-loop mixed region satisfies $\sum_n \alpha_{2n}T^{c\text{-}us}_{2n} = 4z_1(\varepsilon)T^c_{11}T^{us}_{11}$ with $z_1 \neq 1$, and the analogous three-loop relations require two distinct correction factors $z'_{21}(\varepsilon) \neq z'_{22}(\varepsilon)$. Because a single twist of the jet and soft functions produces only a common factor, the two factors cannot both be absorbed, so the multiplicative ansatz fails and a residual $R(\varepsilon,t)$ is necessary.

Load-bearing premise

The conclusion depends on requiring factorization to have the strictly multiplicative form $F_2 = h\,J\,S$; the authors concede that replacing the product with a convolution could reproduce the offending terms, so if convolutions or additional operator factors are allowed, the central claim of violation would not follow.

Editorial extensions

If this is right

  • The double-logarithmic term proportional to $\log^2 t$ still factorizes; the violation sits in subleading poles of order $\varepsilon^{-2}$ and lower, so leading-log resummations remain valid.
  • No momentum-independent redefinition of the hard, jet, and soft functions can restore multiplicative factorization at three loops.
  • The paper suggests that on-shell scattering amplitudes on this Coulomb branch will also resist simple soft-collinear factorization.
  • Any complete factorization statement for this observable must include the additional factor $R(\varepsilon,t)$, whose operator definition is left as an open problem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One could test whether the extra factor $R(\varepsilon,t)$ is already visible in the form factor's finite part at three loops, and whether its $\zeta_4$ and $\zeta_6$ content matches a known cross-talk between jet and soft anomalous dimensions.
  • If a convolution or multi-local operator ansatz is admitted, the paper's no-go result becomes a constructive constraint: the convolution kernel must reproduce $z'_{21}\neq z'_{22}$ and would define the joint ultrasoft-collinear function the authors leave for future work.
  • At four loops, the mixed $c\text{-}c\text{-}us$ regions could be computed numerically to see whether the mismatch factors continue to grow in a systematic pattern, such as simple products of gamma functions, or whether new structures appear.
  • The same region-based analysis could be applied to the massive quark Sudakov form factor in QCD, where the same 'offending' propagator structure appears, to see whether factorization violation is generic or special to $\mathcal{N}=4$ sYM.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies the three-loop Sudakov form factor in N=4 super-Yang-Mills on the Coulomb branch, in the near mass-shell limit t = m^2/Q^2 -> 0. Using the Method of Regions, the contributing Feynman integrals are decomposed into hard, collinear, and ultrasoft regions and evaluated by a combination of Mellin-Barnes techniques, tropical-geometric subtraction, and differential equations. The authors verify that hard-region contributions factorize from the infrared regions through Eqs. (6.9)-(6.12). They find that the multiplicative hard-collinear-ultrasoft ansatz F2 = h(ε) J(ε,√t) S(ε,t) of Eq. (6.5) fails: a two-loop correction factor z1 is needed in Eq. (6.23), and at three loops the required factors z'_21 and z'_22 in Eqs. (6.27)-(6.28) are unequal, so no simultaneous redefinition of J and S can absorb the mismatch. The paper introduces a factorization-restoring factor R(ε,t) in Eq. (6.29) and concludes that ultrasoft and collinear modes remain intertwined.

Significance. If the central claim is read as a statement about the minimal multiplicative product ansatz, the paper is a significant and technically impressive result. It provides the first explicit region-by-region three-loop demonstration that the ultrasoft and collinear sectors of the off-shell Sudakov form factor do not decouple into independent factors, and it does so with parameter-free, first-principles computations: exact gamma-function results for many regions, explicit Laurent expansions in Appendix A, and a nontrivial consistency check against the known all-order exponentiation (2.6). The hard-region factorization identities in Section 6.1 are clean and convincing. However, the broader claim that 'factorization is violated' is not established by the computation, because the paper explicitly leaves open the possibility of a convolution-type or operator-valued generalized factorization and does not provide an operator definition of the joint ultrasoft-collinear function. The significance thus lies in the precise obstruction to the simple product form, not in a proof of the impossibility of all generalized factorizations.

