Leading-Color Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders

B. Page; E. Pascual; S. Abreu; V. Sotnikov

REVIEW

Leading-Color Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders

S. Abreu, B. Page, E. Pascual, V. Sotnikov

High Energy Physics - Phenomenology · Submitted 29 Oct 2020 · 71 references

Pith's verdict: A genuine deliverable: the first analytic all-orders master-integral decomposition of a five-point two-loop QCD amplitude, plus a public code for triphoton NNLO — the physics is solid, the abstract's phase-space wording is the one loose thread.

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arXiv ID2010.15834 (version 2)

Pith IDAVRNYXGE, Pith's permanent record of this paper and its review

CitationB. Page, E. Pascual, S. Abreu, V. Sotnikov. "Leading-Color Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders." arXiv:2010.15834 (2020). https://pith.science/paper/AVRNYXGE

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Referee report

Significance

If the results hold, this is a genuinely important contribution: it is the first publicly available two-loop five-point amplitude suitable for on-the-fly NNLO phenomenology, and the first analytic master-integral decomposition for a physical two-loop five-point process. The paper earns credit for shipping reproducible deliverables (ancillary files with master-integral definitions, amplitude and remainder expressions, an assembly example reproducing Tables 4-6, and a public code), for a validation chain that is unusually complete for this genre, and for quantifying numerical stability rather than asserting it. The results have already been used in ref. [16], which is a meaningful falsifiable test of practical usability. The leading-color approximation and the omission of the non-planar N_f contribution are stated explicitly and their phenomenological adequacy is deferred to ref. [16], which is appropriate.

Minor objections (10)

[Abstract; §4 and Fig. 2] The abstract states that the remainders are 'given in a form that can be efficiently evaluated across the whole physical phase space' (abstract:s3) and that 'numerical stability across phase-space is demonstrated' (abstract:s4), whereas the demonstration in §4 and Fig. 2 uses a single cut set (the ATLAS 8 TeV selection of ref. [16]) and the authors themselves note that a precision-rescue system is deferred to future work. The body is more careful than the abstract, and the limitation is disclosed, so this is a wording issue rather than a physics issue; I would nonetheless ask the authors to soften the abstract to match the evidence, e.g. by stating that stability is demonstrated on a representative phenomenological phase-space sample and that a small fraction of points near spurious-singularity surfaces lose accuracy.
[§4, Fig. 2] The text states that 'less than 0.1% of the sampled points have an accuracy of less than four digits', but the histogram axis extends to d = -4, i.e. some points are wrong at the level of the leading digit. Please quote the actual number of such points, the worst observed value of d, and the fraction of the 90000 points with d below, say, 1 and 2. This would let a reader judge the practical risk more concretely than the single '0.1%' figure does.
[§4, public code] Since the library returns a number without an error estimate and the problematic regions are, as the authors state, explicitly known (surfaces where letters such as s12 + s23 vanish), it would be cheap and useful to document those letters and the associated phase-space regions in the README, even if a full precision-rescue system is left for future work. I would encourage this as a stopgap.
[Table 8] The one-loop 'Pentagon' entry reads MI[``Pentagon'', i, s12, s23, s45, s45, s15], listing s45 twice; presumably the fourth argument should be s34. Please check this against the ancillary master-integral definition files, since a mislabelled permutation representative would be easy for a user to propagate.
[Appendix B, list of collinear configurations] The sentence enumerating the collinear limits reads 'either (gamma,gamma), (q,q), (q,gamma) or (q,gamma)', with the last pair repeated; the final entry should presumably be (qbar,gamma), consistent with the separate '(qbar,gamma) limit' paragraph that follows.
[Eq. (4.3)] The prefactor of the one-loop-squared term is written as 'N_c^2/4 N_c', which is ambiguous as typeset. Please write it unambiguously (e.g. (N_c^2/4) N_c) and state explicitly that the C_F^2 factor of the one-loop amplitude has been replaced by its leading-color value for consistency with the two-loop approximation, as is done in the text for Eq. (4.2).
[§3.3] The claim that 'this is the first time that such a decomposition has been made available for a two-loop five-point QCD amplitude' would be strengthened by one explicit sentence distinguishing it from refs. [46,48], which provide analytic five-gluon amplitudes but, as I understand, only in the form of finite remainders rather than master-integral decompositions valid to all orders in epsilon. As written, a reader may not immediately see why the two claims are compatible.
[§3.2 and Appendix B] A few typographical and typesetting issues should be fixed: 'Aside form the fact' in §3.2; 'heliciy' in the sentence introducing Eq. (B.3); and in §4 the sentence 'the amplitude must stay regular As an example, s12 + s23 is a letter...' is missing a sentence break and a full stop.
[§3.5] The ancillary path is given as 'anc/example assembly.m' with a space; please confirm the actual filename (presumably example_assembly.m) so that readers can locate it.
[§4, code availability] For reproducibility, please state the version of PentagonFunctions++ the library was tested against, and the license under which the code is released.

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The paper

The paper claims

The two-loop QCD corrections to three-photon production are now written down in full analytic form and shipped as runnable code, so a five-point two-loop process can be used in phenomenology without re-deriving it.

The reading

The paper sets out to make the double-virtual NNLO QCD corrections to $qbar q to gammagammagamma$ usable, rather than merely computable. Its authors obtain the two independent helicity amplitudes at two loops in the leading-color approximation, first as an analytic decomposition into master integrals valid to all orders in the dimensional regulator $epsilon$, and then as finite remainders expanded in a basis of pentagon functions. The remainder coefficients are simplified by multivariate partial fractioning so that they can be evaluated quickly, and the whole thing is packaged in a public C++ library that returns the helicity- and color-summed squared remainders. The practical stake is that triphoton production has shown slow perturbative convergence and a tension with LHC Run 1 data at NLO, so a stable two-loop ingredient is what a full NNLO prediction needs. The authors report that the code evaluates the two-loop quantity in a few seconds per phase-space point and agrees with quadruple precision to better than four digits on more than 99.9 percent of 90000 sampled points.

What carries the argument

Two ingredients carry the argument. The first is two-loop numerical unitarity: the integrand is decomposed into master integrands and surface terms, and the coefficients are fixed by solving cut equations built from products of tree amplitudes, evaluated in finite-field arithmetic so that the coefficients are exact rational numbers for rational phase-space points. The second is the pentagon-function basis for five massless particles, which turns the master-integral decomposition into a common function space shared by the one-loop and two-loop amplitudes and by the infrared subtraction operators, so that the finite remainder is again a linear combination of the same functions with rational coefficients. Reconstruction of those rational coefficients proceeds by fixing denominators from a subset of the symbol alphabet letters that are polynomial in the Mandelstam invariants, then fitting the numerators with a multivariate Newton method. The photon's lack of color ordering multiplies the pentagon-box topologies from five to thirty, which is why the non-planar alphabet, closed under all photon permutations, is needed even in the leading-color approximation.

