One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits

D.A. Kosower; D.C. Dunbar; L. Dixon; Z. Bern

REVIEW

One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits

Z. Bern, L. Dixon, D.C. Dunbar, D.A. Kosower

High Energy Physics - Phenomenology · Submitted 4 Mar 1994 · 49 references

Pith's verdict: The all-n one-loop MHV N=4 formula and the unitarity-plus-collinear bootstrap are the real contributions and they hold up; the §6 'proof' is a sketch, but the result is independently checked at n=6.

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CitationD.A. Kosower, D.C. Dunbar, L. Dixon, Z. Bern. "One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits." arXiv:hep-ph/9403226 (1994). https://pith.science/paper/OPIMNWEH

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Editor's note

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Worth your time. The durable contributions here are the all-n one-loop MHV amplitude in N=4 SYM (eqs. 4.3–4.8, equivalently the box sum in 4.19) and the bootstrap machinery itself — unitarity cuts to fix the analytic structure, collinear limits to fix what the cuts miss. That combination became standard, and this is where it starts. Appendix II's one-loop splitting amplitudes are also a resource people still pull off the shelf. The paper is honest about scope in two places the classifier missed: end of §5 (the factorized ansatz fails for the non-MHV six-point amplitude, verified by direct string calculation) and the conclusions (for N=4 the collinear limits aren't actually needed). So the record's 'no limitations found' is wrong; the authors state them plainly. Credit for that — it's the opposite of overclaiming, and it makes the rest easier to trust. §5 is the solid pillar. The ρ² identity (5.4)–(5.6) is right, the Schouten rearrangement into four cut boxes is explicit, and the observation that only Parke-Taylor trees are needed for MHV cuts is the reason this works for arbitrary n. Where it thins is §6. The uniqueness argument — that Σ c_i I^box_i = polynomial forces all c_i = 0 — is worked through for exactly one coefficient, the three-mass box, and then closed with 'one can continue in this way.' The hard two-mass box (I.5c) and the four-mass box (I.5e) are never treated, and degenerate kinematics where invariants coincide aren't addressed. Since the abstract says 'we prove' [abstract:s2], that gap is load-bearing for the claim as worded. It's a rigor gap, not a correctness gap: the structural premise (only scalar boxes survive in N=4) is standard and correct, and the n=6 string-based check is genuine independent evidence. A paragraph or two of enumeration, or softening 'prove' to 'verify for all n given the standard reduction,' closes it. One minor point: the q→qg and g→qq̄ loop splitting amplitudes in appendix II are extracted from ref. [42], listed 'in preparation,' and the g→gg universality leans on ref. [28], a preprint. The N=4 piece (3.3)/(3.5) is derived in-paper from V^g5, so this is minor, but those two entries aren't independently checkable as printed. Who should read it: anyone doing one-loop multi-leg work, and anyone interested in where the modern amplitudes program got its footing. Send it out. Ask the referee specifically to press on §6 and to say whether the enumeration is routine or not.

Referee report

Significance

If correct, this is an important paper: it gives the first all-n one-loop amplitude in a four-dimensional gauge theory beyond the special all-plus helicity case, and it does so without a Feynman-diagram calculation. Several features deserve explicit credit. The result is checked independently at n=6 by a direct string-based calculation (Section 4, p. 13), which is a genuine cross-check rather than a restatement. The box-function representation in Eq. (4.19) is a parameter-free, explicitly cyclic object that later work can compare against directly. The splitting amplitudes in Appendix II, including the supersymmetric ratio r_S^{SUSY} of Eq. (3.5) and the Yukawa corrections of Eqs. (II.14)-(II.15), are reusable inputs for other calculations. The subleading-color formula of Eq. (7.2) is a clean structural statement. The paper is also honest about scope: Section 5 states explicitly that the ansatz fails for non-MHV configurations at six points, verified both by string-based calculation and by a direct cut, and Section 3 flags the cut-free ambiguities of Eqs. (3.6)-(3.7) rather than hiding them. I checked the algebraic identities that carry Section 5, namely the Schouten rearrangement (

Major objections (1)

[Section 6, Eq. (6.2)] The uniqueness step that closes the proof is sketched rather than carried out. The text works through exactly one coefficient, the three-mass box I^{3m} of Eq. (I.8d), using the ln(-t^{[n-r-r'-1]}_{i+r+r'}) term in the t^{[r]}_i cut, and then writes 'One can continue in this way to show that eq. (6.2) has only the trivial solution c_i = 0 for all coefficients.' No enumeration follows. In particular the hard two-mass box F^{2mh} of Eq. (I.5c), whose 1/epsilon^2 three-logarithm structure differs from the easy two-mass case, and the four-mass box F^{4m} of Eq. (I.5e) are never treated, and the possibility that boxes with different mass patterns combine into a cut-free function in every channel simultaneously is excluded only by the single worked example. Since the abstract claims 'we prove' (abstract:s2), this is the load-bearing soft spot. The fix is local: either add a short paragraph enumerating the remaining box topologies and identifying, for each, a kinematic invariant that appears in only that box and in only one cut, or soften the abstract and Section 6 to 'argue' or 'verify to all orders in the class of integrals that can appear.' The direct n=6 string-based check provides independent support but is not an all-n proof, so it does not by itself discharge this point.

Minor objections (7)

[Section 3, Eqs. (3.6)-(3.7)] The paper correctly identifies cut-free functions such as the epsilon-tensor ratio (3.6) and the logarithm sum (3.7) as additive ambiguities of the collinear bootstrap. It would strengthen Section 6 to say explicitly that these functions are excluded not by collinearity but by the scalar-box decomposition, since that is the actual mechanism and the current text leaves the reader to infer it.
[Section 7, Eq. (7.2)] The field-theory argument for the subleading-color formula is also a sketch: the cancellation of diagrams of types 1 and 2 is asserted to be arrangeable pairwise, with the string-based rules doing the real work and the field-theory version described as something 'one can construct.' This is a secondary result and does not affect the central claim, but the status of the field-theory proof should be stated plainly.
[Appendix II, Eqs. (II.10)-(II.15)] The loop-level q -> qg and g -> q qbar splitting amplitudes are extracted partly from ref. [42], listed as 'in preparation,' and the new Yukawa terms r_S^{Yukawa} of Eqs. (II.14)-(II.15) rest on that five-point calculation. The agreement with the published Giele-Glover results [33] is stated, which mitigates the concern, but the provenance of the new terms should be made explicit so a reader can tell which pieces are independently checkable from the cited literature.
[Section 5, below Eq. (5.11)] The sentence 'Since the box only contains only three independent momenta' has a duplicated word, and the argument that the epsilon-tensor term vanishes under Passarino-Veltman reduction deserves one more clause: the point is that a rank-one tensor box with only three independent external momenta reduces to scalar boxes whose coefficients are proportional to the contracted epsilon, which is antisymmetric and therefore zero. Stating this explicitly would make the step checkable at a glance.
[Section 4, Eq. (4.3)] The parity split between the odd-n and even-n forms of D_n and L_n, and the range of the r sum, are correct but take effort to verify. A sentence fixing the branch of the logarithms through the continuation -t^{[r]}_i > 0, and noting explicitly that V_n^g is real and cyclically invariant term by term, would help a reader who wants to check the collinear limit of Eq. (4.2) directly.
[Appendix I, Eqs. (I.5)-(I.8)] The index conventions shift between the main text and the appendix, for example t^{[r]}_{i-1} versus t^{[r]}_i in the arguments of the box functions, and the collinear-limit equations (I.9)-(I.16) relabel legs mod (n-1). A short index-convention table would reduce the chance of misreading these formulas.
[Section 4, opening paragraph] There is a typographical error in 'construct an satz' in the first sentence of Section 4, and a similar typesetting slip in 'The splitting amplitude s Split' in Section 3. These are cosmetic.