major comments (2)
  1. [Section 6.3 and Section 7] The title, abstract, and conclusion claim that 'factorization is violated', but what is actually proved is that the strictly multiplicative ansatz F2 = h(ε) J(ε,√t) S(ε,t) of Eq. (6.5) fails. The authors explicitly concede in Section 6.3 that 'replacing the product with a convolution could reproduce it', and the Conclusion leaves open the operator definition of joint ultrasoft-collinear functions. Since convolution-type factorizations and factorizations with additional operators are standard in TMD and threshold resummation, the broad no-go claim is not established. To make the central claim stand, the paper must either prove that no such generalized factorization can reproduce Eqs. (6.23), (6.25)-(6.26), and (6.29), or it must revise the title, abstract, and conclusions to state the precise result: the simple product ansatz fails, while the possibility of a generalized factorization remains open.
  2. [Section 6.3, Eqs. (6.25)-(6.28)] The three-loop obstruction rests on the inequality z'_21(ε) ≠ z'_22(ε), but the printed region results in Appendix A.3 are given only to O(ε0), while z'_21 and z'_22 are quoted through O(ε6). The inequality appears at order ε4, so the printed data are sufficient for the leading assertion, but the higher-order coefficients and the precise order to which Eq. (6.29) is valid are not independently verifiable from the paper alone. Please state explicitly which ancillary file entries determine each z factor, and spell out the minimum order of each region integral required to prove the inequality.
minor comments (4)
  1. [Figure 2] The caption of Figure 2 says 'divergent rays of a region contributing to diagram T33', but the text introducing the figure says it shows a region contributing to T32; please correct the mismatch.
  2. [Section 6.3, Eqs. (6.29)-(6.32)] The notation z21 and z22 is introduced only after the sentence 'Unfortunately, z21 and z22 are different', which forces the reader to infer that these are the factors appearing in R and not the z'_21 and z'_22 of Eqs. (6.27)-(6.28). Please define all z factors explicitly before the comparison.
  3. [Section 6.3, final paragraph] The statement that the breaking is 'pushed to the subleading order O(ε^{-2})' is ambiguous: at ℓ loops the leading singularity is O(ε^{-2ℓ}), so the violation appears at absolute order ε^{-2}, not at a fixed relative order. Please rephrase to state the absolute order explicitly.
  4. [Section 7] The closing speculation that 'all on-shell scattering amplitudes on this Coulomb branch will follow suit' is an extrapolation from a single observable, and it is especially unsupported given that the generalized factorization question is left open in the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed factorization violation is established by explicit first-principles region integrals against a clearly stated multiplicative ansatz, not by fitting or by importing the conclusion from prior work.

full rationale

The paper's central result is a negative statement about the multiplicative hard-collinear-ultrasoft ansatz F2 = h(ε) J(ε,√t) S(ε,t) in Eq. (6.5). The obstruction is exhibited by direct evaluation of mixed collinear-ultrasoft region integrals: the two-loop identity (6.13) shows α2n T^{c-us}_{2n} ≠ 4 T^c_11 T^us_11, and the three-loop relations (6.25)–(6.28) require two unequal correction factors z'_{21} ≠ z'_{22}, so no simultaneous twist of J and S can repair the product form. These region integrals are computed in Sections 3–5 using Mellin-Barnes methods, tropical geometry, and differential equations, with results quoted in Appendix A.3; no parameter is fitted to the quantity being predicted. The known exponentiation (2.6), cited from the authors' earlier work, is used only as a consistency check of the total sum, not as an input that forces the factorization conclusion. The two-loop twisting device is parameter-free and its prior citation is not load-bearing for the three-loop impossibility. The paper itself flags the true limitation: 'Though replacing the product with a convolution could reproduce it, we will not dwell on it...' (Section 6.3), and the Conclusions state that the operator definition of joint ultrasoft-collinear functions was not addressed. Thus the broad title claim is stronger than what is proven, but this is a scope/overstatement issue, not circularity. The derivation chain does not reduce to its own inputs, and no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the completeness of the Method-of-Regions expansion, the correctness of the previously established all-order exponentiation of the form factor, and standard tools such as Landau equations and the Cheng-Wu theorem. No parameters are fitted and no new entities are introduced.

