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Integral of the double-emission eikonal function for a massive and a massless emitter at an arbitrary angle

T0 review · 0 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper computes the integrated double-emission eikonal function for one massive and one massless emitter at an arbitrary angle, expressing the two soft-gluon and quark-antiquark channels analytically to order $\varepsilon^0$ in terms…

desk verdict Strong, careful NNLO ingredient: the integrated double-soft eikonal for a massive+massless pair is genuinely new, two independent methods agree, and the per-pair convention caveat is real but minor and already flagged by the authors. read the letter →

arxiv 2504.20977 v2 pith:AAZKQDQW submitted 2025-04-29 hep-ph

Dennis Horstmann , Kirill Melnikov , Ming-Ming Long , Andrey Pikelner This is my paper · ORCID
classification hep-ph
keywords QCDcorrectionsNNLOcalculationshadroniccollidersdouble-softlimiteikonalfunctionsmassiveemittersreverseunitaritypolylogarithms
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

When two partons are emitted with very small momenta, the squared QCD amplitude factorizes into a universal eikonal function, and next-to-next-to-leading-order subtraction schemes need that function integrated over the unresolved phase space. The paper supplies this integrated quantity for the still-missing case of one massive and one massless emitter whose momenta form an arbitrary angle, covering both two-gluon and quark-antiquark emission. The result—explicit functions of the massive emitter's velocity and the cosine of the angle, expressed through ordinary logarithms and polylogarithms including $\mathrm{Li}_{2,2}$—holds through the finite part in dimensional regularization. A second, semi-numerical computation in the massive parton's rest frame independently confirms the analytic answer.

What carries the argument

The machinery is reverse unitarity extended to Heaviside functions: the energy-ordering theta-function becomes a delta-function when differentiated, enabling integration-by-parts reduction to 52 master integrals. Differential equations are solved and two elliptic sectors are transformed using complete elliptic integrals before symbol-level simplification removes elliptic functions entirely. A second approach works directly in the massive parton's rest frame, applying soft and collinear subtraction operators to the eikonal integrand so that divergent parts are computed analytically and the finite remainder is a four-dimensional numerical integral.

What would settle it

Numerically evaluate the defining ordered phase-space integrals in higher dimension $d=6$, where they converge, at a kinematic point from Table 2, then use the dimension-shift relation to continue to $d=4-2\varepsilon$; a mismatch with the $\varepsilon^0$ coefficient of Eqs. (4.36)-(4.37) would refute the analytic result.

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Extended reading notes

Core claim

The central claim is that the energy-ordered double-emission integrals $\mathrm{SS}[\widetilde{S}_{ij}]$ and $\mathrm{SS}[\widetilde{I}_{ij}]$ can be evaluated analytically for any relative angle. Equations (4.36) and (4.37) give their expansions through $O(\varepsilon^0)$ as functions of $\beta$ and $y=\cos\theta$ in terms of $\mathrm{Li}_2$, $\mathrm{Li}_3$, $\mathrm{Li}_4$, and the two-variable function $\mathrm{Li}_{2,2}$. All intermediate elliptic integrals cancel in the final combination, so the integrated eikonal has polylogarithmic complexity. The paper also develops an independent rest-frame subtraction computation that extracts all $1/\varepsilon$ poles analytically and leaves a finite numerical remainder; the two methods agree.

Load-bearing premise

The load-bearing premise is that the massive-emitter eikonal function taken from Ref. [63] is the correct soft factor; the paper notes that another reference gives a different expression that agrees only after color summation, and both independent computational routes share this same input, so an error there would invalidate the integrated result.

Editorial extensions

If this is right

  • Removes the missing massless-massive ingredient for extending the nested soft-collinear subtraction scheme to processes with massive final-state particles.
  • The analytic form uses only classical polylogarithms and $\mathrm{Li}_{2,2}$, allowing fast, high-precision evaluation in milliseconds via the supplied C implementation.
  • The rest-frame subtraction method provides an independent check and is expected to be reusable for other measurement constraints, including the massive-massive emitter case.
  • The provided benchmark points and small-$\beta$ expansions give concrete validation targets for future implementations.