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The paper

The paper claims

A one-loop amplitude for any number of gluons can be pinned down by two consistency conditions — unitarity cuts and collinear limits — and the resulting closed form is proved correct in the maximally supersymmetric gauge theory.

The reading

The paper asks whether one can write down the one-loop correction to gluon scattering for an arbitrary number of external particles, without grinding through Feynman diagrams. Its answer is a bootstrap: the imaginary parts of the amplitude (its cuts) are computed exactly from tree-level Parke-Taylor formulas, and the behavior when two external gluons become collinear is imposed as a second constraint. In $N=4$ super-Yang-Mills, with the helicity pattern of the Parke-Taylor tree amplitudes, these two constraints turn out to fix everything, and the authors give the resulting all-$n$ expression. The reason a reader should care is practical: next-to-leading-order predictions for multi-jet production need exactly this kind of many-particle loop amplitude, and here one is obtained in closed form rather than one diagram at a time. The paper also hands over the one-loop 'splitting amplitudes' that govern how any one-loop helicity amplitude degrades as two legs merge, tools that apply well beyond the supersymmetric case.

What carries the argument

Two constraints do the work. The first is the Cutkosky unitarity cut: the discontinuity of the loop amplitude across a kinematic channel equals a phase-space integral of two tree amplitudes. Because the external helicities are MHV, only Parke-Taylor tree amplitudes are needed, and the whole cut reduces to an integral whose loop-momentum dependence sits in four spinor-product denominators — a cut hexagon. Using the Schouten identity and Passarino-Veltman reduction, that hexagon becomes four cut scalar boxes, with triangles and bubbles cancelling. The second constraint is the collinear limit: as two neighbouring legs merge, the one-loop amplitude must obey $A^{loop}_{n;1} → Σ_λ [ Split^{tree} A^{loop}_{n-1;1} + Split^{loop} A^{tree}_{n-1} ]$, and in $N=4$ the loop splitting function is proportional to the tree one with a helicity-independent ratio $r^{SUSY}_S(z,s) = -ε^{-2}(μ^2/(z(1-z)(-s)))^ε + 2ln z ln(1-z) - π^2/6$. Imposing this on the box decomposition is what forces the specific set of boxes that appears.

Extended reading notes

Core claim

The central claim is that the leading-color one-loop MHV amplitude for $n$ gluons in $N=4$ super-Yang-Mills factorizes as $A^{N=4}_{n;1} = c_Γ A^{tree}_n V^g_n$, where $A^{tree}_n$ is the Parke-Taylor tree amplitude and $V^g_n$ is a universal function of the momentum invariants $t^{[r]}_i = (k_i+...+k_{i+r-1})^2$ given explicitly in eqs. (4.3)-(4.8) — logarithms at every $n$, plus dilogarithms that first appear at six points. The same function is rewritten in eq. (4.19) as a sum of scalar box integrals, restricted to one-mass boxes and 'easy two-mass' boxes with only two diagonally opposite massless legs. The authors then prove the ansatz rather than merely propose it: they evaluate every unitarity cut directly, show the cut integrand collapses to a single cut hexagon that partial-fractions into four cut boxes, and argue that in $N=4$ the amplitude is a sum of scalar boxes alone, so the cuts determine every coefficient uniquely. The six-point case is checked independently by a direct string-based calculation.

Load-bearing premise

The proof that the cuts determine the amplitude uniquely rests on the claim that in $N=4$ super-Yang-Mills the amplitude is a sum of scalar box integrals only, with no triangle or bubble integrals and no cut-free rational terms left over.

Editorial extensions

If this is right

Any one-loop $n$-gluon MHV amplitude in $N=4$ super-Yang-Mills can be written down immediately from the tree amplitude and the universal function $V^g_n$, with no diagram-by-diagram work.
The same $V^g_n$ can be recycled: because a QCD gluon loop decomposes into an $N=4$ piece, an $N=1$ chiral piece, and a scalar piece, this result supplies one of the three ingredients of the corresponding QCD amplitude.
Supersymmetry Ward identities extend the result to amplitudes where some external gluons are replaced by gluinos or scalars, since the loop-level identities hold as well as the tree-level ones.
The subleading-color partial amplitudes $A_{n;c>1}$ are not independent: section 7 gives them as an explicit alternating sum over permutations of the leading-color amplitude, for any adjoint-representation loop.
The one-loop splitting amplitudes collected in appendix II let anyone check a candidate one-loop amplitude, or bootstrap a higher-point one, from known lower-point results.

Reading between the lines

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

The bootstrap pattern here — cuts fix the analytic structure, collinear limits fix the rational remainder — looks like a template that later became standard for computing multi-leg loop amplitudes; the paper is an early demonstration that the two constraints are complementary rather than redundant.
The uniqueness argument is really a statement about the integrand's tensor structure, not about the cuts: if one could show that any $N=4$ amplitude is a sum of scalar boxes by a route independent of string-based power counting, the proof would become self-contained rather than resting on prior machinery.
The explicit statement that the factorized form $A = c_Γ A^{tree} V^g_n$ fails for non-MHV six-point amplitudes ($1^-,2^-,3^-,4^+,5^+,6^+$) marks a sharp boundary: whatever structure protects MHV amplitudes does not extend to general helicity, and characterizing that failure would be a natural next question.
Since the collinear-bootstrap ambiguities the authors flag are cut-free rational functions such as $ε(1,2,3,4)/\prod\langle i\,i+1\rangle$, a systematic classification of which of those can actually appear in gauge theory would turn the bootstrap from an ansatz generator into a decision procedure.

Original abstract

We present a technique which utilizes unitarity and collinear limits to construct ansatze for one-loop amplitudes in gauge theory. As an example, we obtain the one-loop contribution to amplitudes for $n$ gluon scattering in $N=4$ supersymmetric Yang-Mills theory with the helicity configuration of the Parke-Taylor tree amplitudes. We prove that our $N=4$ ansatz is correct using general properties of the relevant one-loop $n$-point integrals. We also give the ``splitting amplitudes'' which govern the collinear behavior of one-loop helicity amplitudes in gauge theories.

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

We present a technique which utilizes unitarity and collinear limits to construct ansatze for one-loop amplitudes in gauge theory. As an example, we obtain the one-loop contribution to amplitudes for $n$ gluon scattering in $N=4$ supersymmetric Yang-Mills theory with the helicity configuration of the Parke-Taylor tree amplitudes. We prove that our $N=4$ ansatz is correct using general properties of the relevant one-loop $n$-point integrals. We also give the ``splitting amplitudes'' which govern the collinear behavior of one-loop helicity amplitudes in gauge theories.