Extended reading notes

Core claim

The central claim is a pair of firsts, stated in the abstract and again in section 3.3: analytic master-integral coefficients for a two-loop five-point QCD amplitude, and a five-point two-loop process released publicly in a form ready for immediate phenomenological use. Concretely, the authors compute $A^{(2,0)}_{-++}$, $A^{(2,N_f)}_{-++}$, $A^{(2,0)}_{+++}$ and $A^{(2,N_f)}_{+++}$ by two-loop numerical unitarity with finite-field arithmetic and functional reconstruction, keeping the full $epsilon$ dependence in the master-integral decomposition. They then convert to finite remainders in the pentagon-function basis of the five-massless-particle case, where the coefficient functions are rational in the Mandelstam invariants and $mathrm{tr}_5$ and have a simpler denominator structure than the master-integral coefficients themselves. The paper quantifies the gain: the largest remainder needs 171 independent rational functions against 1320 for the corresponding all-order coefficients, and the pole in $mathrm{tr}_5^2$ present in the coefficients is absent from the remainders. The released library evaluates the squared remainders $H^{(1)}$ and $H^{(2)}$ and was already used in a full NNLO triphoton calculation.

Load-bearing premise

The claim that the released code is trustworthy rests on a sample statistic rather than a guarantee: fewer than 0.1 percent of 90000 points under one specific set of LHC cuts fell below four correct digits, the library returns no per-point error estimate, and the authors explicitly defer a precision-rescue system to future work.

Editorial extensions

If this is right

A complete NNLO QCD prediction for triphoton production becomes feasible on the same footing as diphoton, since the double-virtual ingredient can now be called as a library rather than reconstructed by interpolation.
The all-order-in-$epsilon$ master-integral decomposition gives a starting point for defining the three-loop remainder that N$^3$LO corrections would require.
Because the remainders have fewer spurious denominators than the master-integral coefficients, they are the better object to hand to a Monte Carlo, and the paper's own timing (seconds per point in double precision) supports on-the-fly use.
The same reconstruction pipeline, with photons treated as unordered colorless particles, extends to other five-point processes with more than one colorless final-state particle.
The observed correlation between unstable evaluations and proximity to spurious singularity surfaces gives a concrete diagnostic for building precision-rescue logic around the shipped code.

Reading between the lines

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

The paper's stability evidence is a population statistic, not a per-point bound: the library returns a number with no error flag, so a user running cuts that concentrate near surfaces where letters like $s_{12}+s_{23}$ vanish could silently receive a wrong value. A wrapper that evaluates at perturbed points and checks the spread would close this gap cheaply.
The claim of a first rests on a comparison the authors cannot make directly, since the earlier interpolation-based triphoton study does not publish explicit two-loop amplitudes; independent agreement is therefore established only indirectly, through the NNLO cross section it feeds. A direct numerical comparison at a handful of published phase-space points would make the claim much harder to disput
The reference values in tables 4 to 6, together with the assembly script, are the natural anchor for anyone reimplementing this: agreement at the stated phase-space point is a cheap and sharp test of an independent pipeline.
If the spurious-singularity explanation of the unstable points holds, then the same diagnostic should apply to the five-parton two-loop remainders already in the literature, which would make it a general tool rather than a fix specific to this process.

Original abstract

We compute the two-loop helicity amplitudes for the production of three photons at hadron colliders in QCD at leading-color. Using the two-loop numerical unitarity method coupled with analytic reconstruction techniques, we obtain the decomposition of the two-loop amplitudes in terms of master integrals in analytic form. These expressions are valid to all orders in the dimensional regulator. We use them to compute the two-loop finite remainders, which are given in a form that can be efficiently evaluated across the whole physical phase space. We further package these results in a public code which assembles the helicity-summed squared two-loop remainders, whose numerical stability across phase-space is demonstrated. This is the first time that a five-point two-loop process is publicly available for immediate phenomenological applications.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,882 input tokens.

What the abstract itself states, as probabilities: claims first 0.93 · states limitation 0.27 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.24 · mentions code or data availability 0.98.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We compute the two-loop helicity amplitudes for the production of three photons at hadron colliders in QCD at leading-color. Using the two-loop numerical unitarity method coupled with analytic reconstruction techniques, we obtain the decomposition of the two-loop amplitudes in terms of master integrals in analytic form. These expressions are valid to all orders in the dimensional regulator. We use them to compute the two-loop finite remainders, which are given in a form that can be efficiently evaluated across the whole physical phase space. We further package these results in a public code which assembles the helicity-summed squared two-loop remainders, whose numerical stability across phase-space is demonstrated. This is the first time that a five-point two-loop process is publicly available for immediate phenomenological applications.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We compute the two-loop helicity amplitudes for the production of three photons at hadron colliders in QCD at leading-color.	contribution 1.00	0.97	0.20	asserted 1.00
s1	Using the two-loop numerical unitarity method coupled with analytic reconstruction techniques, we obtain the decomposition of the two-loop amplitudes in terms of master integrals in analytic form.	method 0.97	0.96	0.11	asserted 1.00
s2	These expressions are valid to all orders in the dimensional regulator.	result 0.84	0.75	0.07	asserted 1.00
s3	We use them to compute the two-loop finite remainders, which are given in a form that can be efficiently evaluated across the whole physical phase space.	result 0.80	0.96	0.14	asserted 1.00
s4	We further package these results in a public code which assembles the helicity-summed squared two-loop remainders, whose numerical stability across phase-space is demonstrated.	contribution 0.96	0.96	0.15	asserted 1.00
s5	This is the first time that a five-point two-loop process is publicly available for immediate phenomenological applications.	interpretation 0.37	0.82	0.46	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.91 · evidence from simulation or computation 0.69.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Representative diagrams for each of the contributions in eq. (2.13). Photons (γ) are denoted by wavy lines, gluon by curly lines, and quarks by straight lines. In summary, the main goal of this paper consists in the calculation of four functions: A (2,0) +++, A (2,Nf ) +++ , A (2,0) −++ and A (2,Nf ) −++ . We note that since A (0) +++ = 0, only the latter two are required to compute NNLO QCD corrections to the process in eq. (2.1). 2.2 Finite Remainders At NNLO, the two-loop amplitudes contribute to the cross section only through finite remainders (see e.g. [53]), which can b

**Figure 2.** The logarithmic distribution of correct decimal digits (as defined in eq. (4.5)) for 90000 double-precision evaluations of the H(2) function. The phase-space points are sampled from a distribution representative of typical phenomenological studies. It is interesting to note that a good understanding of the physical properties that govern the analytic structure of scattering amplitudes can be used to explain and improve the behavior of numerical algorithms. In particular, we recall that our analytic representation of the remainders (and thus of the H(i) ) only has poles that are…

**Figure 3.** Graphical representation of equation (B.5). White blobs represent the splitting amplitudes and hatched blobs represent the four-point amplitudes. The number of contained circles represents the loop order of the object. The types of particles are omitted from this representation, the external and internal lines representing only momenta. We can then expand both sides of (B.3), and we are particularly interested in the order (α 0 s ) 2 contributions: A¯(2)(i hi , jhj , . . .) i||j −−→ X h Split(2) P h (i hi , jhj )A¯(0)(P −h , . . .) + Split(1) P h (i hi , jhj )A¯

Figures reproduced from arXiv:2010.15834 by the authors.