Reading between the lines

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

  • The cancellation of elliptic sectors suggests that the integrated eikonal is simpler than the master integrals from which it is built; a direct derivation that avoids elliptic functions might exist for this quantity.
  • The rest-frame subtraction strategy likely applies to the remaining two-massive-emitter arbitrary-angle case, where conventional reduction is even harder.
  • Because the energy-ordering constraint is what forces the laboratory frame, replacing it with another slicing variable may change the polylogarithmic content; testing this would clarify the role of the Heaviside function in the final complexity.
  • Embedding $\mathrm{SS}[\widetilde{S}_{ij}]$ and $\mathrm{SS}[\widetilde{I}_{ij}]$ in a subtraction code for a specific massive process and verifying infrared pole cancellation would provide a direct phenomenological validation.
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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

0 major / 3 minor

Summary. The paper presents an analytic calculation of the integrated double-emission eikonal functions SS[\tilde{S}_{ij}] and SS[\tilde{I}_{ij}] for one massive and one massless emitter at arbitrary relative angle, which are needed to extend the nested soft-collinear subtraction scheme to processes with massive final-state particles. The main computation uses reverse unitarity extended to integrals with Heaviside functions, IBP reduction to 52 master integrals, and differential equations; the final result through O(ε^0) is expressed in terms of classical polylogarithms and the function Li_{2,2}. A complementary semi-numerical method is developed in the massive parton rest frame, where all divergences are extracted analytically and the finite remainder is given as a numerically integrable expression. The two methods agree, and additional checks include a small-β expansion up to β^4, benchmark points, and public C code for fast evaluation.

Significance. The result is a relevant technical ingredient for NNLO QCD calculations with massive final-state particles and appears to be correct. Strengths of the paper are the two independent computational routes (differential-equation/IBP and rest-frame subtraction) that agree, analytic boundary conditions at β=0, the simplification of the elliptic sectors, and the provided fast C code and ancillary files. The combination of these validation steps makes the central result credible and useful. The calculation is conditional on the accepted soft factorization input for the eikonal functions, which is standard and clearly referenced.

minor comments (3)
  1. [Section 2 (footnote 1) and Section 6] The per-pair integrated functions in Eqs. (4.36) and (4.37) are defined through Eq. (2.9), i.e. the Ref. [63] expression for S^m_ij. Since footnote 1 notes that Ref. [41] gives a different expression for S^m_ij that agrees only after color summation, the pairwise integrated functions are convention-dependent, with only the color-summed combination entering the physical double-soft contribution being invariant. Please add an explicit sentence to this effect, so that readers do not apply the pairwise results in a different convention.
  2. [Eq. (2.20)] There is a typo in the integrand: 'Ξij({km,k n)' should read 'Ξij(km,kn)' with the braces removed.
  3. [Abstract and Section 5] The abstract calls the computation 'analytic', while Section 5 is semi-numerical and uses numerical integration for finite terms. Adjust the wording to indicate that an analytic result is provided and complemented by a semi-numerical cross-check.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the integrated eikonal results are computed from stated external eikonal inputs by independent analytic and semi-numerical methods.

full rationale

The paper's central objects, SS[S̃_ij] and SS[Ĩ_ij], are defined as phase-space integrals of fixed eikonal functions taken from established factorization results, namely the double-soft factorization of Catani and Grazzini and the massive-emitter eikonal function S^m_ij from Ref. [63]. No physical parameter is fitted to the target integrals, and no result is renamed as a prediction. The boundary conditions at β = 0 are genuine constants obtained by direct integration of master integrals, not quantities chosen to reproduce the final answer. The PSLQ step is used only to identify mathematical constants such as π, ζ(3), log(2), and Li_4(1/2) by matching an equivalent GPL representation; this is a simplification of the analytic form, not an input that forces the physical content. The two computational strategies, reverse unitarity with IBP and differential equations in Section 4 and the rest-frame subtraction method in Section 5, share the same eikonal integrand but proceed through entirely different reduction and integration routes; their agreement is a substantive check. The footnote-1 caveat about the differing expression for S^m_ij in Ref. [41] is an explicitly acknowledged convention question about the pairwise tilde functions, and it is a correctness or interpretation concern rather than a circular one, since the paper does not claim to derive that input from its own result. Prior self-citations are methodological and computational, and none is used as the sole justification for the claimed master-integral results. The derivation is therefore self-contained with respect to its stated inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No physical free parameters or invented entities are used. The kinematic variables beta and cos(theta) are independent phase-space variables, not fitted; epsilon is the dimensional regulator. In Section 4.3, PSLQ is used to express constants in the simplified polylog representation, but those constants are known mathematical constants such as pi, zeta3, log(2), and Li4(1/2), fixed by equality with the original GPL expression and checked at multiple test points, so they are not free physical parameters. The calculation rests on standard QCD soft-factorization and on prior computational methods for Heaviside integrals.