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 present a technique which utilizes unitarity and collinear limits to construct ansatze for one-loop amplitudes in gauge theory.	method 0.68	0.94	0.26	asserted 1.00
s1	As an example, we obtain the one-loop contribution to amplitudes for $n$ gluon scattering in $N=4$ supersymmetric Yang-Mills theory with the helicity configuration of the Parke-Taylor tree amplitudes.	result 0.70	0.94	0.14	asserted 1.00
s2	We prove that our $N=4$ ansatz is correct using general properties of the relevant one-loop $n$-point integrals.	contribution 0.45	0.95	0.40	asserted 1.00
s3	We also give the ``splitting amplitudes'' which govern the collinear behavior of one-loop helicity amplitudes in gauge theories.	result 0.55	0.92	0.09	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.91 · evidence from proof or derivation 1.00.

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arXiv listing abstract v2

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Figures

**Figure 1.** To obtain all-n expressions we impose a variety of constraints summarized here

Showing 3 of the paper's 9 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9403226 by the authors.

References

Works cited (49)

[42]

Z. Bern, L. Dixon and D. A. Kosower, in preparation

View on Pith
[28]

Z. Bern, G. Chalmers, L. Dixon and D. A. Kosower, SLAC-PU B-6409, hep-ph/9312333

View on Pith Pith review arXiv
[1]

M. Mangano and S.J. Parke, Phys. Rep. 200:301 (1991)

View on Pith 1991
[2]

S.J. Parke and T.R. Taylor, Phys. Rev. Lett. 56:2459 (198 6)

View on Pith
[3]

F.A. Berends and W.T. Giele, Nucl. Phys. B306:759 (1988)

View on Pith 1988
[4]

D. A. Kosower, Nucl. Phys. B335:23 (1990)

View on Pith 1990
[5]

M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, P hys. Rev. D15:996 (1977); M.T. Grisaru and H.N. Pendleton, Nucl. Phys. B124:81 (1977) ; S.J. Parke and T. Taylor, Phys. Lett. B157:81 (1985); Z. Kunszt, Nucl. Phys. B271:333 (1986)

View on Pith 1977
[6]

S.J. Parke and T. Taylor, Nucl. Phys. B269:410 (1986); Z. Kunszt, Nucl. Phys. B271:333 (1986); J.F. Gunion and J. Kalinowski, Phys. Rev. D34:2119 (1986); F.A. Berends and W.T. Giele, Nucl. Phys. B294:700 (1987)

View on Pith 1986
[7]

R. Kleiss and H. Kuijf, Nucl. Phys. B312:616 (1989)

View on Pith 1989
[8]

M. Mangano, S. Parke, and Z. Xu, Nucl. Phys. B298:653 (198 8)

View on Pith
[9]

Z. Kunszt and W.J. Stirling, Phys. Rev. D37:2439 (1988); C.J. Maxwell, Phys. Lett. B192:190 (1987)

View on Pith 1988
[10]

V.P. Nair, Phys. Lett. B214:215 (1988)

View on Pith 1988
[11]

F.A. Berends and W.T. Giele, Nucl. Phys. B313:595 (1989 )

View on Pith 1989
[12]

G. Altarelli and G. Parisi, Nucl. Phys. B126:298 (1977)

View on Pith 1977
[13]

Z. Bern, L. Dixon and D. A. Kosower, Phys. Rev. Lett. 70:2 677 (1993)

View on Pith 1993
[14]

Z. Bern and D. A. Kosower, Nucl. Phys. B379:451 (1992)

View on Pith 1992
[15]

Z. Bern and D. A. Kosower, Phys. Rev. Lett. 66:1669 (1991 ); Z. Bern and D. A. Kosower, in Proceedings of the PASCOS-91 Symposium , eds. P. Nath and S. Reucroft (World Scientiﬁc, 1992); Z. Bern, Phys. Lett. 296B:85 (1992); Z. Bern, D.C. Dunbar and T. Shimada, Phys. Lett. B312:277 (19 93)

View on Pith 1992
[16]

Z. Bern and D.C. Dunbar, Nucl. Phys. B379:562 (1992)

View on Pith 1992
[17]

L.D. Landau, Nucl. Phys. 13:181 (1959); S. Mandelstam, Phys. Rev. 112:1344 (1958), 115:1741 (1959) ; R.E. Cutkosky, J. Math. Phys. 1:429 (1960)

View on Pith 1959
[18]

W. Siegel, Phys. Lett. 84B:193 (1979); D.M. Capper, D.R.T. Jones and P. van Nieuwenhuizen, Nucl. Ph ys. B167:479 (1980); L.V. Avdeev and A.A. Vladimirov, Nucl. Phys. B219:262 (1983 ). 41

View on Pith 1979
[19]

Z. Bern, hep-ph/9304249, in Proceedings of Theoretical Advanced Study Institute in High Energy Physics (TASI 92) , eds. J. Harvey and J. Polchinski (World Scientiﬁc, 1993)

View on Pith Pith review arXiv 1993
[20]

Z. Kunszt, A. Signer and Z. Trocsanyi, Nucl. Phys. B411: 397 (1994)

View on Pith 1994
[21]

M. B. Green, J. H. Schwarz and L. Brink, Nucl. Phys. B198: 472 (1982)

View on Pith 1982
[22]

Z. Bern and A. Morgan, hep-ph/9312218, to appear in Phys . Rev. D

View on Pith Pith review arXiv
[23]

S.J. Gates, M.T. Grisaru, M. Rocek and W. Siegel, Superspace, (Benjamin/Cummings, 1983), pages 390-391

View on Pith 1983
[24]

L.M. Brown and R.P. Feynman, Phys. Rev. 85 (1952) 231; G. Passarino and M. Veltman, Nucl. Phys. B160 (1979) 151; G. ’t Hooft and M. Veltman, Nucl. Phys. B153:365 (1979); R. G. Stuart, Comp. Phys. Comm. 48:367 (1988); R. G. Stuart and A. Gongora, Comp. Phys. Comm. 56:337 (1990)

View on Pith 1952
[25]

D. B. Melrose, Il Nuovo Cimento 40A:181 (1965); W. van Neerven and J. A. M. Vermaseren, Phys. Lett. 137B:241 ( 1984); G. J. van Oldenborgh and J. A. M. Vermaseren, Z. Phys. C46:425 (1990); G. J. van Oldenborgh, PhD thesis, University of Amsterdam (1 990); A. Aeppli, PhD thesis, University of Zurich (1992)

View on Pith 1965
[26]

Z. Bern, L. Dixon and D. A. Kosower, Phys. Lett. B302:299 (1993); erratum B318:649 (1993); Nucl. Phys. B412:751 (1994)

View on Pith 1993
[27]

Z. Bern, L. Dixon and D. A. Kosower, Proceedings of Strin gs 1993, May 24-29, Berke- ley, CA, hep-th/9311026

View on Pith Pith review arXiv 1993
[29]

G.D. Mahlon, preprint Fermilab-Pub-93/389-T, hep-ph /9312276

View on Pith
[30]

L. Dixon and O. Puzyrko, in preparation

View on Pith
[31]

G.D. Mahlon, preprint Fermilab-Pub-93/327-T, hep-ph /9311213

View on Pith
[32]

H. Goldberg and R. Rosenfeld, hep-ph/9304238; V. Del Duca, Phys. Rev. D48:5133 (1993)

View on Pith Pith review arXiv 1993
[33]

W.T. Giele and E. W. N. Glover, Phys. Rev. D46:1980 (1992 )

View on Pith 1980
[34]

W.T. Giele, E. W. N. Glover and D. A. Kosower, Nucl. Phys. B403:633 (1993)

View on Pith 1993
[35]

Z. Kunszt and D. Soper, Phys. Rev. D46:192 (1992)

View on Pith 1992
[36]

Z. Kunszt, A. Signer, and Z. Trocsanyi, preprint ETH-TH –94–03, hep-ph/9401294

View on Pith Pith review arXiv
[37]

F. A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans and T. T. Wu, Phys. Lett. 103B:124 (1981); P. De Causmaeker, R. Gastmans, W. Troost and T. T. Wu, Nucl. Ph ys. B206:53 (1982); 42 R. Kleiss and W. J. Stirling, Nucl. Phys. B262:235 (1985); J. F. Gunion and Z. Kunszt, Phys. Lett. 161B:333 (1985); R. Gastmans and T. T. Wu, The Ubiquitous Photon: Helic...