References

Works cited (71)

[16]

S. Kallweit, V. Sotnikov and M. Wiesemann, Triphoton production at hadron colliders in NNLO QCD, Phys. Lett. B 812 (2021) 136013 [ 2010.04681]

View on Pith Pith review arXiv 2021
[15]

H. A. Chawdhry, M. L. Czakon, A. Mitov and R. Poncelet, NNLO QCD corrections to three-photon production at the LHC , JHEP 02 (2020) 057 [ 1911.00479]

View on Pith Pith review arXiv 2020
[49]

S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, B. Page and V. Sotnikov, Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD , JHEP 05 (2019) 084 [1904.00945]

View on Pith Pith review arXiv 2019
[23]

D. Chicherin and V. Sotnikov, Pentagon Functions for Scattering of Five Massless Particles , JHEP 20 (2020) 167 [ 2009.07803]

View on Pith Pith review arXiv 2020
[10]

E. Glover and M. Tejeda-Yeomans, Two loop QCD helicity amplitudes for massless quark massless gauge boson scattering , JHEP 06 (2003) 033 [ hep-ph/0304169]

View on Pith Pith review arXiv 2003
[1]

S. Amoroso et al., Les Houches 2019: Physics at TeV Colliders: Standard Model Working Group Report, in 11th Les Houches Workshop on Physics at TeV Colliders: PhysTeV Les Houches, 3, 2020, 2003.01700

View on Pith Pith review arXiv 2019
[2]

ATLAS collaboration, G. Aad et al., Search for new phenomena in events with at least three photons collected in pp collisions at√s = 8 TeV with the ATLAS detector , Eur. Phys. J. C 76 (2016) 210 [ 1509.05051]

View on Pith Pith review arXiv 2016
[3]

H. Denizli, K. Oyulmaz and A. Senol, Testing for observability of Higgs eﬀective couplings in triphoton production at FCC-hh , J. Phys. G 46 (2019) 105007 [ 1901.04784]

View on Pith Pith review arXiv 2019
[4]

H. Denizli and A. Senol, Sensitivity reach on anomalous Higgs couplings via triphoton production for the post-LHC circular high-energy hadron colliders , 2005.08760

View on Pith Pith review arXiv 2005
[5]

J. Aguilar-Saavedra, J. Cano and J. No, More light on Higgs ﬂavor at the LHC: Higgs couplings to light quarks through h +γ production, 2008.12538

View on Pith Pith review arXiv 2008
[6]

A. Zerwekh, C. Dib and R. Rosenfeld, Triple photon production at the Tevatron in technicolor models, Phys. Lett. B 549 (2002) 154 [ hep-ph/0207270]

View on Pith Pith review arXiv 2002
[7]

N. Toro and I. Yavin, Multiphotons and photon jets from new heavy vector bosons , Phys. Rev. D 86 (2012) 055005 [ 1202.6377]

View on Pith Pith review arXiv 2012
[8]

G. Das and P. Mathews, Neutral Triple Vector Boson Production in Randall-Sundrum Model at the LHC , Phys. Rev. D 92 (2015) 094034 [ 1507.08857]

View on Pith Pith review arXiv 2015
[9]

C. Anastasiou, E. Glover and M. Tejeda-Yeomans, Two loop QED and QCD corrections to massless fermion boson scattering , Nucl. Phys. B 629 (2002) 255 [ hep-ph/0201274]

View on Pith Pith review arXiv 2002
[11]

S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, Diphoton production at hadron colliders: a fully-diﬀerential QCD calculation at NNLO , Phys. Rev. Lett. 108 (2012) 072001 [1110.2375]

View on Pith Pith review arXiv 2012
[12]

J. M. Campbell, R. Ellis, Y. Li and C. Williams, Predictions for diphoton production at the LHC through NNLO in QCD , JHEP 07 (2016) 148 [ 1603.02663]

View on Pith Pith review arXiv 2016
[13]

T. Gehrmann, N. Glover, A. Huss and J. Whitehead, Scale and isolation sensitivity of diphoton distributions at the LHC , 2009.11310

View on Pith Pith review arXiv 2009
[14]

S. Alioli, A. Broggio, A. Gavardi, S. Kallweit, M. A. Lim, R. Nagar et al., Precise predictions for photon pair production matched to parton showers in GENEVA , 2010.10498

View on Pith Pith review arXiv 2010
[17]

ATLAS collaboration, M. Aaboud et al., Measurement of the production cross section of three isolated photons in pp collisions at√s = 8 TeV using the ATLAS detector , Phys. Lett. B 781 (2018) 55 [ 1712.07291]

View on Pith Pith review arXiv 2018
[18]

C. G. Papadopoulos, D. Tommasini and C. Wever, The Pentabox Master Integrals with the Simpliﬁed Diﬀerential Equations approach, JHEP 04 (2016) 078 [ 1511.09404]. – 25 –

View on Pith Pith review arXiv 2016
[19]

T. Gehrmann, J. Henn and N. Lo Presti, Analytic form of the two-loop planar ﬁve-gluon all-plus-helicity amplitude in QCD , Phys. Rev. Lett. 116 (2016) 062001 [ 1511.05409]

View on Pith Pith review arXiv 2016
[20]

S. Abreu, L. J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop ﬁve-point amplitude inN = 4 super-Yang-Mills theory, Phys. Rev. Lett. 122 (2019) 121603 [ 1812.08941]

View on Pith Pith review arXiv 2019
[21]

D. Chicherin, T. Gehrmann, J. Henn, P. Wasser, Y. Zhang and S. Zoia, All Master Integrals for Three-Jet Production at Next-to-Next-to-Leading Order , Phys. Rev. Lett. 123 (2019) 041603 [1812.11160]

View on Pith Pith review arXiv 2019
[22]

T. Gehrmann, J. Henn and N. Lo Presti, Pentagon functions for massless planar scattering amplitudes, JHEP 10 (2018) 103 [ 1807.09812]

View on Pith Pith review arXiv 2018
[24]

K. J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93 (2016) 041701 [ 1511.01071]

View on Pith Pith review arXiv 2016
[25]

H. Ita, Two-loop Integrand Decomposition into Master Integrals and Surface Terms , Phys. Rev. D 94 (2016) 116015 [ 1510.05626]

View on Pith Pith review arXiv 2016
[26]

T. Peraro, Scattering amplitudes over ﬁnite ﬁelds and multivariate functional reconstruction , JHEP 12 (2016) 030 [ 1608.01902]

View on Pith Pith review arXiv 2016
[27]

A. Georgoudis, K. J. Larsen and Y. Zhang, Azurite: An algebraic geometry based package for ﬁnding bases of loop integrals , Comput. Phys. Commun. 221 (2017) 203 [ 1612.04252]