assumptions (4)
  • domain assumption The double-soft factorization formulas in Eqs (2.2) and (2.3), with the eikonal functions S_ij, S^0_ij, S^m_ij and I_ij from Refs [61,63], are the correct integrands for the massive-massless double-soft limit.
    Entered in Section 2, Eqs (2.2) to (2.10). Both computational methods integrate these functions; the paper does not re-derive the factorization.
  • domain assumption The energy-ordered phase-space integral with the Emax cutoff, Eqs (2.16) and (2.21), is the quantity required by the nested soft-collinear subtraction scheme.
    This definition is taken from Ref [43]; if the scheme used a different ordering or cutoff prescription, the integrated function would differ.
  • standard math The extension of reverse unitarity and IBP to integrands with Heaviside functions, used through Eq (4.2), is valid for the integral families considered.
    The method is established in Ref [57] and in the zero-jettiness soft-function calculations [73-76], not re-proven here. It supports the reduction to 52 master integrals in Section 4.1.
  • standard math Boundary values at beta=0, computed in Appendix B, together with the differential equations in Eq (4.23), uniquely determine the master integrals in the physical domain.
    The paper relies on analytic continuation of solutions of first-order systems from a regular point beta=0 to all beta in [0,1) and y in [-1,1].

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Pith. "Pith review of Integral of the double-emission eikonal function for a massive and a massless emitter at an arbitrary angle." pith.science (2026). https://pith.science/paper/AAZKQDQW

@misc{pith2026250420977,
  author       = {Pith},
  title        = {Pith review of: Integral of the double-emission eikonal function for a massive and a massless emitter at an arbitrary angle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AAZKQDQW}},
  note         = {Machine review of arXiv:2504.20977}
}
read the original abstract

We present an analytic calculation of the integrated double-emission eikonal function of a massive and a massless emitter whose momenta are at an arbitrary angle to each other. This quantity provides one of the required ingredients for extending the nested soft-collinear subtraction scheme to processes with massive final-state particles. To calculate it, we use the standard methodology involving reverse unitarity and its extension to cases with Heaviside functions, integration-by-parts technology and reduction to master integrals, and differential equations. In addition, we also describe a semi-numerical method based on the subtraction of infra-red and collinear singularities from the eikonal function, allowing us to extract divergences of the integrated eikonal function analytically, and to derive a simple integral representation for the finite remainder.

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Works this paper leans on

95 extracted references · 5 canonical work pages

  1. [63]

    Catani, D

    S. Catani, D. Colferai and A. Torrini,Triple (and quadruple) soft-gluon radiation in QCD hard scattering,JHEP01(2020) 118 [1908.01616]

    arXiv 2020

  2. [41]

    Czakon,Double-real radiation in hadronic top quark pair production as a proof of a certain concept,Nucl

    M. Czakon,Double-real radiation in hadronic top quark pair production as a proof of a certain concept,Nucl. Phys. B849(2011) 250 [1101.0642]

    arXiv 2011

  3. [1]

    X. Chen, T. Gehrmann, E. W. N. Glover and M. Jaquier,Precise QCD predictions for the production of Higgs + jet final states,Phys. Lett. B740(2015) 147 [1408.5325]

    arXiv 2015

  4. [2]

    Boughezal, F

    R. Boughezal, F. Caola, K. Melnikov, F. Petriello and M. Schulze,Higgs boson production in association with a jet at next-to-next-to-leading order,Phys. Rev. Lett.115(2015) 082003 [1504.07922]

    arXiv 2015

  5. [3]

    Caola, K

    F. Caola, K. Melnikov and M. Schulze,Fiducial cross sections for Higgs boson production in association with a jet at next-to-next-to-leading order in QCD,Phys. Rev. D92(2015) 074032 [1508.02684]

    arXiv 2015

  6. [4]

    X. Chen, J. Cruz-Martinez, T. Gehrmann, E. W. N. Glover and M. Jaquier,NNLO QCD corrections to Higgs boson production at large transverse momentum,JHEP10(2016) 066 [1607.08817]

    arXiv 2016

  7. [5]