View on Pith 1981
[38]

Z. Bern and D. A. Kosower, Nucl. Phys. B362:389 (1991)

View on Pith 1991
[39]

M.J. Strassler, Nucl. Phys. B385:145 (1992); M. G. Schmidt and C. Schubert, Phys. Lett. B318:1993 438; D. Fliegner, M.G. Schmidt and C. Schubert, HD-THEP-93-44, h ep-ph/9401221; F. Bastianelli, USITP-93-17, hep-th/9308041

View on Pith Pith review arXiv 1992
[40]

S. Mandelstam, Nucl. Phys. B213:149 (1983)

View on Pith 1983
[41]

M. Mangano and S. J. Parke, Nucl. Phys. B299:673 (1988)

View on Pith 1988
[43]

J. Collins, G. Sterman and D. Soper in Perturbative Quantum Chromodynamics , ed. A. Mueller (World Scientiﬁc, 1989)

View on Pith 1989
[44]

G. Curci, W. Furmanski and R. Petronzio, Nucl. Phys. B17 5:27 (1980); W. Furmanski and R. Petronzio, Phys. Lett. B97:437 (1980)

View on Pith 1980
[45]

J. Kalinowski, K. Konishi, P.N. Scharbach and T.R. Tayl or, Nucl. Phys. B181:253 (1981); J. F. Gunion and J. Kalinowski, Phys. Rev. D32:2303 (1985)

View on Pith 1981
[46]

L. Lewin, Dilogarithms and Associated Functions (Macdonald, 1958)

View on Pith 1958
[47]

J. E. Paton and H. M. Chan, Nucl. Phys. B10:516 (1969)

View on Pith 1969
[48]

R. K. Ellis and J. C. Sexton, Nucl. Phys. B269:445 (1986)

View on Pith 1986
[49]

A. Denner, U. Nierste, and R. Scharf, Nucl. Phys. B367:6 37 (1991); N.I. Usyukina and A.I. Davydychev, Phys. Lett. B298:363 (19 93); Phys. Lett. B305:136 (1993). 43 Figure Captions Fig. 1: To obtain all- n expressions we impose a variety of constraints summarized h ere. Fig. 2: In the collinear limit of a one-loop amplitude we obtain two t ype of terms: t...

View on Pith Pith review arXiv 1991

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Environmental Effects in Post-Minkowskian Dynamics: Effective Field Theory, Feynman Rules, and Ward Identities for Compact Objects in Relativistic Fluids
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An effective field theory for environmental effects on compact objects in fluids is formulated using phonon-graviton interactions, with Feynman rules and Ward identities verified through four points.
A six-gluon obstruction to dimension-independent local color-kinematics duality at one loop
hep-th 2026-09 accept novelty 8.0 of 10

No dimension-independent local color-kinematics dual representation exists for the one-loop six-gluon amplitude in pure Yang-Mills theory.

Show 55 more citing papers

Reimagining Gravity: Generalized Symmetries, Double Copy, and Spinning Black Holes
hep-th 2026-08 accept novelty 8.0 of 10

A thesis presenting original results on gravitational generalized symmetries, the double copy, and the effective dynamics of spinning black holes, including a new symmetry and exact equations of motion.
Gravitational bremsstrahlung waveform at the eighth post-Minkowskian order in the extreme-mass-ratio limit
gr-qc 2026-07 conditional novelty 8.0 of 10

New high-order perturbative results for extreme-mass-ratio gravitational scattering waveforms, radiated angular momentum, and radiation-reaction deflection, claimed at eighth but explicitly computed to seventh post-Mi...
Spacelike-Collinear Scattering by the Method of Regions
hep-ph 2026-07 conditional novelty 8.0 of 10

The kinematic factorisation-violating part of the two-loop spacelike-collinear splitting amplitude comes entirely from a single hidden region with soft and Glauber loop momenta.
Bootstrapping two-loop six-gluon amplitudes in QCD
hep-ph 2026-06 unverdicted novelty 8.0 of 10

A symbol bootstrap using leading singularities determines the maximal-weight symbol of the planar two-loop six-gluon amplitude in massless QCD for MHV configurations.
Geometry of soft scalars at one loop
hep-th 2025-04 conditional novelty 8.0 of 10

The geometric soft theorem for scalar effective field theories is unchanged at one loop in derivative-coupled theories, and receives a universal leading correction when potential interactions are present.
Classical worldlines from scattering amplitudes
hep-th 2024-12 conditional novelty 8.0 of 10

A new 'quantum worldline' representation makes the classical limit of scattering amplitudes exactly match worldline field theory integrands at two loops.
Tropical Monte Carlo quadrature for Feynman integrals
math-ph 2020-08 conditional novelty 8.0 of 10

Tropical geometry yields a stratified Monte Carlo sampler that evaluates parametric Feynman integrals up to 17 loops, with preprocessing cost exponential in the number of edges but sampling cost independent of the sec...
A Conformal Dispersion Relation: Correlations from Absorption
hep-th 2019-10 accept novelty 8.0 of 10

A four-point correlator in a conformal field theory can be reconstructed from its double discontinuity via an explicitly computed, theory-independent kernel, giving the CFT analogue of Kramers-Kronig dispersion relations.
Two-Loop Scattering Amplitudes: Double-Forward Limit and Colour-Kinematics Duality
hep-th 2019-08 conditional novelty 8.0 of 10

Two-loop Yang-Mills and gravity integrands are constructed from tree-level trivalent diagrams via a double-forward limit, with multiplicity factors that make the loop-level colour-kinematics duality manifest.
Multi-Quark Colour Decompositions from Unitarity
hep-ph 2019-08 accept novelty 8.0 of 10

New tree-level colour decompositions for multi-quark QCD amplitudes with arbitrary fixed pair of partons, derived from unitarity factorisation, with explicit closed-form colour factors.
Black Hole Binary Dynamics from the Double Copy and Effective Theory
hep-th 2019-08 accept novelty 8.0 of 10

The authors derive the complete conservative two-body Hamiltonian for spinless black holes at third post-Minkowskian order using scattering amplitudes and effective field theory.
Analytic form of the full two-loop five-gluon all-plus helicity amplitude
hep-ph 2019-05 accept novelty 8.0 of 10

The full-colour two-loop five-gluon all-plus helicity amplitude is obtained analytically in the physical scattering region, and collapses to a compact expression built from one-mass box dilogarithms times rational prefactors.
Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order
hep-th 2019-01 conditional novelty 8.0 of 10

Using generalized unitarity and double copy, the authors derive the two-loop classical gravitational amplitude and the conservative spinless two-body Hamiltonian at O(G^3), exact in velocity.
A first look at two-loop five-gluon scattering in QCD
hep-ph 2017-12 conditional novelty 8.0 of 10