assumptions (3)
  • domain assumption Method of Regions yields the complete and correct t to 0 asymptotic expansion of the relevant Feynman integrals.
    Stated as experimental mathematics in Section 2; used to enumerate all hard, collinear, and ultrasoft contributions. If a region were missing, the factorization conclusions could change.
  • domain assumption The exact all-order form log F2 = minus 1/2 Gamma_oct(g) log^2 t minus D(g) (Eq. 2.6) from Refs. [43,44] is correct.
    Used as a benchmark in Section 6 to verify that the sum of region contributions reproduces the known result.
  • standard math Landau equations and the Cheng-Wu theorem justify the pinch-surface classification and allow simplification of parametric integrals.
    Invoked in Sections 2 and 3 to identify IR regions and apply Feynman parameter integration.

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Cite this review

Pith. "Pith review of Off-shell form factor: factorization is violated." pith.science (2026). https://pith.science/paper/S3DP5WA2

@misc{pith2026250522595,
  author       = {Pith},
  title        = {Pith review of: Off-shell form factor: factorization is violated},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S3DP5WA2}},
  note         = {Machine review of arXiv:2505.22595}
}
read the original abstract

We study the Sudakov form factor on the Coulomb branch of N=4 sYM, which endows only external states with masses, and implies that the former is off-shell in the traditional sense. Our consideration is performed at three-loop order in the near mass-shell limit. We use a combination of tools to perform required calculations centered around the Method of Regions as the main go-to formalism for the asymptotic expansion of emerging parametric Feynman integrals. Explicit separation of quantum loops in terms of hard, collinear, and ultrasoft modes allows us to explore the factorization properties of this infrared-sensitive quantity. While the hard region is cleanly separated from the rest, the ultrasoft-collinear modes remain intertwined. We exhibit effects of factorization violation explicitly in the momentum space making use of the infrared power counting.

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Reference graph

Works this paper leans on

80 extracted references · 25 canonical work pages · cited by 2 Pith papers

  1. [34]

    A. V. Belitsky, L. V. Bork and V. A. Smirnov, Pinching Sudakov , 2409.05945. – 28 –

    arXiv

  2. [1]

    V. V. Sudakov, Vertex parts at very high-energies in quantum electrodynamics , Sov. Phys. JETP 3 (1956) 65–71

    work page 1956

  3. [2]

    L. D. Landau, On analytic properties of vertex parts in quantum field theory , Nucl. Phys. 13 (1959) 181–192

    1959

  4. [3]

    Coleman and R

    S. Coleman and R. E. Norton, Singularities in the physical region , Nuovo Cim. 38 (1965) 438–442

    work page 1965

  5. [4]

    Sen, Asymptotic Behavior of the Sudakov Form-Factor in QCD , Phys

    A. Sen, Asymptotic Behavior of the Sudakov Form-Factor in QCD , Phys. Rev. D 24 (1981) 3281

    work page 1981

  6. [5]

    Sen, Asymptotic Behavior of the Wide Angle On-Shell Quark Scattering Amplitudes in Nonabelian Gauge Theories , Phys

    A. Sen, Asymptotic Behavior of the Wide Angle On-Shell Quark Scattering Amplitudes in Nonabelian Gauge Theories , Phys. Rev. D 28 (1983) 860

    work page 1983

  7. [6]

    J. C. Collins, D. E. Soper and G. F. Sterman, Factorization of Hard Processes in QCD , Adv. Ser. Direct. High Energy Phys. 5 (1989) 1–91, [ hep-ph/0409313]

    arXiv 1989

  8. [7]