    J. M. Campbell, R. K. Ellis and S. Seth,H + 1 jet production revisited,JHEP10(2019) 136 [1906.01020]

    arXiv 2019

  8. [6]

    Cacciari, F

    M. Cacciari, F. A. Dreyer, A. Karlberg, G. P. Salam and G. Zanderighi,Fully Differential Vector-Boson-Fusion Higgs Production at Next-to-Next-to-Leading Order,Phys. Rev. Lett. 115(2015) 082002 [1506.02660]

    arXiv 2015

Show all 95 references
  1. [7]

    Cruz-Martinez, T

    J. Cruz-Martinez, T. Gehrmann, E. W. N. Glover and A. Huss,Second-order QCD effects in Higgs boson production through vector boson fusion,Phys. Lett. B781(2018) 672 [1802.02445]

    2018 arXiv

  2. [8]

    Gauld, A

    R. Gauld, A. Gehrmann-De Ridder, E. W. N. Glover, A. Huss and I. Majer,VH + jet production in hadron-hadron collisions up to orderα 3 s in perturbative QCD,JHEP03(2022) 008 [2110.12992]

    2022 arXiv

  3. [9]

    Catani, S

    S. Catani, S. Devoto, M. Grazzini, S. Kallweit, J. Mazzitelli and C. Savoini,Higgs Boson Production in Association with a Top-Antitop Quark Pair in Next-to-Next-to-Leading Order QCD,Phys. Rev. Lett.130(2023) 111902 [2210.07846]

    2023 arXiv

  4. [10]

    H. A. Chawdhry, M. L. Czakon, A. Mitov and R. Poncelet,NNLO QCD corrections to three-photon production at the LHC,JHEP02(2020) 057 [1911.00479]

    2020 arXiv

  5. [11]

    H. A. Chawdhry, M. Czakon, A. Mitov and R. Poncelet,NNLO QCD corrections to diphoton production with an additional jet at the LHC,JHEP09(2021) 093 [2105.06940]

    2021 arXiv

  6. [12]

    Czakon, A

    M. Czakon, A. Mitov, M. Pellen and R. Poncelet,NNLO QCD predictions for W+c-jet production at the LHC,JHEP06(2021) 100 [2011.01011]

    2021 arXiv

  7. [13]

    Gauld, A

    R. Gauld, A. Gehrmann-De Ridder, E. W. N. Glover, A. Huss, A. R. Garcia and G. Stagnitto,NNLO QCD predictions for Z-boson production in association with a charm jet within the LHCb fiducial region,Eur. Phys. J. C83(2023) 336 [2302.12844]

    2023 arXiv

  8. [14]

    Currie, A

    J. Currie, A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover, A. Huss and J. Pires, Precise predictions for dijet production at the LHC,Phys. Rev. Lett.119(2017) 152001 [1705.10271]. – 38 –

    2017 arXiv

  9. [15]

    X. Chen, T. Gehrmann, E. W. N. Glover, A. Huss and J. Mo,NNLO QCD corrections in full colour for jet production observables at the LHC,JHEP09(2022) 025 [2204.10173]

    2022 arXiv

  10. [16]

    Badger, M

    S. Badger, M. Czakon, H. B. Hartanto, R. Moodie, T. Peraro, R. Poncelet et al.,Isolated photon production in association with a jet pair through next-to-next-to-leading order in QCD,2304.06682

    arXiv

  11. [17]

    Czakon, A

    M. Czakon, A. Mitov and R. Poncelet,Next-to-Next-to-Leading Order Study of Three-Jet Production at the LHC,Phys. Rev. Lett.127(2021) 152001 [2106.05331]

    2021 arXiv

  12. [18]

    Czakon, D

    M. Czakon, D. Heymes and A. Mitov,High-precision differential predictions for top-quark pairs at the LHC,Phys. Rev. Lett.116(2016) 082003 [1511.00549]

    2016 arXiv

  13. [19]

    Catani, S

    S. Catani, S. Devoto, M. Grazzini, S. Kallweit and J. Mazzitelli,Top-quark pair production at the LHC: Fully differential QCD predictions at NNLO,JHEP07(2019) 100 [1906.06535]

    2019 arXiv

  14. [20]

    Buonocore, S

    L. Buonocore, S. Devoto, M. Grazzini, S. Kallweit, J. Mazzitelli, L. Rottoli et al.,Associated production of a W boson with a top-antitop quark pair: next-to-next-to-leading order QCD predictions for the LHC,2306.16311

    arXiv

  15. [21]