The leading-colour two-loop five-gluon helicity amplitudes in massless QCD are computed and evaluated numerically at one kinematic point, giving the first benchmark numbers for a 2-to-3 process at two loops.
High-energy evolution to three loops
hep-ph 2016-04 accept novelty 8.0 of 10

The three-loop Balitsky-Kovchegov and non-global-logarithm evolution kernels are derived in planar N=4 super Yang-Mills, and the linearized eigenvalue agrees with an independent integrability prediction.
High-precision differential predictions for top-quark pairs at the LHC
hep-ph 2015-11 conditional novelty 8.0 of 10

The first complete next-to-next-to-leading-order QCD calculation of differential top-quark pair distributions at the LHC shows that higher-order corrections soften the top transverse momentum spectrum while leaving th...
Scattering Amplitudes and the Positive Grassmannian
hep-th 2012-12 accept novelty 8.0 of 10

Scattering amplitudes in planar N=4 super Yang-Mills are organized by on-shell diagrams whose invariant content is a cell of the positive Grassmannian, labeled by a permutation.
Four-Jet Production at the Large Hadron Collider at Next-to-Leading Order in QCD
hep-ph 2011-12 accept novelty 8.0 of 10

First next-to-leading-order QCD prediction for four-jet production at a hadron collider, agreeing with ATLAS 7 TeV data for three and four jets and giving a four-to-three-jet ratio of 0.102 against the measured 0.098.
The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
hep-th 2010-08 conditional novelty 8.0 of 10

A recursive formula generates the all-loop integrand for planar N=4 SYM scattering amplitudes in a form that keeps the full Yangian symmetry manifest.
Perturbative Quantum Gravity as a Double Copy of Gauge Theory
hep-th 2010-04 accept novelty 8.0 of 10

Gravity loop amplitudes can be built as a double copy of gauge-theory diagram numerators once those numerators are made to obey the same algebraic relations as color factors.
The Ultraviolet Behavior of N=8 Supergravity at Four Loops
hep-th 2009-05 conditional novelty 8.0 of 10

The complete four-loop four-graviton amplitude of N=8 supergravity is ultraviolet finite in four and five dimensions, ruling out a predicted five-dimensional counterterm.
The S-Matrix in Twistor Space
hep-th 2009-03 conditional novelty 8.0 of 10

In (2,2) signature, scattering amplitudes of N=4 SYM and N=8 supergravity become simple sign functions in twistor space, and the BCFW recursion turns into a quadratic equation for an S-matrix functional.
What is the Simplest Quantum Field Theory?
hep-th 2008-08 accept novelty 8.0 of 10

N=8 supergravity's one-loop amplitudes contain only scalar box integrals, proved via an on-shell superspace that also yields recursion relations for every tree amplitude.
Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
hep-th 2008-07 accept novelty 8.0 of 10

N=4 super-Yang-Mills scattering amplitudes are shown to carry a dual superconformal symmetry, yielding a compact Lorentz-covariant formula for tree-level next-to-MHV amplitudes and evidence that the symmetry survives ...
The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
hep-th 2008-03 accept novelty 8.0 of 10

The two-loop six-gluon MHV amplitude in N=4 super-Yang-Mills has a nonzero remainder beyond the BDS ansatz, and that remainder equals the hexagonal Wilson loop remainder to numerical precision.
The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude
hep-th 2007-12 conditional novelty 8.0 of 10

The two-loop hexagon Wilson loop differs from the BDS prediction by a non-trivial function of three conformal cross-ratios, so either BDS or the amplitude/Wilson loop duality fails at six points.
Three-Loop Superfiniteness of N=8 Supergravity
hep-th 2007-02 accept novelty 8.0 of 10

The three-loop four-point amplitude of N=8 supergravity is ultraviolet finite in four dimensions and no more divergent in D dimensions than the corresponding N=4 super-Yang-Mills amplitude.
The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
hep-th 2006-10 conditional novelty 8.0 of 10

A direct four-loop computation in N=4 super Yang-Mills gives a cusp anomalous dimension of -29.34 ± 0.05, refuting the Eden-Staudacher Bethe-ansatz prediction of -26.40.
Supersymmetric Gauge Theories in Twistor Space
hep-th 2006-04 conditional novelty 8.0 of 10

A twistor-space action combining holomorphic Chern-Simons theory with a log-determinant term is shown to be perturbatively equivalent to N=4 super Yang-Mills, without conformal supergravity.
Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
hep-th 2005-05 accept novelty 8.0 of 10

A direct three-loop computation in maximally supersymmetric Yang-Mills shows the planar four-point amplitude iterates from lower-loop amplitudes, fixes the three-loop cusp anomalous dimension, and motivates an all-loo...
Antenna Subtraction at NNLO
hep-ph 2005-05 accept novelty 8.0 of 10

A complete antenna-subtraction framework for NNLO e+e- jet calculations, with all colour-ordered antenna functions and integrated forms, and an analytic demonstration that infrared poles cancel for three-jet production.
New Recursion Relations for Tree Amplitudes of Gluons
hep-th 2004-12 conditional novelty 8.0 of 10

A new on-shell recursion relation computes any tree-level gluon amplitude as a sum of products of two smaller physical amplitudes divided by a Feynman propagator.
Generalized Unitarity and One-Loop Amplitudes in N=4 Super-Yang-Mills
hep-th 2004-12 conditional novelty 8.0 of 10

Cutting all four internal lines of a one-loop box diagram at once yields its coefficient directly as a product of four tree-level amplitudes, for every box type in N=4 gluon amplitudes.
Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory
hep-th 2003-09 accept novelty 8.0 of 10

In planar N=4 super-Yang-Mills, the two-loop four-point amplitude equals one half the one-loop amplitude squared plus a fixed function of the one-loop amplitude at doubled epsilon, and the authors conjecture this hold...
An Amplitudes Approach to Spinning Black Hole Scattering
hep-th 2026-11 accept novelty 7.0 of 10

Computes the classical two-black-hole scattering amplitude through quartic order in spin at third post-Minkowskian order and shows a spin-shift symmetry constrains the probe radial action.
Light-Transformed Carrollian OPEs and the Four-Gluon Spectral Bootstrap
hep-th 2026-10 conditional novelty 7.0 of 10

The four-gluon tree amplitude's massive two-particle spectrum is uniquely determined by its two collinear Carrollian operator products, crossing and Carroll-time scaling, up to one subtraction constant.
Quantum Love Numbers are Non-Zero
hep-th 2026-09 conditional novelty 7.0 of 10

Quantum loop corrections necessarily generate non-zero static Love numbers for Schwarzschild black holes, suppressed by the square of the Planck length divided by the horizon radius squared, breaking the classical vanishing.
The geometry of multiloop Feynman Integrals from the Scattering Facet
hep-th 2026-09 accept novelty 7.0 of 10

The scattering facet is a Cayley polytope whose fibers are deformed graphical zonotopes, yielding a geometric derivation of causal and spanning tree representations of Feynman energy integrals.
Five-point spinor-helicity Compton amplitudes in gauge theory and gravity
hep-th 2026-08 conditional novelty 7.0 of 10