    G. F. Sterman and M. E. Tejeda-Yeomans, Multiloop amplitudes and resummation , Phys. Lett. B 552 (2003) 48–56, [ hep-ph/0210130]

    arXiv 2003

Show all 80 references
  1. [8]

    S. M. Aybat, L. J. Dixon and G. F. Sterman, The Two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole , Phys. Rev. D 74 (2006) 074004, [hep-ph/0607309]

    2006 arXiv

  2. [9]

    L. J. Dixon, L. Magnea and G. F. Sterman, Universal structure of subleading infrared poles in gauge theory amplitudes , JHEP 08 (2008) 022, [ 0805.3515]

    2008 arXiv

  3. [10]

    Becher and M

    T. Becher and M. Neubert, On the Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP 06 (2009) 081, [ 0903.1126]

    2009 arXiv

  4. [11]

    Agarwal, L

    N. Agarwal, L. Magnea, C. Signorile-Signorile and A. Tripathi, The infrared structure of perturbative gauge theories, Phys. Rept. 994 (2023) 1–120, [ 2112.07099]

    2023 arXiv

  5. [12]

    C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, An Effective field theory for collinear and soft gluons: Heavy to light decays , Phys. Rev. D 63 (2001) 114020, [ hep-ph/0011336]

    2001 arXiv

  6. [13]

    Beneke, A

    M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power , Nucl. Phys. B 643 (2002) 431–476, [hep-ph/0206152]

    2002 arXiv

  7. [14]

    Becher, A

    T. Becher, A. Broggio and A. Ferroglia, Introduction to Soft-Collinear Effective Theory , vol. 896. Springer, 2015, 10.1007/978-3-319-14848-9. – 27 –

    2015 doi

  8. [15]

    Feige and M

    I. Feige and M. D. Schwartz, An on-shell approach to factorization , Phys. Rev. D 88 (2013) 065021, [1306.6341]

    2013 arXiv

  9. [16]

    Feige and M

    I. Feige and M. D. Schwartz, Hard-Soft-Collinear Factorization to All Orders , Phys. Rev. D 90 (2014) 105020, [ 1403.6472]

    2014 arXiv

  10. [17]

    J. C. Collins and G. F. Sterman, Soft Partons in QCD , Nucl. Phys. B 185 (1981) 172–188

    1981

  11. [18]

    G. T. Bodwin, S. J. Brodsky and G. P. Lepage, Initial State Interactions and the Drell-Yan Process, Phys. Rev. Lett. 47 (1981) 1799

    1981

  12. [19]

    J. C. Collins, D. E. Soper and G. F. Sterman, All Order Factorization for Drell-Yan Cross-sections, Phys. Lett. B 134 (1984) 263

    1984

  13. [20]

    I. Z. Rothstein and I. W. Stewart, An Effective Field Theory for Forward Scattering and Factorization Violation, JHEP 08 (2016) 025, [ 1601.04695]

    2016 arXiv

  14. [21]

    M. D. Schwartz, K. Yan and H. X. Zhu, Collinear factorization violation and effective field theory, Phys. Rev. D 96 (2017) 056005, [ 1703.08572]

    2017 arXiv

  15. [22]

    J. C. Collins, D. E. Soper and G. F. Sterman, Soft Gluons and Factorization , Nucl. Phys. B 308 (1988) 833–856

    1988

  16. [23]

    S. M. Aybat and G. F. Sterman, Soft-Gluon Cancellation, Phases and Factorization with Initial-State Partons , Phys. Lett. B 671 (2009) 46–50, [ 0811.0246]

    2009 arXiv

  17. [24]

    M. D. Schwartz, K. Yan and H. X. Zhu, Factorization Violation and Scale Invariance , Phys. Rev. D 97 (2018) 096017, [ 1801.01138]

    2018 arXiv

  18. [25]