    Brucherseifer, F

    M. Brucherseifer, F. Caola and K. Melnikov,On the NNLO QCD corrections to single-top production at the LHC,Phys. Lett. B736(2014) 58 [1404.7116]

    2014 arXiv

  16. [22]

    E. L. Berger, J. Gao, C. P. Yuan and H. X. Zhu,NNLO QCD Corrections to t-channel Single Top-Quark Production and Decay,Phys. Rev. D94(2016) 071501 [1606.08463]

    2016 arXiv

  17. [23]

    Campbell, T

    J. Campbell, T. Neumann and Z. Sullivan,Single-top-quark production in thet-channel at NNLO,JHEP02(2021) 040 [2012.01574]

    2021 arXiv

  18. [24]

    Gehrmann, M

    T. Gehrmann, M. Jaquier, E. Glover and A. Koukoutsakis,Two-Loop QCD Corrections to the Helicity Amplitudes forH→3 partons,JHEP1202(2012) 056 [1112.3554]

    2012 arXiv

  19. [25]

    Caola, J

    F. Caola, J. M. Henn, K. Melnikov, A. V. Smirnov and V. A. Smirnov,Two-loop helicity amplitudes for the production of two off-shell electroweak bosons in quark-antiquark collisions,JHEP11(2014) 041 [1408.6409]

    2014 arXiv

  20. [26]

    Abreu, J

    S. Abreu, J. Dormans, F. Febres Cordero, H. Ita and B. Page,Analytic Form of Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD,Phys. Rev. Lett.122(2019) 082002 [1812.04586]

    2019 arXiv

  21. [27]

    H. A. Chawdhry, M. Czakon, A. Mitov and R. Poncelet,Two-loop leading-color helicity amplitudes for three-photon production at the LHC,JHEP06(2021) 150 [2012.13553]

    2021 arXiv

  22. [28]

    Agarwal, F

    B. Agarwal, F. Buccioni, A. von Manteuffel and L. Tancredi,Two-Loop Helicity Amplitudes for Diphoton Plus Jet Production in Full Color,Phys. Rev. Lett.127(2021) 262001 [2105.04585]

    2021 arXiv

  23. [29]

    Badger, H

    S. Badger, H. B. Hartanto, J. Kry´ s and S. Zoia,Two-loop leading-colour QCD helicity amplitudes for Higgs boson production in association with a bottom-quark pair at the LHC, JHEP11(2021) 012 [2107.14733]

    2021 arXiv

  24. [30]

    Agarwal, G

    B. Agarwal, G. Heinrich, S. P. Jones, M. Kerner, S. Y. Klein, J. Lang et al.,Two-loop amplitudes fort tHproduction: the quark-initiated N f -part,JHEP05(2024) 013 [2402.03301]

    2024 arXiv

  25. [31]

    Anastasiou and G

    C. Anastasiou and G. Sterman,Locally finite two-loop QCD amplitudes from IR universality for electroweak production,JHEP05(2023) 242 [2212.12162]. – 39 –

    2023 arXiv

  26. [32]

    J. J. Aguilera-Verdugo, F. Driencourt-Mangin, R. J. Hern´ andez-Pinto, J. Plenter, S. Ramirez-Uribe, A. E. Renteria Olivo et al.,Open Loop Amplitudes and Causality to All Orders and Powers from the Loop-Tree Duality,Phys. Rev. Lett.124(2020) 211602 [2001.03564]

    2020 arXiv

  27. [33]

    Capatti, V

    Z. Capatti, V. Hirschi, A. Pelloni and B. Ruijl,Local Unitarity: a representation of differential cross-sections that is locally free of infrared singularities at any order,JHEP04 (2021) 104 [2010.01068]

    2021 arXiv

  28. [34]

    Kermanschah and M

    D. Kermanschah and M. Vicini,N f -contribution to the virtual correction for electroweak vector boson production at NNLO,2407.18051

  29. [35]

    Frixione and M

    S. Frixione and M. Grazzini,Subtraction at NNLO,JHEP06(2005) 010 [hep-ph/0411399]

    2005 arXiv

  30. [36]

    Gehrmann-De Ridder, T

    A. Gehrmann-De Ridder, T. Gehrmann and E. W. N. Glover,Antenna subtraction at NNLO,JHEP09(2005) 056 [hep-ph/0505111]

    2005 arXiv

  31. [37]