View on Pith Pith review arXiv 2017
[28]

A. Georgoudis, K. J. Larsen and Y. Zhang, Cristal and Azurite: new tools for integration-by-parts reductions, PoS RADCOR2017 (2017) 020 [ 1712.07510]

View on Pith Pith review arXiv 2017
[29]

H. A. Chawdhry, M. A. Lim and A. Mitov, Two-loop ﬁve-point massless QCD amplitudes within the integration-by-parts approach, Phys. Rev. D 99 (2019) 076011 [ 1805.09182]

View on Pith Pith review arXiv 2019
[30]

J. B¨ ohm, A. Georgoudis, K. J. Larsen, H. Sch¨ onemann and Y. Zhang,Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections , JHEP 09 (2018) 024 [ 1805.01873]

View on Pith Pith review arXiv 2018
[31]

T. Peraro, FiniteFlow: multivariate functional reconstruction using ﬁnite ﬁelds and dataﬂow graphs, JHEP 07 (2019) 031 [ 1905.08019]

View on Pith Pith review arXiv 2019
[32]

T. Peraro and L. Tancredi, Physical projectors for multi-leg helicity amplitudes , JHEP 07 (2019) 114 [ 1906.03298]

View on Pith Pith review arXiv 2019
[33]

D. Bendle, J. B¨ ohm, W. Decker, A. Georgoudis, F.-J. Pfreundt, M. Rahn et al., Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space , JHEP 02 (2020) 079 [ 1908.04301]

View on Pith Pith review arXiv 2020
[34]

Y. Wang, Z. Li and N. Ul Basat, Direct reduction of multiloop multiscale scattering amplitudes, Phys. Rev. D 101 (2020) 076023 [ 1901.09390]

View on Pith Pith review arXiv 2020
[35]

X. Guan, X. Liu and Y.-Q. Ma, Complete reduction of two-loop ﬁve-light-parton scattering amplitudes, Chin. Phys. C 44 (2020) 9 [ 1912.09294]

View on Pith Pith review arXiv 2020
[36]

S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, M. Kraus, B. Page et al., Caravel: A C++ Framework for the Computation of Multi-Loop Amplitudes with Numerical Unitarity , 2009.11957

View on Pith Pith review arXiv 2009
[37]

A. von Manteuﬀel and R. M. Schabinger, A novel approach to integration by parts reduction , Phys. Lett. B 744 (2015) 101 [ 1406.4513]. – 26 –

View on Pith Pith review arXiv 2015
[38]

J. Klappert and F. Lange, Reconstructing rational functions with FireFly , Comput. Phys. Commun. 247 (2020) 106951 [ 1904.00009]

View on Pith Pith review arXiv 2020
[39]

J. Klappert, S. Y. Klein and F. Lange, Interpolation of Dense and Sparse Rational Functions and other Improvements in FireFly, 2004.01463

View on Pith Pith review arXiv 2004
[40]

J. Klappert, F. Lange, P. Maierh¨ ofer and J. Usovitsch, Integral Reduction with Kira 2.0 and Finite Field Methods , 2008.06494

View on Pith Pith review arXiv 2008
[41]

S. Badger, H. Frellesvig and Y. Zhang, A Two-Loop Five-Gluon Helicity Amplitude in QCD , JHEP 12 (2013) 045 [ 1310.1051]

View on Pith Pith review arXiv 2013
[42]

S. Badger, C. Brønnum-Hansen, H. B. Hartanto and T. Peraro, First look at two-loop ﬁve-gluon scattering in QCD , Phys. Rev. Lett. 120 (2018) 092001 [ 1712.02229]

View on Pith Pith review arXiv 2018
[43]

S. Abreu, F. Febres Cordero, H. Ita, B. Page and M. Zeng, Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity , Phys. Rev. D 97 (2018) 116014 [ 1712.03946]

View on Pith Pith review arXiv 2018
[44]

S. Badger, C. Brønnum-Hansen, T. Gehrmann, H. B. Hartanto, J. Henn, N. A. Lo Presti et al., Applications of integrand reduction to two-loop ﬁve-point scattering amplitudes in QCD, PoS LL2018 (2018) 006 [ 1807.09709]

View on Pith Pith review arXiv 2018
[45]

S. Abreu, F. Febres Cordero, H. Ita, B. Page and V. Sotnikov, Planar Two-Loop Five-Parton Amplitudes from Numerical Unitarity , JHEP 11 (2018) 116 [ 1809.09067]

View on Pith Pith review arXiv 2018
[46]

S. Badger, C. Brønnum-Hansen, H. B. Hartanto and T. Peraro, Analytic helicity amplitudes for two-loop ﬁve-gluon scattering: the single-minus case , JHEP 01 (2019) 186 [ 1811.11699]

View on Pith Pith review arXiv 2019
[47]

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]

View on Pith Pith review arXiv 2019
[48]

S. Badger, D. Chicherin, T. Gehrmann, G. Heinrich, J. Henn, T. Peraro et al., Analytic form of the full two-loop ﬁve-gluon all-plus helicity amplitude , Phys. Rev. Lett. 123 (2019) 071601 [1905.03733]

View on Pith Pith review arXiv 2019
[50]

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

View on Pith Pith review arXiv 2018
[51]

S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page and M. Zeng, Two-Loop Four-Gluon Amplitudes from Numerical Unitarity , Phys. Rev. Lett. 119 (2017) 142001 [ 1703.05273]

View on Pith Pith review arXiv 2017
[52]

https://gitlab.com/five-point-amplitudes/FivePointAmplitudes-cpp.git

View on Pith
[53]

S. Weinzierl, Does one need the O(epsilon)- and O(epsilonˆ2)-terms of one-loop amplitudes in an NNLO calculation ? , Phys. Rev. D 84 (2011) 074007 [ 1107.5131]

View on Pith Pith review arXiv 2011
[54]

S. Catani, The Singular behavior of QCD amplitudes at two loop order , Phys. Lett. B 427 (1998) 161 [ hep-ph/9802439]

View on Pith Pith review arXiv 1998
[55]

S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier and B. Page, Subleading Poles in the Numerical Unitarity Method at Two Loops , Phys. Rev. D 95 (2017) 096011 [ 1703.05255]

View on Pith Pith review arXiv 2017
[56]

A. Ochirov and B. Page, Full Colour for Loop Amplitudes in Yang-Mills Theory , JHEP 02 (2017) 100 [ 1612.04366]. – 27 –

View on Pith Pith review arXiv 2017
[57]

A. Ochirov and B. Page, Multi-Quark Colour Decompositions from Unitarity , JHEP 10 (2019) 058 [ 1908.02695]

View on Pith Pith review arXiv 2019
[58]

F. Anger and V. Sotnikov, On the Dimensional Regularization of QCD Helicity Amplitudes With Quarks, 1803.11127

View on Pith Pith review arXiv
[59]