    J. R. Forshaw, A. Kyrieleis and M. H. Seymour, Super-leading logarithms in non-global observables in QCD , JHEP 08 (2006) 059, [ hep-ph/0604094]

    2006 arXiv

  19. [26]

    Becher, M

    T. Becher, M. Neubert and D. Y. Shao, Resummation of Super-Leading Logarithms, Phys. Rev. Lett. 127 (2021) 212002, [ 2107.01212]

    2021 arXiv

  20. [27]

    B¨ oer, P

    P. B¨ oer, P. Hager, M. Neubert, M. Stillger and X. Xu, Renormalization-group improved resummation of super-leading logarithms , JHEP 08 (2024) 035, [ 2405.05305]

    2024 arXiv

  21. [28]

    Catani, D

    S. Catani, D. de Florian and G. Rodrigo, Space-like (versus time-like) collinear limits in QCD: Is factorization violated? , JHEP 07 (2012) 026, [ 1112.4405]

    2012 arXiv

  22. [29]

    J. R. Forshaw, M. H. Seymour and A. Siodmok, On the Breaking of Collinear Factorization in QCD , JHEP 11 (2012) 066, [ 1206.6363]

    2012 arXiv

  23. [30]

    Beneke and V

    M. Beneke and V. A. Smirnov, Asymptotic expansion of Feynman integrals near threshold , Nucl. Phys. B 522 (1998) 321–344, [ hep-ph/9711391]

    1998 arXiv

  24. [31]

    V. A. Smirnov, Applied asymptotic expansions in momenta and masses , Springer Tracts Mod. Phys. 177 (2002) 1–262

    2002

  25. [32]

    V. A. Smirnov, Analytic tools for Feynman integrals , Springer Tracts Mod. Phys. 250 (2012) 1–296

    2012

  26. [33]

    V. A. Smirnov, Expansion by Regions: An Overview . 2021. 2406.11475

    2021 arXiv

  27. [35]

    Dennen, Y.-t

    T. Dennen, Y.-t. Huang and W. Siegel, Supertwistor space for 6D maximal super Yang-Mills, JHEP 04 (2010) 127, [ 0910.2688]

    2010 arXiv

  28. [36]

    Z. Bern, J. J. Carrasco, T. Dennen, Y.-t. Huang and H. Ita, Generalized Unitarity and Six-Dimensional Helicity , Phys. Rev. D 83 (2011) 085022, [ 1010.0494]

    2011 arXiv

  29. [37]

    A. V. Belitsky, Collinear anatomy , 2412.11886

    arXiv

  30. [38]

    Caron-Huot and F

    S. Caron-Huot and F. Coronado, Ten dimensional symmetry of N = 4 SYM correlators , JHEP 03 (2022) 151, [ 2106.03892]

    2022 arXiv

  31. [39]

    Jackiw, Dynamics at high momentum and the vertex function of spinor electrodynamics , Annals Phys

    R. Jackiw, Dynamics at high momentum and the vertex function of spinor electrodynamics , Annals Phys. 48 (1968) 292–321

    1968

  32. [40]

    W. L. van Neerven, Infrared Behavior of On-shell Form-factors in a N = 4 Supersymmetric Yang-Mills Field Theory , Z. Phys. C 30 (1986) 595

    1986

  33. [41]

    Gehrmann, J

    T. Gehrmann, J. M. Henn and T. Huber, The three-loop form factor in N=4 super Yang-Mills, JHEP 03 (2012) 101, [ 1112.4524]

    2012 arXiv

  34. [42]

    L. F. Alday, J. M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of N=4 super Yang-Mills , JHEP 01 (2010) 077, [ 0908.0684]

    2010 arXiv

  35. [43]

    A. V. Belitsky, L. V. Bork, A. F. Pikelner and V. A. Smirnov, Exact Off Shell Sudakov Form Factor in N=4 Supersymmetric Yang-Mills Theory , Phys. Rev. Lett. 130 (2023) 091605, [2209.09263]