Exact five-point tree-level Compton amplitudes for minimally coupled spinning matter, valid for spins up to 1 in gauge theory and 2 in gravity, are derived with spurious poles manifestly canceled.
Gravitational Faraday rotation, gravitational spin Hall effect, and spin-refined causality analysis from Magnusian matrix in effective field theories of gravity
hep-th 2026-08 conditional novelty 7.0 of 10

In modified-gravity theories, spin Hall deflection of light and gravitational waves by a spinning black hole becomes non-commuting, so a ray disperses into a blob, and black-hole spin direction can tighten causality b...
$N$-Photon Amplitudes in EFT from Recursion Relations and Effective Vertices
hep-th 2026-08 conditional novelty 7.0 of 10

A CSW-like recursion plus a 'contact Lagrangian' computes arbitrary tree-level N-photon amplitudes in general EFTs, with results through 10 photons in Born-Infeld theory.
Kerr Soft Dressing and the $w_{1+\infty}$ Frame Algebra at Null Infinity
hep-th 2026-07 conditional novelty 7.0 of 10

The Kerr soft exponent generates a parity-alternating hierarchy of null-infinity frame generators—displacement memory at s=0, spin memory at s=1, and higher moments alternating by Kerr multipole parity—realized as the...
NLSM amplitudes from a quartic two-derivative theory
hep-th 2026-07 conditional novelty 7.0 of 10

A two-field scalar theory with a single quartic two-derivative vertex reproduces all planar NLSM tree amplitudes and, up to scaleless terms, all loop integrands.
Scattering Amplitudes as Programs: Self-Evolving Search for Theory and Event Generation
hep-ph 2026-07 accept novelty 7.0 of 10

Self-evolving program search over scattering amplitudes discovers known and hybrid evaluation structures, cutting counted arithmetic ~48x and reaching within ~14x of optimized MadGraph at n=6 while beating the tested ...
Quantum anomalies from three-point on-shell bootstrap
hep-th 2026-06 unverdicted novelty 7.0 of 10

On-shell three-point bootstrap recovers Weyl, chiral, diffeomorphism and Lorentz anomalies up to constant prefactors and reproduces gauge-anomaly cancellation while excluding Pontryagin densities from the Weyl anomaly.
Perturbative construction of amplitudes from on-shell trees with vacuum pairs: the all-plus four-gluon amplitude through order $g^6$
hep-th 2026-06 unverdicted novelty 7.0 of 10

An on-shell construction using BCFW trees plus vacuum-pair phase-space integrals with inclusion-exclusion signs reproduces the known one- and two-loop all-plus four-gluon amplitudes.
On-Shell Bootstrap of Loop Inflation Correlators with Spectral Dispersion
hep-th 2026-06 unverdicted novelty 7.0 of 10

Introduces spectral dispersion bootstrap combining dS spectral decomposition and dispersion relations to compute 3- and 4-point loop correlators with massive scalar and vector exchanges.
Five legs @ three loops: N=4 sYM amplitude near mass-shell
hep-th 2026-05 unverdicted novelty 7.0 of 10

Three-loop five-leg amplitude in planar N=4 sYM near mass shell is computed via 6D unitarity cuts and dimensional reduction, confirming IR exponentiation governed by octagon anomalous dimension with each of three kine...
Two-loop leading-color QCD corrections for Higgs plus two-jet production in the heavy-top limit
hep-ph 2026-05 unverdicted novelty 7.0 of 10

Two-loop leading-color helicity amplitudes for H+2 jets in the heavy-top limit are computed analytically and validated, enabling NNLO phenomenology and revealing an anomalous threshold.
Negative running of gravitational positivity
hep-th 2026-03 unverdicted novelty 7.0 of 10

Non-minimal three-point interactions induce negative one-loop running of Wilson coefficients in gravitational EFTs, yet graviton loops generate positive IR contributions that dominate the bounds after smearing if the ...
QCD Scattering Amplitudes and Prescriptive Unitarity
hep-th 2026-02 conditional novelty 7.0 of 10

The maximally-transcendental part of planar two-loop six-gluon MHV QCD amplitudes is bootstrapped at symbol level and expressed in a 137-letter alphabet.
Resumming Scattering Amplitudes for Waveforms
hep-th 2026-01 unverdicted novelty 7.0 of 10

A new projector-based formalism determines effective potentials from perturbative amplitudes and resums them to compute non-perturbative gravitational waveforms for generic two-body trajectories.
Twisted Feynman Integrals: from generating functions to spin-resummed post-Minkowskian dynamics
hep-th 2025-12 unverdicted novelty 7.0 of 10

Twisted Feynman integrals are introduced with graded Symanzik polynomials, classified as exponential periods, and shown to have geometry not inferable from generalized Baikov leading singularities.
Two-loop all-plus helicity amplitudes for self-dual Higgs boson with gluons via unitarity cut constraints
hep-ph 2025-11 unverdicted novelty 7.0 of 10

Two-loop all-plus helicity amplitudes for self-dual Higgs plus gluons are obtained via four-dimensional unitarity cuts into one-loop and tree amplitudes plus finite-field tensor reduction.
Generalized Unitarity Method for Worldline Field Theory
hep-th 2025-10 conditional novelty 7.0 of 10

Worldline gravitational observables can be bootstrapped from locality, unitarity, gauge invariance, and a soft theorem once worldline energies are complexified, reproducing the known O(G^5/2) waveform and O(G^3) on-sh...

See all 348 Pith citations

How later papers used it

593 later papers cite it

Evidence: substantial. 555 of the citing papers share no author with it, and 131 of those carry a confident label from Pith's citation classifier.

Pith read 39 of the 946 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

9 later papers by other authors are listed below: 3 used its method or tool, 1 used its data or values, 4 compared their own results with it and 1 cited it as support. 4 more papers by the same authors are listed: 4 used its method or tool.

Uses its method or tool 7 of 123 papers with this label · 4 by the same authors

Computed the integral coefficients using this paper's unitarity method and iterated two-particle cuts.

Higher point MHV amplitudes in N=4 Supersymmetric Yang-Mills Theory (2009) · 1.00 · paraphrased: the licence does not allow quoting
Constructed the integrand by gluing tree-level amplitudes on a unitarity cut.

Supersymmetry and the Celestial Jacobi Identity (2023) · 1.00 · paraphrased: the licence does not allow quoting
Computed one-loop amplitudes analytically using the generalized unitarity approach.

$H \rightarrow b\overline{b}j$ at Next-to-Next-to-Leading Order Accuracy (2019) · 1.00 · paraphrased: the licence does not allow quoting

By the same authors

Constructed its numerators using the method of maximal cuts, based on this paper's unitarity method.

The Five-Loop Four-Point Amplitude of N=4 super-Yang-Mills Theory (2012) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
Used the generalized unitarity method to construct the parts of the one-loop amplitude needed to extract the classical interaction potential between spinning particles.

Spinning Black Hole Binary Dynamics, Scattering Amplitudes and Effective Field Theory (2020) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
Used generalized unitarity to construct the two-loop integrand from tree amplitudes of gravitons and massive scalars obtained from the double-copy construction.

Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order (2019) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
Used generalized unitarity, a refinement of this paper's unitarity method, to find the desired representation.

D = 5 maximally supersymmetric Yang-Mills theory diverges at six loops (2012) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

Uses its data or values 1 paper

Took the square of this paper's one-loop results for the q to q g splitting amplitudes and summed over helicities to reproduce the expression for H_+.