V. Sotnikov, Scattering amplitudes with the multi-loop numerical unitarity method , Ph.D. thesis, Freiburg U., 9, 2019. 10.6094/UNIFR/151540

View on Pith doi:10.6094/unifr/151540 2019
[60]

F. A. Berends and W. Giele, Recursive Calculations for Processes with n Gluons , Nucl. Phys. B 306 (1988) 759

View on Pith 1988
[61]

A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes , JHEP 05 (2013) 135 [0905.1473]

View on Pith Pith review arXiv 2013
[62]

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]

View on Pith Pith review arXiv 2010
[63]

C. Duhr, H. Gangl and J. R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [ 1110.0458]

View on Pith Pith review arXiv 2012
[64]

C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes , JHEP 08 (2012) 043 [ 1203.0454]

View on Pith Pith review arXiv 2012
[65]

D. Chicherin, J. Henn and V. Mitev, Bootstrapping pentagon functions, JHEP 05 (2018) 164 [1712.09610]

View on Pith Pith review arXiv 2018
[66]

F. Buccioni, J.-N. Lang, J. M. Lindert, P. Maierh¨ ofer, S. Pozzorini, H. Zhang et al., OpenLoops 2, Eur. Phys. J. C 79 (2019) 866 [ 1907.13071]

View on Pith Pith review arXiv 2019
[67]

B. Ruijl, T. Ueda and J. Vermaseren, FORM version 4.2 , 1707.06453

View on Pith Pith review arXiv
[68]

J. Kuipers, T. Ueda and J. Vermaseren, Code Optimization in FORM , Comput. Phys. Commun. 189 (2015) 1 [ 1310.7007]

View on Pith Pith review arXiv 2015
[69]

W. T. Giele, Z. Kunszt and K. Melnikov, Full one-loop amplitudes from tree amplitudes , JHEP 04 (2008) 049 [ 0801.2237]

View on Pith Pith review arXiv 2008
[70]

M. Hidding, DiﬀExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions , 2006.05510

View on Pith Pith review arXiv 2006
[71]

Z. Bern, V. Del Duca, W. B. Kilgore and C. R. Schmidt, The infrared behavior of one loop QCD amplitudes at next-to-next-to leading order , Phys. Rev. D 60 (1999) 116001 [hep-ph/9903516]. – 28 –

View on Pith Pith review arXiv 1999

Forward citations

Cited by 14 Pith papers

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Intersection numbers between dual forms and Feynman integrands yield master-integral coefficients directly from unitarity cuts, demonstrated on one-loop four- and five-gluon amplitudes.
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How later papers used it

47 later papers cite it

Evidence: some. 40 of the citing papers share no author with it, and 2 of those carry a confident label from Pith's citation classifier.

Pith read 13 of the 69 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 papers by the same authors are listed below: 3 used its data or values and 4 compared their own results with it. 3 of them appear under more than one label.

Uses its data or values 3 papers · 3 by the same authors

By the same authors

[ref] for the kinematic invariants.

Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders (2023) (same authors) · Reference Values · 0.99
In order to show the benefits of our approach in a two-loop five-point context, we applied it to the finite remainders of the two-loop three-photon production amplitudes of Ref. [ref] .

Constructing Compact Ans\"atze for Scattering Amplitudes (2022) (same authors) · Application to Leading-Color Two-Loop Tri-Photon Production Amplitudes · 0.82
As a proof-of-concept application of our approach, we reconsider the collection of pentagon-function remainder coefficients for the leading-color process $0→qq̅γγγ$ at two loops, obtained in Ref. [ref] .

Ans\"atze for Scattering Amplitudes from $p$-adic Numbers and Algebraic Geometry (2022) (same authors) · Application to Two-Loop $0 →qq̅γγγ$ Finite-Remainder Coefficients · 0.88

Compares with it 4 of 6 papers with this label · 4 by the same authors

By the same authors

For the planar contributions $R^(2,0)$ and $R^(2,)$, we find full agreement between our result and a previous calculation [ref] .

Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders (2023) (same authors) · Validation · 1.00
Comparing the size of the Ansatz used in Ref. [ref] , $dim(ℳ_d̃,0⃗)$, with the size of Ansatz in spinor variables taking into account numerator vanishing constraints, $dim(ℳ_d,ϕ⃗) ∩𝔍$, we see an improvement of a factor of $73$ in the worst case.

Constructing Compact Ans\"atze for Scattering Amplitudes (2022) (same authors) · Application to Leading-Color Two-Loop Tri-Photon Production Amplitudes · 0.95
We note nevertheless that, compared to e.g. the case of triphoton production [ref] , the evaluation of $H^(2)$ for five-gluon channels receives contributions from a large number of partial amplitudes, and therefore the time spent in the evaluation of the other components in <ref> becomes noticeable.

Leading-Color Two-Loop QCD Corrections for Three-Jet Production at Hadron Colliders (2021) (same authors) · Numerical Evaluation · 0.94
Finally, we note that the rational numbers appearing in $M_ji$ are smaller than those appearing in the analogous matrix in the work of Ref. [ref] , with the largest denominators being $∼10^7$ and $∼10^9$ respectively.

Ans\"atze for Scattering Amplitudes from $p$-adic Numbers and Algebraic Geometry (2022) (same authors) · Application to Two-Loop $0 →qq̅γγγ$ Finite-Remainder Coefficients · 0.99

How the literature uses it

46 arXiv papers cite this in 68 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We calculate all planar contributions to the two-loop massless helicity amplitudes for the process $q\bar q\to \gamma\gamma\gamma$.

Two-loop leading-color helicity amplitudes for three-photon production at the LHC · 0.81

Papers citing it · 44 sentences without a confident label

Uses its method or tool 1 paper

Using a package such as [ref] it is straightforward to match at a random phase-space point the product $r̃_j M_ji$ to the functions $r_i$ as given in Ref. [ref] .

Ans\"atze for Scattering Amplitudes from $p$-adic Numbers and Algebraic Geometry · Implementation and Results · 0.92

Uses its data or values 3 papers

[ref] for the kinematic invariants.

Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders · Reference Values · 0.99
In order to show the benefits of our approach in a two-loop five-point context, we applied it to the finite remainders of the two-loop three-photon production amplitudes of Ref. [ref] .

Constructing Compact Ans\"atze for Scattering Amplitudes · Application to Leading-Color Two-Loop Tri-Photon Production Amplitudes · 0.82
As a proof-of-concept application of our approach, we reconsider the collection of pentagon-function remainder coefficients for the leading-color process $0→qq̅γγγ$ at two loops, obtained in Ref. [ref] .

Ans\"atze for Scattering Amplitudes from $p$-adic Numbers and Algebraic Geometry · Application to Two-Loop $0 →qq̅γγγ$ Finite-Remainder Coefficients · 0.88

Compares with it 6 papers

[ref] solves the IBP equations numerically for each point in which the amplitude is evaluated and being reconstructed.