    2023 arXiv

  36. [44]

    A. V. Belitsky, L. V. Bork and V. A. Smirnov, Off-shell form factor in N =4 sYM at three loops, JHEP 11 (2023) 111, [ 2306.16859]

    2023 arXiv

  37. [45]

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

    2013 arXiv

  38. [46]

    Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes , Nucl. Phys. B 435 (1995) 59–101, [ hep-ph/9409265]

    1995 arXiv

  39. [47]

    Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits , Nucl. Phys. B 425 (1994) 217–260, [hep-ph/9403226]

    1994 arXiv

  40. [48]

    A. V. Belitsky and G. P. Korchemsky, Exact null octagon , JHEP 05 (2020) 070, [1907.13131]

    2020 arXiv

  41. [49]

    T. Y. Semenova, A. V. Smirnov and V. A. Smirnov, On the status of expansion by regions , Eur. Phys. J. C 79 (2019) 136, [ 1809.04325]

    2019 arXiv

  42. [50]

    V. A. Smirnov, Problems of the strategy of regions , Phys. Lett. B 465 (1999) 226–234, [hep-ph/9907471]

    1999 arXiv

  43. [51]

    Pak and A

    A. Pak and A. Smirnov, Geometric approach to asymptotic expansion of Feynman integrals , Eur. Phys. J. C 71 (2011) 1626, [ 1011.4863]

    2011 arXiv

  44. [52]

    Salvatori, The Tropical Geometry of Subtraction Schemes , 2406.14606

    G. Salvatori, The Tropical Geometry of Subtraction Schemes , 2406.14606. – 29 –

    arXiv

  45. [53]

    Jantzen, A

    B. Jantzen, A. V. Smirnov and V. A. Smirnov, Expansion by regions: revealing potential and Glauber regions automatically , Eur. Phys. J. C 72 (2012) 2139, [ 1206.0546]

    2012 arXiv

  46. [54]

    A. V. Belitsky and V. A. Smirnov, An off-shell Wilson loop , JHEP 04 (2023) 071, [2110.13206]

    2023 arXiv

  47. [55]

    V. A. Smirnov and F. Wunder, Expansion by regions meets angular integrals , JHEP 08 (2024) 138, [ 2405.13120]

    2024 arXiv

  48. [56]

    A. V. Smirnov, FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions, Comput. Phys. Commun. 185 (2014) 2090–2100, [ 1312.3186]

    2014 arXiv

  49. [57]

    A. V. Smirnov, N. D. Shapurov and L. I. Vysotsky, FIESTA5: Numerical high-performance Feynman integral evaluation, Comput. Phys. Commun. 277 (2022) 108386, [ 2110.11660]

    2022 arXiv

  50. [58]

    Cheng and T

    H. Cheng and T. T. Wu, EXPANDING PROTONS: SCATTERING AT HIGH-ENERGIES. 1987

    1987

  51. [59]

    A. V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158–164

    1991

  52. [60]

    Gehrmann and E

    T. Gehrmann and E. Remiddi, Differential equations for two-loop four-point functions , Nucl. Phys. B 580 (2000) 485–518, [ hep-ph/9912329]

    2000 arXiv

  53. [61]

    A. V. Belitsky, A. V. Smirnov and V. A. Smirnov, MB tools reloaded, Nucl. Phys. B 986 (2023) 116067, [ 2211.00009]

    2023 arXiv

  54. [62]

    Czakon, Automatized analytic continuation of Mellin-Barnes integrals , Comput

    M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals , Comput. Phys. Commun. 175 (2006) 559–571, [ hep-ph/0511200]

    2006 arXiv

  55. [63]

    A. V. Smirnov and V. A. Smirnov, On the Resolution of Singularities of Multiple Mellin-Barnes Integrals, Eur. Phys. J. C 62 (2009) 445–449, [ 0901.0386]

    2009 arXiv

  56. [64]

    J. B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs , Phys. Lett. B 469 (1999) 225–234, [ hep-ph/9909506]

    1999 arXiv

  57. [65]