Factorization and Resummation for Dijet Invariant Mass Spectra (2011) · 0.82 · paraphrased: the licence does not allow quoting

Compares with it 4 of 24 papers with this label

Verified its own expression for the finite part of the amplitude at five-point by comparison with this paper.

One-Loop Soft Theorems via Dual Superconformal Symmetry (2015) · 1.00 · paraphrased: the licence does not allow quoting
Compared its own findings to the corresponding factorisation formula for massless fermions extracted from this paper's QCD results.

Universal structure of radiative QED amplitudes at one loop (2021) · 1.00 · paraphrased: the licence does not allow quoting
Did its gauge theory calculation to facilitate comparison with this paper.

Generating Tree Amplitudes in N=4 SYM and N=8 SG (2008) · 1.00 · paraphrased: the licence does not allow quoting
It is useful to compare our findings to the corresponding factorisation formula for massless fermions that can be extracted from the QCD results of [ref] .

Muon-Electron Scattering at NNLO (2022) · Collinear factorisation at one loop · 1.00

Cites it as support 1 of 3 papers with this label

Said that this is supported by the one-loop computation in this paper.

The two-loop MHV amplitudes in N=4 supersymmetric Yang-Mills theory (2009) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

579 arXiv papers cite this in 929 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 citing it · 551 sentences without a confident label

Uses its method or tool 117 papers

To construct the numerators $N_i$, we use the method of maximal cuts [ref] , based on the unitarity method [ref] .

The Five-Loop Four-Point Amplitude of N=4 super-Yang-Mills Theory · Introduction · 1.00
In this talk, we focus on an approach [ref] that exploits the unitarity method [ref] in order to numerically compute multi-loop amplitudes directly from gauge-invariant building blocks.

Five-Point Two-Loop Amplitudes from Numerical Unitarity · Introduction · 0.89
We construct these coefficients out of the Compton amplitude in <ref>[The same-helicity Compton amplitude does not contribute to classical scattering at 2PM when <ref> is exhibited [ref] .] and the three-point amplitude describing a Kerr black hole [ref] using generalized unitarity [ref] (see also refs.

Classical gravitational spinning-spinless scattering at $\mathcal{O}(G^{2} S^{\infty})$ · All-spin scattering · 0.92

Uses its data or values 1 paper

Taking the square of the one-loop results for the $q→q g$ splitting amplitudes from Ref. [ref] and summing over helicities reproduces the expression for $H_+$ in H3Plus.

Factorization and Resummation for Dijet Invariant Mass Spectra · Factorization · 0.82

Compares with it 22 papers

When rewritten in terms of these functions, the right-hand side of 1loopnn coincides with twice the one-loop $n$-gluon MHV amplitude (see Eq. (4.19) in [ref] ) [equation] In conclusion, the correlator calculated by means of a Lagrangian insertion and in dual infrared dimensional regularization, reproduces the amplitude in terms of momentum integrals.

From correlation functions to scattering amplitudes · From correlators to amplitudes: $n$ points at one loop · 0.80
We compare with the analytic expressions of [ref] ; the dimension-shift relation to self-dual and supersymmetric amplitudes is described in [ref] .

An adaptive inverse-problem framework for one-loop five-gluon BCJ numerators · Helicity-amplitude benchmarks and factorization targets · 1.00
To the best of our understanding, this fact corresponds to the no-triangle hypothesis for the 1-loop graphs in =8 supergravity as formulated in [ref] , despite the difference in the two approaches.

The Effective Action of N=8 Supergravity · Discussion · 0.95

Cites it as support 2 papers

This is supported by the one-loop computation in Ref. [ref] and also by the leg addition rule of Ref. [ref] .

The two-loop MHV amplitudes in N=4 supersymmetric Yang-Mills theory · Description of the computational procedure · 1.00
Furthermore, for $𝒩 ≥ 1$, the amplitudes are cut-constructible [ref] , which suggests the validity of a loop recursive structure.

On-shell diagrammatics and the perturbative structure of planar gauge theories · On-shell processes and scattering amplitudes · 0.98

Cites it as background 192 papers

Originally, they were discovered [ref] by comparing infrared divergences of one-loop amplitudes with tree-level amplitudes [ref] .

An Introduction to On-shell Recursion Relations · On-shell recursion relations · 0.87
The one-loop finite rational amplitudes were worked out in [ref] , while the two-loop four-gluon case and the complete two-loop helicity amplitudes are given in [ref] .

Perturbative construction of amplitudes from on-shell trees with vacuum pairs: the all-plus four-gluon amplitude through order $g^6$ · Discussion and outlook · 0.87
At NNLO, one is concerned with the double unresolved limits of tree-level matrix elements [ref] , as well as the single unresolved limits of one-loop matrix elements [ref] .

Idealised Antenna Functions for Higher Order QCD Calculations · Infrared Divergences · 0.87

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] [42]

Z. Bern, L. Dixon and D. A. Kosower, in preparation

View on Pith

[2] [28]

Z. Bern, G. Chalmers, L. Dixon and D. A. Kosower, SLAC-PU B-6409, hep-ph/9312333

View on Pith Pith review arXiv

[3] [1]

M. Mangano and S.J. Parke, Phys. Rep. 200:301 (1991)

View on Pith 1991

[4] [2]

S.J. Parke and T.R. Taylor, Phys. Rev. Lett. 56:2459 (198 6)

View on Pith

[5] [3]

F.A. Berends and W.T. Giele, Nucl. Phys. B306:759 (1988)

View on Pith 1988

[6] [4]

D. A. Kosower, Nucl. Phys. B335:23 (1990)

View on Pith 1990

[7] [5]

M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, P hys. Rev. D15:996 (1977); M.T. Grisaru and H.N. Pendleton, Nucl. Phys. B124:81 (1977) ; S.J. Parke and T. Taylor, Phys. Lett. B157:81 (1985); Z. Kunszt, Nucl. Phys. B271:333 (1986)

View on Pith 1977

[8] [6]

S.J. Parke and T. Taylor, Nucl. Phys. B269:410 (1986); Z. Kunszt, Nucl. Phys. B271:333 (1986); J.F. Gunion and J. Kalinowski, Phys. Rev. D34:2119 (1986); F.A. Berends and W.T. Giele, Nucl. Phys. B294:700 (1987)

View on Pith 1986

[9] [7]

R. Kleiss and H. Kuijf, Nucl. Phys. B312:616 (1989)

View on Pith 1989

[10] [8]

M. Mangano, S. Parke, and Z. Xu, Nucl. Phys. B298:653 (198 8)

View on Pith

[11] [9]

Z. Kunszt and W.J. Stirling, Phys. Rev. D37:2439 (1988); C.J. Maxwell, Phys. Lett. B192:190 (1987)

View on Pith 1988

[12] [10]

V.P. Nair, Phys. Lett. B214:215 (1988)

View on Pith 1988

[13] [11]

F.A. Berends and W.T. Giele, Nucl. Phys. B313:595 (1989 )

View on Pith 1989

[14] [12]

G. Altarelli and G. Parisi, Nucl. Phys. B126:298 (1977)

View on Pith 1977

[15] [13]

Z. Bern, L. Dixon and D. A. Kosower, Phys. Rev. Lett. 70:2 677 (1993)

View on Pith 1993

[16] [14]

Z. Bern and D. A. Kosower, Nucl. Phys. B379:451 (1992)

View on Pith 1992

[17] [15]