    Currie, E

    J. Currie, E. W. N. Glover and S. Wells,Infrared Structure at NNLO Using Antenna Subtraction,JHEP04(2013) 066 [1301.4693]

    2013 arXiv

  32. [38]

    Somogyi, Z

    G. Somogyi, Z. Trocsanyi and V. Del Duca,Matching of singly- and doubly-unresolved limits of tree-level QCD squared matrix elements,JHEP06(2005) 024 [hep-ph/0502226]

    2005 arXiv

  33. [39]

    Somogyi and Z

    G. Somogyi and Z. Trocsanyi,A Subtraction scheme for computing QCD jet cross sections at NNLO: Regularization of real-virtual emission,JHEP01(2007) 052 [hep-ph/0609043]

    2007 arXiv

  34. [40]

    Czakon,A novel subtraction scheme for double-real radiation at NNLO,Phys

    M. Czakon,A novel subtraction scheme for double-real radiation at NNLO,Phys. Lett. B 693(2010) 259 [1005.0274]

    2010 arXiv

  35. [42]

    Anastasiou, K

    C. Anastasiou, K. Melnikov and F. Petriello,A new method for real radiation at NNLO, Phys. Rev. D69(2004) 076010 [hep-ph/0311311]

    2004 arXiv

  36. [43]

    Caola, K

    F. Caola, K. Melnikov and R. R¨ ontsch,Nested soft-collinear subtractions in NNLO QCD computations,Eur. Phys. J. C77(2017) 248 [1702.01352]

    2017 arXiv

  37. [44]

    Catani and M

    S. Catani and M. Grazzini,An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC,Phys. Rev. Lett.98(2007) 222002 [hep-ph/0703012]

    2007 arXiv

  38. [45]

    Grazzini, S

    M. Grazzini, S. Kallweit and M. Wiesemann,Fully differential NNLO computations with MATRIX,Eur. Phys. J. C78(2018) 537 [1711.06631]

    2018 arXiv

  39. [46]

    Boughezal, K

    R. Boughezal, K. Melnikov and F. Petriello,A subtraction scheme for NNLO computations, Phys. Rev. D85(2012) 034025 [1111.7041]

    2012 arXiv

  40. [47]

    Gaunt, M

    J. Gaunt, M. Stahlhofen, F. J. Tackmann and J. R. Walsh,N-jettiness Subtractions for NNLO QCD Calculations,JHEP09(2015) 058 [1505.04794]

    2015 arXiv

  41. [48]

    G. F. R. Sborlini, F. Driencourt-Mangin and G. Rodrigo,Four-dimensional unsubtraction with massive particles,JHEP10(2016) 162 [1608.01584]

    2016 arXiv

  42. [49]

    Herzog,Geometric IR subtraction for final state real radiation,JHEP08(2018) 006 [1804.07949]

    F. Herzog,Geometric IR subtraction for final state real radiation,JHEP08(2018) 006 [1804.07949]

    2018 arXiv

  43. [50]

    Magnea, E

    L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati,Local analytic sector subtraction at NNLO,JHEP12(2018) 107 [1806.09570]. – 40 –

    2018 arXiv

  44. [51]

    Bertolotti, L

    G. Bertolotti, L. Magnea, G. Pelliccioli, A. Ratti, C. Signorile-Signorile, P. Torrielli et al., NNLO subtraction for any massless final state: a complete analytic expression,2212.11190

    arXiv

  45. [52]

    Capatti, V

    Z. Capatti, V. Hirschi, D. Kermanschah and B. Ruijl,Loop-Tree Duality for Multiloop Numerical Integration,Phys. Rev. Lett.123(2019) 151602 [1906.06138]

    2019 arXiv

  46. [53]

    W. J. Torres Bobadilla et al.,May the four be with you: Novel IR-subtraction methods to tackle NNLO calculations,Eur. Phys. J. C81(2021) 250 [2012.02567]

    2021 arXiv

  47. [54]

    Caola, M

    F. Caola, M. Delto, H. Frellesvig and K. Melnikov,The double-soft integral for an arbitrary angle between hard radiators,Eur. Phys. J. C78(2018) 687 [1807.05835]

    2018 arXiv

  48. [55]