    V. A. Smirnov, Analytical result for dimensionally regularized massless on shell double box , Phys. Lett. B 460 (1999) 397–404, [ hep-ph/9905323]

    1999 arXiv

  58. [66]

    H. R. P. Ferguson, D. Bailey and S. Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comput. 68 (1999) 351–369

    1999

  59. [67]

    Arkani-Hamed, A

    N. Arkani-Hamed, A. Hillman and S. Mizera, Feynman polytopes and the tropical geometry of UV and IR divergences , Phys. Rev. D 105 (2022) 125013, [ 2202.12296]

    2022 arXiv

  60. [68]

    Hillman, On the Infrared and Ultraviolet Behavior of Scattering Amplitudes and Wavefunctions

    A. Hillman, On the Infrared and Ultraviolet Behavior of Scattering Amplitudes and Wavefunctions. PhD thesis, Princeton U., 2023

    2023

  61. [69]

    Berkesch, J

    C. Berkesch, J. Forsg ˚ ard and M. Passare,Euler-mellin integrals and a-hypergeometric functions, Michigan Mathematical Journal 63 (Mar., 2014) 101–123

    2014

  62. [70]

    Nilsson and M

    L. Nilsson and M. Passare, Mellin transforms of multivariate rational functions , arXiv e-prints (Oct., 2010) arXiv:1010.5060, [ 1010.5060]. – 30 –

    2010 arXiv

  63. [71]

    ’t Hooft and M

    G. ’t Hooft and M. J. G. Veltman, Regularization and Renormalization of Gauge Fields , Nucl. Phys. B 44 (1972) 189–213

    1972

  64. [72]

    Gawrilow and M

    E. Gawrilow and M. Joswig, polymake: a Framework for Analyzing Convex Polytopes . 2000. 10.1007/978-3-0348-8438-9-2

    2000 doi

  65. [73]

    Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales , JHEP 03 (2014) 071, [ 1401.4361]

    E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales , JHEP 03 (2014) 071, [ 1401.4361]

    2014 arXiv

  66. [74]

    Gehrmann, G

    T. Gehrmann, G. Heinrich, T. Huber and C. Studerus, Master integrals for massless three-loop form-factors: One-loop and two-loop insertions , Phys. Lett. B 640 (2006) 252–259, [hep-ph/0607185]

    2006 arXiv

  67. [75]

    Heinrich, T

    G. Heinrich, T. Huber and D. Maitre, Master integrals for fermionic contributions to massless three-loop form-factors, Phys. Lett. B 662 (2008) 344–352, [ 0711.3590]

    2008 arXiv

  68. [76]

    P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnov and M. Steinhauser, Quark and gluon form factors to three loops , Phys. Rev. Lett. 102 (2009) 212002, [ 0902.3519]

    2009 arXiv

  69. [77]

    Heinrich, T

    G. Heinrich, T. Huber, D. A. Kosower and V. A. Smirnov, Nine-Propagator Master Integrals for Massless Three-Loop Form Factors, Phys. Lett. B 678 (2009) 359–366, [0902.3512]

    2009 arXiv

  70. [78]

    R. N. Lee and V. A. Smirnov, Analytic Epsilon Expansions of Master Integrals Corresponding to Massless Three-Loop Form Factors and Three-Loop g-2 up to Four-Loop Transcendentality Weight, JHEP 02 (2011) 102, [ 1010.1334]

    2011 arXiv

  71. [79]

    Pikelner, Three-loop vertex integrals at symmetric point , JHEP 06 (2021) 083, [2104.06958]

    A. Pikelner, Three-loop vertex integrals at symmetric point , JHEP 06 (2021) 083, [2104.06958]

    2021 arXiv

  72. [80]

    Meyer, Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA , Comput

    C. Meyer, Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA , Comput. Phys. Commun. 222 (2018) 295–312, [ 1705.06252]. – 31 –

    2018 arXiv

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