Two-loop leading-color helicity amplitudes for three-photon production at the LHC · Results · 0.83
For the planar contributions $R^(2,0)$ and $R^(2,)$, we find full agreement between our result and a previous calculation [ref] .

Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders · Validation · 1.00
Comparing the size of the Ansatz used in Ref. [ref] , $dim(ℳ_d̃,0⃗)$, with the size of Ansatz in spinor variables taking into account numerator vanishing constraints, $dim(ℳ_d,ϕ⃗) ∩𝔍$, we see an improvement of a factor of $73$ in the worst case.

Constructing Compact Ans\"atze for Scattering Amplitudes · Application to Leading-Color Two-Loop Tri-Photon Production Amplitudes · 0.95

Cites it as background 11 papers

This can be achieved either computing individual Feynman diagrams or using recursion relations; * reduction of two-loop amplitudes in terms of Master Integrals [ref] .

Two-Loop Amplitude Reduction with HELAC · Introduction · 0.92
Thanks to all these developments, full colour two-loop analytic results [ref] are now available for all massless five-particle scattering processes of interest for the LHC, superseding the leading colour results [ref] .

Full-colour double-virtual amplitudes for associated production of a Higgs boson with a bottom-quark pair at the LHC · Introduction · 0.80
The remarkable progress in the computation of massless two-loop five-particle scattering amplitudes in the last few years [ref] has recently culminated in the first NNLO theoretical predictions for $2→3$ processes [ref] .

Pentagon Functions for One-Mass Planar Scattering Amplitudes · Introduction · 0.96

What it cites · 55 sentences without a confident label

Uses the method or tool of 16 papers

The approach starts from a parametrization of the integrand of the amplitude $A^(2,j)_h(ℓ_l)$ in terms of master integrands and surface terms [ref] ($ℓ_l$ denotes the loop momenta).

Two-loop Integrand Decomposition into Master Integrals and Surface Terms · Two-loop Numerical Unitarity · 0.99
To compute the functions $A^(2,0)_+++$, $A^(2,)_+++$, $A^(2,0)_-++$ and $A^(2,)_-++$ defined in <ref>, we use the framework of two-loop numerical unitarity [ref] coupled with functional reconstruction techniques.

Subleading Poles in the Numerical Unitarity Method at Two Loops · Calculation of Helicity Amplitudes · 1.00
To compute the functions $A^(2,0)_+++$, $A^(2,)_+++$, $A^(2,0)_-++$ and $A^(2,)_-++$ defined in <ref>, we use the framework of two-loop numerical unitarity [ref] coupled with functional reconstruction techniques.

Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity · Calculation of Helicity Amplitudes · 1.00

Uses data or values from 2 papers

To demonstrate the stability of our results, we compare the numerical evaluation of $H^(2)$ in double precision, which we denote $H^(2)_double$, with the evaluation in quadruple precision, which we denote $H^(2)_quad$, on a sample of 90000 phase-space points from the distribution employed in [ref] for the computation of predictions at the center-of-mass energy of 8 TeV.

Triphoton production at hadron colliders in NNLO QCD · Numerical Evaluation · 0.88
The phase-space cuts correspond to the ones used in the ATLAS 8 TeV measurement [ref] (we refer to the table 1 of [ref] for the explicit definitions).

Measurement of the production cross section of three isolated photons in $pp$ collisions at $\sqrt{s}$ = 8 TeV using the ATLAS detector · Numerical Evaluation · 0.98

Compares with 4 papers

For the five-point amplitudes, we reproduced the results obtained from [ref] to order $ϵ^0$ and numerically verified all the relations in <ref>.

OpenLoops 2 · Validation · 1.00
[ref] , but the complexity of the final result is much higher than in the five-parton remainder case (where the number of independent rational functions was at most $𝒪(120)$).

Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD · Functional Reconstruction of Master Integral Coefficients · 0.99
[ref] , and we found full agreement with the quoted remainders.

Two-loop QCD helicity amplitudes for massless quark-massless gauge boson scattering · Validation · 1.00

Cites as background 26 papers

A basis of master integrals for planar five-point massless amplitudes is known [ref] .

Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD · Pentagon Functions · 0.80
A basis of master integrals for planar five-point massless amplitudes is known [ref] .

The Pentabox Master Integrals with the Simplified Differential Equations approach · Pentagon Functions · 0.80
A basis of master integrals relevant for the scattering of five massless particles has been known for a number of years in the planar limit [ref] , and more recently including non-planar topologies [ref] .

The two-loop five-point amplitude in $\mathcal{N} =4$ super-Yang-Mills theory · Introduction · 0.95

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [16]

S. Kallweit, V. Sotnikov and M. Wiesemann, Triphoton production at hadron colliders in NNLO QCD, Phys. Lett. B 812 (2021) 136013 [ 2010.04681]

View on Pith Pith review arXiv 2021

[2] [15]

H. A. Chawdhry, M. L. Czakon, A. Mitov and R. Poncelet, NNLO QCD corrections to three-photon production at the LHC , JHEP 02 (2020) 057 [ 1911.00479]

View on Pith Pith review arXiv 2020

[3] [49]

S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, B. Page and V. Sotnikov, Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD , JHEP 05 (2019) 084 [1904.00945]

View on Pith Pith review arXiv 2019

[4] [23]

D. Chicherin and V. Sotnikov, Pentagon Functions for Scattering of Five Massless Particles , JHEP 20 (2020) 167 [ 2009.07803]

View on Pith Pith review arXiv 2020

[5] [10]

E. Glover and M. Tejeda-Yeomans, Two loop QCD helicity amplitudes for massless quark massless gauge boson scattering , JHEP 06 (2003) 033 [ hep-ph/0304169]

View on Pith Pith review arXiv 2003

[6] [1]

S. Amoroso et al., Les Houches 2019: Physics at TeV Colliders: Standard Model Working Group Report, in 11th Les Houches Workshop on Physics at TeV Colliders: PhysTeV Les Houches, 3, 2020, 2003.01700

View on Pith Pith review arXiv 2019

[7] [2]

ATLAS collaboration, G. Aad et al., Search for new phenomena in events with at least three photons collected in pp collisions at√s = 8 TeV with the ATLAS detector , Eur. Phys. J. C 76 (2016) 210 [ 1509.05051]

View on Pith Pith review arXiv 2016

[8] [3]

H. Denizli, K. Oyulmaz and A. Senol, Testing for observability of Higgs eﬀective couplings in triphoton production at FCC-hh , J. Phys. G 46 (2019) 105007 [ 1901.04784]

View on Pith Pith review arXiv 2019

[9] [4]

H. Denizli and A. Senol, Sensitivity reach on anomalous Higgs couplings via triphoton production for the post-LHC circular high-energy hadron colliders , 2005.08760

View on Pith Pith review arXiv 2005

[10] [5]

J. Aguilar-Saavedra, J. Cano and J. No, More light on Higgs ﬂavor at the LHC: Higgs couplings to light quarks through h +γ production, 2008.12538

View on Pith Pith review arXiv 2008

[11] [6]