    Bizo´ n and M

    W. Bizo´ n and M. Delto,Analytic double-soft integrated subtraction terms for two massive emitters in a back-to-back kinematics,JHEP07(2020) 011 [2004.01663]

    2020 arXiv

  49. [56]

    Anastasiou and K

    C. Anastasiou and K. Melnikov,Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys.B646(2002) 220 [hep-ph/0207004]

    2002 arXiv

  50. [57]

    Baranowski, M

    D. Baranowski, M. Delto, K. Melnikov and C.-Y. Wang,On phase-space integrals with Heaviside functions,JHEP02(2022) 081 [2111.13594]

    2022 arXiv

  51. [58]

    Tkachov,A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions,Phys.Lett.B100(1981) 65

    F. Tkachov,A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions,Phys.Lett.B100(1981) 65

    1981

  52. [59]

    Chetyrkin and F

    K. Chetyrkin and F. Tkachov,Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops,Nucl.Phys.B192(1981) 159

    1981

  53. [60]

    Agarwal, K

    P. Agarwal, K. Melnikov and I. Pedron,N-jettiness soft function at next-to-next-to-leading order in perturbative QCD,JHEP05(2024) 005 [2403.03078]

    2024 arXiv

  54. [61]

    Catani and M

    S. Catani and M. Grazzini,Infrared factorization of tree level QCD amplitudes at the next-to-next-to-leading order and beyond,Nucl. Phys. B570(2000) 287 [hep-ph/9908523]

    2000 arXiv

  55. [62]

    Catani and M

    S. Catani and M. H. Seymour,The Dipole formalism for the calculation of QCD jet cross-sections at next-to-leading order,Phys. Lett. B378(1996) 287 [hep-ph/9602277]

    1996 arXiv

  56. [64]

    Somogyi,Angular integrals in d dimensions,J

    G. Somogyi,Angular integrals in d dimensions,J. Math. Phys.52(2011) 083501 [1101.3557]

    2011 arXiv

  57. [65]

    Kotikov,Differential equations method: New technique for massive Feynman diagrams calculation,Phys.Lett.B254(1991) 158

    A. Kotikov,Differential equations method: New technique for massive Feynman diagrams calculation,Phys.Lett.B254(1991) 158

    1991

  58. [66]

    Gehrmann and E

    T. Gehrmann and E. Remiddi,Differential equations for two loop four point functions, Nucl.Phys.B580(2000) 485 [hep-ph/9912329]

    2000 arXiv

  59. [67]

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

    2013 arXiv

  60. [68]

    Brønnum-Hansen, K

    C. Brønnum-Hansen, K. Melnikov, J. Quarroz, C. Signorile-Signorile and C.-Y. Wang, Non-factorisable contribution to t-channel single-top production,JHEP06(2022) 061 [2204.05770]

    2022 arXiv

  61. [69]

    Alioli, P

    S. Alioli, P. Nason, C. Oleari and E. Re,A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX,JHEP1006(2010) 043 [1002.2581]. – 41 –

    2010 arXiv

  62. [70]

    Catani,The Singular behavior of QCD amplitudes at two loop order,Phys

    S. Catani,The Singular behavior of QCD amplitudes at two loop order,Phys. Lett. B427 (1998) 161 [hep-ph/9802439]

    1998 arXiv

  63. [71]

    R. N. Lee,Presenting LiteRed: a tool for the Loop InTEgrals REDuction,1212.2685

    arXiv

  64. [72]

    Klappert, F

    J. Klappert, F. Lange, P. Maierh¨ ofer and J. Usovitsch,Integral reduction with Kira 2.0 and finite field methods,Comput. Phys. Commun.266(2021) 108024 [2008.06494]

    2021 arXiv

  65. [73]

    Baranowski, M

    D. Baranowski, M. Delto, K. Melnikov and C.-Y. Wang,Same-hemisphere three-gluon-emission contribution to the zero-jettiness soft function at N3LO QCD,Phys. Rev. D106(2022) 014004 [2204.09459]

    2022 arXiv

  66. [74]

    Baranowski, M

    D. Baranowski, M. Delto, K. Melnikov, A. Pikelner and C.-Y. Wang,One-loop corrections to the double-real emission contribution to the zero-jettiness soft function at N3LO in QCD, JHEP04(2024) 114 [2401.05245]

    2024 arXiv

  67. [75]