Z. Bern and D. A. Kosower, Phys. Rev. Lett. 66:1669 (1991 ); Z. Bern and D. A. Kosower, in Proceedings of the PASCOS-91 Symposium , eds. P. Nath and S. Reucroft (World Scientiﬁc, 1992); Z. Bern, Phys. Lett. 296B:85 (1992); Z. Bern, D.C. Dunbar and T. Shimada, Phys. Lett. B312:277 (19 93)

View on Pith 1992

[18] [16]

Z. Bern and D.C. Dunbar, Nucl. Phys. B379:562 (1992)

View on Pith 1992

[19] [17]

L.D. Landau, Nucl. Phys. 13:181 (1959); S. Mandelstam, Phys. Rev. 112:1344 (1958), 115:1741 (1959) ; R.E. Cutkosky, J. Math. Phys. 1:429 (1960)

View on Pith 1959

[20] [18]

W. Siegel, Phys. Lett. 84B:193 (1979); D.M. Capper, D.R.T. Jones and P. van Nieuwenhuizen, Nucl. Ph ys. B167:479 (1980); L.V. Avdeev and A.A. Vladimirov, Nucl. Phys. B219:262 (1983 ). 41

View on Pith 1979

[21] [19]

Z. Bern, hep-ph/9304249, in Proceedings of Theoretical Advanced Study Institute in High Energy Physics (TASI 92) , eds. J. Harvey and J. Polchinski (World Scientiﬁc, 1993)

View on Pith Pith review arXiv 1993

[22] [20]

Z. Kunszt, A. Signer and Z. Trocsanyi, Nucl. Phys. B411: 397 (1994)

View on Pith 1994

[23] [21]

M. B. Green, J. H. Schwarz and L. Brink, Nucl. Phys. B198: 472 (1982)

View on Pith 1982

[24] [22]

Z. Bern and A. Morgan, hep-ph/9312218, to appear in Phys . Rev. D

View on Pith Pith review arXiv

[25] [23]

S.J. Gates, M.T. Grisaru, M. Rocek and W. Siegel, Superspace, (Benjamin/Cummings, 1983), pages 390-391

View on Pith 1983

[26] [24]

L.M. Brown and R.P. Feynman, Phys. Rev. 85 (1952) 231; G. Passarino and M. Veltman, Nucl. Phys. B160 (1979) 151; G. ’t Hooft and M. Veltman, Nucl. Phys. B153:365 (1979); R. G. Stuart, Comp. Phys. Comm. 48:367 (1988); R. G. Stuart and A. Gongora, Comp. Phys. Comm. 56:337 (1990)

View on Pith 1952

[27] [25]

D. B. Melrose, Il Nuovo Cimento 40A:181 (1965); W. van Neerven and J. A. M. Vermaseren, Phys. Lett. 137B:241 ( 1984); G. J. van Oldenborgh and J. A. M. Vermaseren, Z. Phys. C46:425 (1990); G. J. van Oldenborgh, PhD thesis, University of Amsterdam (1 990); A. Aeppli, PhD thesis, University of Zurich (1992)

View on Pith 1965

[28] [26]

Z. Bern, L. Dixon and D. A. Kosower, Phys. Lett. B302:299 (1993); erratum B318:649 (1993); Nucl. Phys. B412:751 (1994)

View on Pith 1993

[29] [27]

Z. Bern, L. Dixon and D. A. Kosower, Proceedings of Strin gs 1993, May 24-29, Berke- ley, CA, hep-th/9311026

View on Pith Pith review arXiv 1993

[30] [29]

G.D. Mahlon, preprint Fermilab-Pub-93/389-T, hep-ph /9312276

View on Pith

[31] [30]

L. Dixon and O. Puzyrko, in preparation

View on Pith

[32] [31]

G.D. Mahlon, preprint Fermilab-Pub-93/327-T, hep-ph /9311213

View on Pith

[33] [32]

H. Goldberg and R. Rosenfeld, hep-ph/9304238; V. Del Duca, Phys. Rev. D48:5133 (1993)

View on Pith Pith review arXiv 1993

[34] [33]

W.T. Giele and E. W. N. Glover, Phys. Rev. D46:1980 (1992 )

View on Pith 1980

[35] [34]

W.T. Giele, E. W. N. Glover and D. A. Kosower, Nucl. Phys. B403:633 (1993)

View on Pith 1993

[36] [35]

Z. Kunszt and D. Soper, Phys. Rev. D46:192 (1992)

View on Pith 1992

[37] [36]

Z. Kunszt, A. Signer, and Z. Trocsanyi, preprint ETH-TH –94–03, hep-ph/9401294

View on Pith Pith review arXiv

[38] [37]

F. A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans and T. T. Wu, Phys. Lett. 103B:124 (1981); P. De Causmaeker, R. Gastmans, W. Troost and T. T. Wu, Nucl. Ph ys. B206:53 (1982); 42 R. Kleiss and W. J. Stirling, Nucl. Phys. B262:235 (1985); J. F. Gunion and Z. Kunszt, Phys. Lett. 161B:333 (1985); R. Gastmans and T. T. Wu, The Ubiquitous Photon: Helic...

View on Pith 1981

[39] [38]

Z. Bern and D. A. Kosower, Nucl. Phys. B362:389 (1991)

View on Pith 1991

[40] [39]

M.J. Strassler, Nucl. Phys. B385:145 (1992); M. G. Schmidt and C. Schubert, Phys. Lett. B318:1993 438; D. Fliegner, M.G. Schmidt and C. Schubert, HD-THEP-93-44, h ep-ph/9401221; F. Bastianelli, USITP-93-17, hep-th/9308041

View on Pith Pith review arXiv 1992

[41] [40]

S. Mandelstam, Nucl. Phys. B213:149 (1983)

View on Pith 1983

[42] [41]

M. Mangano and S. J. Parke, Nucl. Phys. B299:673 (1988)

View on Pith 1988

[43] [43]

J. Collins, G. Sterman and D. Soper in Perturbative Quantum Chromodynamics , ed. A. Mueller (World Scientiﬁc, 1989)

View on Pith 1989

[44] [44]

G. Curci, W. Furmanski and R. Petronzio, Nucl. Phys. B17 5:27 (1980); W. Furmanski and R. Petronzio, Phys. Lett. B97:437 (1980)

View on Pith 1980

[45] [45]

J. Kalinowski, K. Konishi, P.N. Scharbach and T.R. Tayl or, Nucl. Phys. B181:253 (1981); J. F. Gunion and J. Kalinowski, Phys. Rev. D32:2303 (1985)

View on Pith 1981

[46] [46]

L. Lewin, Dilogarithms and Associated Functions (Macdonald, 1958)

View on Pith 1958

[47] [47]

J. E. Paton and H. M. Chan, Nucl. Phys. B10:516 (1969)

View on Pith 1969

[48] [48]

R. K. Ellis and J. C. Sexton, Nucl. Phys. B269:445 (1986)

View on Pith 1986

[49] [49]

A. Denner, U. Nierste, and R. Scharf, Nucl. Phys. B367:6 37 (1991); N.I. Usyukina and A.I. Davydychev, Phys. Lett. B298:363 (19 93); Phys. Lett. B305:136 (1993). 43 Figure Captions Fig. 1: To obtain all- n expressions we impose a variety of constraints summarized h ere. Fig. 2: In the collinear limit of a one-loop amplitude we obtain two t ype of terms: t...

View on Pith Pith review arXiv 1991