A. Zerwekh, C. Dib and R. Rosenfeld, Triple photon production at the Tevatron in technicolor models, Phys. Lett. B 549 (2002) 154 [ hep-ph/0207270]

View on Pith Pith review arXiv 2002

[12] [7]

N. Toro and I. Yavin, Multiphotons and photon jets from new heavy vector bosons , Phys. Rev. D 86 (2012) 055005 [ 1202.6377]

View on Pith Pith review arXiv 2012

[13] [8]

G. Das and P. Mathews, Neutral Triple Vector Boson Production in Randall-Sundrum Model at the LHC , Phys. Rev. D 92 (2015) 094034 [ 1507.08857]

View on Pith Pith review arXiv 2015

[14] [9]

C. Anastasiou, E. Glover and M. Tejeda-Yeomans, Two loop QED and QCD corrections to massless fermion boson scattering , Nucl. Phys. B 629 (2002) 255 [ hep-ph/0201274]

View on Pith Pith review arXiv 2002

[15] [11]

S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, Diphoton production at hadron colliders: a fully-diﬀerential QCD calculation at NNLO , Phys. Rev. Lett. 108 (2012) 072001 [1110.2375]

View on Pith Pith review arXiv 2012

[16] [12]

J. M. Campbell, R. Ellis, Y. Li and C. Williams, Predictions for diphoton production at the LHC through NNLO in QCD , JHEP 07 (2016) 148 [ 1603.02663]

View on Pith Pith review arXiv 2016

[17] [13]

T. Gehrmann, N. Glover, A. Huss and J. Whitehead, Scale and isolation sensitivity of diphoton distributions at the LHC , 2009.11310

View on Pith Pith review arXiv 2009

[18] [14]

S. Alioli, A. Broggio, A. Gavardi, S. Kallweit, M. A. Lim, R. Nagar et al., Precise predictions for photon pair production matched to parton showers in GENEVA , 2010.10498

View on Pith Pith review arXiv 2010

[19] [17]

ATLAS collaboration, M. Aaboud et al., Measurement of the production cross section of three isolated photons in pp collisions at√s = 8 TeV using the ATLAS detector , Phys. Lett. B 781 (2018) 55 [ 1712.07291]

View on Pith Pith review arXiv 2018

[20] [18]

C. G. Papadopoulos, D. Tommasini and C. Wever, The Pentabox Master Integrals with the Simpliﬁed Diﬀerential Equations approach, JHEP 04 (2016) 078 [ 1511.09404]. – 25 –

View on Pith Pith review arXiv 2016

[21] [19]

T. Gehrmann, J. Henn and N. Lo Presti, Analytic form of the two-loop planar ﬁve-gluon all-plus-helicity amplitude in QCD , Phys. Rev. Lett. 116 (2016) 062001 [ 1511.05409]

View on Pith Pith review arXiv 2016

[22] [20]

S. Abreu, L. J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop ﬁve-point amplitude inN = 4 super-Yang-Mills theory, Phys. Rev. Lett. 122 (2019) 121603 [ 1812.08941]

View on Pith Pith review arXiv 2019

[23] [21]

D. Chicherin, T. Gehrmann, J. Henn, P. Wasser, Y. Zhang and S. Zoia, All Master Integrals for Three-Jet Production at Next-to-Next-to-Leading Order , Phys. Rev. Lett. 123 (2019) 041603 [1812.11160]

View on Pith Pith review arXiv 2019

[24] [22]

T. Gehrmann, J. Henn and N. Lo Presti, Pentagon functions for massless planar scattering amplitudes, JHEP 10 (2018) 103 [ 1807.09812]

View on Pith Pith review arXiv 2018

[25] [24]

K. J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93 (2016) 041701 [ 1511.01071]

View on Pith Pith review arXiv 2016

[26] [25]

H. Ita, Two-loop Integrand Decomposition into Master Integrals and Surface Terms , Phys. Rev. D 94 (2016) 116015 [ 1510.05626]

View on Pith Pith review arXiv 2016

[27] [26]

T. Peraro, Scattering amplitudes over ﬁnite ﬁelds and multivariate functional reconstruction , JHEP 12 (2016) 030 [ 1608.01902]

View on Pith Pith review arXiv 2016

[28] [27]

A. Georgoudis, K. J. Larsen and Y. Zhang, Azurite: An algebraic geometry based package for ﬁnding bases of loop integrals , Comput. Phys. Commun. 221 (2017) 203 [ 1612.04252]

View on Pith Pith review arXiv 2017

[29] [28]

A. Georgoudis, K. J. Larsen and Y. Zhang, Cristal and Azurite: new tools for integration-by-parts reductions, PoS RADCOR2017 (2017) 020 [ 1712.07510]

View on Pith Pith review arXiv 2017

[30] [29]

H. A. Chawdhry, M. A. Lim and A. Mitov, Two-loop ﬁve-point massless QCD amplitudes within the integration-by-parts approach, Phys. Rev. D 99 (2019) 076011 [ 1805.09182]

View on Pith Pith review arXiv 2019

[31] [30]

J. B¨ ohm, A. Georgoudis, K. J. Larsen, H. Sch¨ onemann and Y. Zhang,Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections , JHEP 09 (2018) 024 [ 1805.01873]

View on Pith Pith review arXiv 2018

[32] [31]

T. Peraro, FiniteFlow: multivariate functional reconstruction using ﬁnite ﬁelds and dataﬂow graphs, JHEP 07 (2019) 031 [ 1905.08019]

View on Pith Pith review arXiv 2019

[33] [32]

T. Peraro and L. Tancredi, Physical projectors for multi-leg helicity amplitudes , JHEP 07 (2019) 114 [ 1906.03298]

View on Pith Pith review arXiv 2019

[34] [33]

D. Bendle, J. B¨ ohm, W. Decker, A. Georgoudis, F.-J. Pfreundt, M. Rahn et al., Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space , JHEP 02 (2020) 079 [ 1908.04301]

View on Pith Pith review arXiv 2020

[35] [34]

Y. Wang, Z. Li and N. Ul Basat, Direct reduction of multiloop multiscale scattering amplitudes, Phys. Rev. D 101 (2020) 076023 [ 1901.09390]

View on Pith Pith review arXiv 2020

[36] [35]

X. Guan, X. Liu and Y.-Q. Ma, Complete reduction of two-loop ﬁve-light-parton scattering amplitudes, Chin. Phys. C 44 (2020) 9 [ 1912.09294]

View on Pith Pith review arXiv 2020

[37] [36]

S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, M. Kraus, B. Page et al., Caravel: A C++ Framework for the Computation of Multi-Loop Amplitudes with Numerical Unitarity , 2009.11957

View on Pith Pith review arXiv 2009

[38] [37]

A. von Manteuﬀel and R. M. Schabinger, A novel approach to integration by parts reduction , Phys. Lett. B 744 (2015) 101 [ 1406.4513]. – 26 –

View on Pith Pith review arXiv 2015

[39] [38]