    Baranowski, M

    D. Baranowski, M. Delto, K. Melnikov, A. Pikelner and C.-Y. Wang,Zero-jettiness soft function to third order in perturbative QCD,2409.11042

    arXiv

  68. [76]

    Baranowski, M

    D. Baranowski, M. Delto, K. Melnikov, A. Pikelner and C.-Y. Wang,Triple real-emission contribution to the zero-jettiness soft function at N3LO in QCD,2412.14001

    arXiv

  69. [77]

    Adams, E

    L. Adams, E. Chaubey and S. Weinzierl,Simplifying Differential Equations for Multiscale Feynman Integrals beyond Multiple Polylogarithms,Phys. Rev. Lett.118(2017) 141602 [1702.04279]

    2017 arXiv

  70. [78]

    Adams and S

    L. Adams and S. Weinzierl,Theε-form of the differential equations for Feynman integrals in the elliptic case,Phys. Lett. B781(2018) 270 [1802.05020]

    2018 arXiv

  71. [79]

    Argeri, S

    M. Argeri, S. Di Vita, P. Mastrolia, E. Mirabella, J. Schlenk et al.,Magnus and Dyson Series for Master Integrals,JHEP1403(2014) 082 [1401.2979]

    2014 arXiv

  72. [80]

    Chen,Iterated path integrals,Bull

    K.-T. Chen,Iterated path integrals,Bull. Am. Math. Soc.83(1977) 831

    1977

  73. [81]

    Ablinger,Computer Algebra Algorithms for Special Functions in Particle Physics, Ph.D

    J. Ablinger,Computer Algebra Algorithms for Special Functions in Particle Physics, Ph.D. thesis, Linz U., 4, 2012.1305.0687

    2012

  74. [82]

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

    1998 arXiv

  75. [83]

    A. B. Goncharov,Multiple polylogarithms and mixed Tate motives,math/0103059

    arXiv

  76. [84]

    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]

    2010 arXiv

  77. [85]

    C. Duhr, H. Gangl and J. R. Rhodes,From polygons and symbols to polylogarithmic functions,JHEP10(2012) 075 [1110.0458]

    2012 arXiv

  78. [86]

    A. B. Goncharov,Galois symmetries of fundamental groupoids and noncommutative geometry,Duke Math. J.128(2005) 209 [math/0208144]

    2005 arXiv

  79. [87]

    Duhr,Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP1208(2012) 043 [1203.0454]

    C. Duhr,Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP1208(2012) 043 [1203.0454]

    2012 arXiv

  80. [88]

    Ferguson and D

    H. Ferguson and D. Bailey,A polynomial time, numerically stable integer relation algorithm,

  81. [89]

    Duhr and F

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

    2019 arXiv

  82. [90]

    Maitre,HPL, a mathematica implementation of the harmonic polylogarithms, Comput.Phys.Commun.174(2006) 222 [hep-ph/0507152]

    D. Maitre,HPL, a mathematica implementation of the harmonic polylogarithms, Comput.Phys.Commun.174(2006) 222 [hep-ph/0507152]. – 42 –

    2006 arXiv

  83. [91]

    Maitre,Extension of HPL to complex arguments,Comput.Phys.Commun.183(2012) 846 [hep-ph/0703052]

    D. Maitre,Extension of HPL to complex arguments,Comput.Phys.Commun.183(2012) 846 [hep-ph/0703052]

    2012 arXiv

  84. [92]

    C. W. Bauer, A. Frink and R. Kreckel,Introduction to the GiNaC framework for symbolic computation within the C++ programming language,cs/0004015

    arXiv

  85. [93]

    Vollinga and S

    J. Vollinga and S. Weinzierl,Numerical evaluation of multiple polylogarithms, Comput.Phys.Commun.167(2005) 177 [hep-ph/0410259]

    2005 arXiv

  86. [94]

    Frellesvig, D

    H. Frellesvig, D. Tommasini and C. Wever,On the reduction of generalized polylogarithms to Lin and Li 2,2 and on the evaluation thereof,JHEP03(2016) 189 [1601.02649]

    2016 arXiv

  87. [95]

    Devoto, K

    F. Devoto, K. Melnikov, R. R¨ ontsch, C. Signorile-Signorile and D. M. Tagliabue,A fresh look at the nested soft-collinear subtraction scheme: NNLO QCD corrections to N-gluon final states inq qannihilation,JHEP02(2024) 016 [2310.17598]. – 43 –

    2024 arXiv

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