J. Klappert and F. Lange, Reconstructing rational functions with FireFly , Comput. Phys. Commun. 247 (2020) 106951 [ 1904.00009]

View on Pith Pith review arXiv 2020

[40] [39]

J. Klappert, S. Y. Klein and F. Lange, Interpolation of Dense and Sparse Rational Functions and other Improvements in FireFly, 2004.01463

View on Pith Pith review arXiv 2004

[41] [40]

J. Klappert, F. Lange, P. Maierh¨ ofer and J. Usovitsch, Integral Reduction with Kira 2.0 and Finite Field Methods , 2008.06494

View on Pith Pith review arXiv 2008

[42] [41]

S. Badger, H. Frellesvig and Y. Zhang, A Two-Loop Five-Gluon Helicity Amplitude in QCD , JHEP 12 (2013) 045 [ 1310.1051]

View on Pith Pith review arXiv 2013

[43] [42]

S. Badger, C. Brønnum-Hansen, H. B. Hartanto and T. Peraro, First look at two-loop ﬁve-gluon scattering in QCD , Phys. Rev. Lett. 120 (2018) 092001 [ 1712.02229]

View on Pith Pith review arXiv 2018

[44] [43]

S. Abreu, F. Febres Cordero, H. Ita, B. Page and M. Zeng, Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity , Phys. Rev. D 97 (2018) 116014 [ 1712.03946]

View on Pith Pith review arXiv 2018

[45] [44]

S. Badger, C. Brønnum-Hansen, T. Gehrmann, H. B. Hartanto, J. Henn, N. A. Lo Presti et al., Applications of integrand reduction to two-loop ﬁve-point scattering amplitudes in QCD, PoS LL2018 (2018) 006 [ 1807.09709]

View on Pith Pith review arXiv 2018

[46] [45]

S. Abreu, F. Febres Cordero, H. Ita, B. Page and V. Sotnikov, Planar Two-Loop Five-Parton Amplitudes from Numerical Unitarity , JHEP 11 (2018) 116 [ 1809.09067]

View on Pith Pith review arXiv 2018

[47] [46]

S. Badger, C. Brønnum-Hansen, H. B. Hartanto and T. Peraro, Analytic helicity amplitudes for two-loop ﬁve-gluon scattering: the single-minus case , JHEP 01 (2019) 186 [ 1811.11699]

View on Pith Pith review arXiv 2019

[48] [47]

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]

View on Pith Pith review arXiv 2019

[49] [48]

S. Badger, D. Chicherin, T. Gehrmann, G. Heinrich, J. Henn, T. Peraro et al., Analytic form of the full two-loop ﬁve-gluon all-plus helicity amplitude , Phys. Rev. Lett. 123 (2019) 071601 [1905.03733]

View on Pith Pith review arXiv 2019

[50] [50]

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

View on Pith Pith review arXiv 2018

[51] [51]

S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page and M. Zeng, Two-Loop Four-Gluon Amplitudes from Numerical Unitarity , Phys. Rev. Lett. 119 (2017) 142001 [ 1703.05273]

View on Pith Pith review arXiv 2017

[52] [52]

https://gitlab.com/five-point-amplitudes/FivePointAmplitudes-cpp.git

View on Pith

[53] [53]

S. Weinzierl, Does one need the O(epsilon)- and O(epsilonˆ2)-terms of one-loop amplitudes in an NNLO calculation ? , Phys. Rev. D 84 (2011) 074007 [ 1107.5131]

View on Pith Pith review arXiv 2011

[54] [54]

S. Catani, The Singular behavior of QCD amplitudes at two loop order , Phys. Lett. B 427 (1998) 161 [ hep-ph/9802439]

View on Pith Pith review arXiv 1998

[55] [55]

S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier and B. Page, Subleading Poles in the Numerical Unitarity Method at Two Loops , Phys. Rev. D 95 (2017) 096011 [ 1703.05255]

View on Pith Pith review arXiv 2017

[56] [56]

A. Ochirov and B. Page, Full Colour for Loop Amplitudes in Yang-Mills Theory , JHEP 02 (2017) 100 [ 1612.04366]. – 27 –

View on Pith Pith review arXiv 2017

[57] [57]

A. Ochirov and B. Page, Multi-Quark Colour Decompositions from Unitarity , JHEP 10 (2019) 058 [ 1908.02695]

View on Pith Pith review arXiv 2019

[58] [58]

F. Anger and V. Sotnikov, On the Dimensional Regularization of QCD Helicity Amplitudes With Quarks, 1803.11127

View on Pith Pith review arXiv

[59] [59]

V. Sotnikov, Scattering amplitudes with the multi-loop numerical unitarity method , Ph.D. thesis, Freiburg U., 9, 2019. 10.6094/UNIFR/151540

View on Pith doi:10.6094/unifr/151540 2019

[60] [60]

F. A. Berends and W. Giele, Recursive Calculations for Processes with n Gluons , Nucl. Phys. B 306 (1988) 759

View on Pith 1988

[61] [61]

A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes , JHEP 05 (2013) 135 [0905.1473]

View on Pith Pith review arXiv 2013

[62] [62]

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]

View on Pith Pith review arXiv 2010

[63] [63]

C. Duhr, H. Gangl and J. R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [ 1110.0458]

View on Pith Pith review arXiv 2012

[64] [64]

C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes , JHEP 08 (2012) 043 [ 1203.0454]

View on Pith Pith review arXiv 2012

[65] [65]

D. Chicherin, J. Henn and V. Mitev, Bootstrapping pentagon functions, JHEP 05 (2018) 164 [1712.09610]

View on Pith Pith review arXiv 2018

[66] [66]

F. Buccioni, J.-N. Lang, J. M. Lindert, P. Maierh¨ ofer, S. Pozzorini, H. Zhang et al., OpenLoops 2, Eur. Phys. J. C 79 (2019) 866 [ 1907.13071]

View on Pith Pith review arXiv 2019

[67] [67]

B. Ruijl, T. Ueda and J. Vermaseren, FORM version 4.2 , 1707.06453

View on Pith Pith review arXiv

[68] [68]

J. Kuipers, T. Ueda and J. Vermaseren, Code Optimization in FORM , Comput. Phys. Commun. 189 (2015) 1 [ 1310.7007]

View on Pith Pith review arXiv 2015

[69] [69]

W. T. Giele, Z. Kunszt and K. Melnikov, Full one-loop amplitudes from tree amplitudes , JHEP 04 (2008) 049 [ 0801.2237]

View on Pith Pith review arXiv 2008

[70] [70]

M. Hidding, DiﬀExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions , 2006.05510

View on Pith Pith review arXiv 2006

[71] [71]

Z. Bern, V. Del Duca, W. B. Kilgore and C. R. Schmidt, The infrared behavior of one loop QCD amplitudes at next-to-next-to leading order , Phys. Rev. D 60 (1999) 116001 [hep-ph/9903516]. – 28 –

View on Pith Pith review arXiv 1999