{"id":"2ed37955-0ee4-4ae2-af1d-336aacdfa059","arxiv_id":"hep-ph/9403226","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A unitarity-and-collinear-limit bootstrap yields, and proves, a closed-form all-n one-loop MHV gluon amplitude in N=4 super-Yang-Mills theory.","lead":"This paper introduces a bootstrap that uses unitarity cuts and collinear limits to compute one-loop gauge-theory amplitudes, and applies it to derive a closed-form all-n expression for maximally helicity-violating gluon scattering in N=4 super-Yang-Mills. The technique and the all-n formula became templates for the modern amplitudes program used to predict LHC backgrounds.","discovery_kind":"new_method","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The load-bearing step is §6's uniqueness claim: that eq. (6.2), Σ c_i I^box_i = polynomial, has only the trivial solution. It is demonstrated for one three-mass box and then closed with 'one can continue in this way'; the hard two-mass and four-mass boxes are never treated.","rationale":"I agree with the reader that §6 is the load-bearing section and that the paper's honesty about scope (end of §5, conclusions) is a point in its favor. I disagree, partially, on where the softness sits. The reader points to the imported m-4 power-counting premise (scalar boxes only, no triangles/bubbles). That premise is actually well-supported within the paper: §2 gives two independent physical arguments (string-based cancellation of σF insertions between gluon and fermion loops; superspace requirement of four D insertions after ghost/matter cancellation), and it is the standard basis of N=4 finiteness. The sharper soft spot is the uniqueness demonstration itself: §6 works one coefficient and then asserts the rest. That is the step that converts 'ansatz with correct cuts' into 'the amplitude,' and it is the step the abstract's word 'prove' depends on. The gap is a rigor gap, not a correctness gap — the result is right and the argument is plausible — so the verdict should not move. I would keep ACCEPT and record the §6 sketch as a derivation-gap note, exactly as the reader did. Two record corrections: the classifier's 'Limitations: none found' is wrong (the manuscript states limitations at the end of §5 and in the conclusions), and 'Methods: none found' is also wrong (the unitarity/collinear bootstrap is described at length in §§3–5). These are classifier errors, not paper errors, and they do not affect the verdict.","tokens_in":36764,"tokens_out":7364,"duration_ms":147360,"concrete_test":"Independently verify the linear independence of the scalar box functions in appendix I (eqs. I.5a–e) modulo polynomials. For each box type — one-mass, easy two-mass, hard two-mass, three-mass, four-mass — compute its discontinuity in each momentum channel and check whether any nontrivial coefficient vector {c_i} makes Σ c_i Disc(I^box_i) vanish in every channel. A targeted version: compute the t^{[r]}_i-channel discontinuity of the hard two-mass box F^{2mh} (I.5c) and test whether it is linearly independent of the discontinuities of the easy two-mass, one-mass, three-mass, and four-mass boxes. If it is not, the coefficient of F^{2mh} is not fixed by the cuts and §6's uniqueness step has a genuine hole.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the all-n MHV N=4 amplitude has two pillars. Pillar (i), §5, computes all cuts and shows they match the cuts of the box sum in eq. (4.19); this is solid — the ρ² identity (5.4)–(5.6) is correct and the reduction to four cut boxes is explicit. Pillar (ii), §6, argues the cuts uniquely fix the amplitude because in N=4 only scalar boxes survive and eq. (6.2) has only the trivial solution. This is where the argument thins. The text works through exactly one coefficient (the three-mass box I^{3m}, via 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 systematic enumeration follows. In particular: (a) the hard two-mass box F^{2mh} (I.5c), whose 1/ε² three-log structure differs from the easy two-mass box, is never shown to be distinguishable by cuts; (b) the four-mass box F^{4m} (I.5e) is never discussed; (c) the possibility that a combination of boxes with different mass patterns conspires to be cut-free in every channel simultaneously is excluded only by the single worked example; (d) degenerate kinematic regions where invariants coincide are not addressed. Since the abstract's claim is 'we prove' [abstract:s2], and the proof's final step is a sketch, this is the load-bearing soft spot. It is a rigor gap, not a correctness gap: the result is right and the argument is plausible, but as written §6 does not fully discharge the uniqueness claim.","agreement_with_reader":"partial"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"The paper presents a method for constructing one-loop gauge-theory amplitudes from two inputs: unitarity (Cutkosky rules applied at the amplitude level) and collinear limits. The worked example is the leading-color, maximally helicity-violating n-gluon amplitude in N=4 super-Yang-Mills. The authors propose that the amplitude factorizes as c_Gamma times the Parke-Taylor tree amplitude times a universal function V_n^g, give V_n^g explicitly in Eqs. (4.3)-(4.8) as a sum of logarithms and dilogarithms of the invariants t^{[r]}_i, and give an equivalent representation in Eq. (4.19) as a sum of one-mass and easy two-mass scalar box functions. They then argue the result is correct for all n in two steps: a direct evaluation of all cuts in Section 5, which reduces to four cut box integrals via the Schouten identity, and a uniqueness argument in Section 6 based on the claim that in N=4 only scalar boxes survive and their coefficients are fixed by the cuts. Two appendices collect the scalar box functions and the one-loop splitting amplitudes, and Section 7 gives a formula expressing subleading-color partial amplitudes as permutation sums over leading-color ones.","tokens_in":37173,"tokens_out":4945,"duration_ms":108793,"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_comments":[{"comment":"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.","section":"Section 6, Eq. (6.2)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"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":"Section 3, Eqs. (3.6)-(3.7)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Section 7, Eq. (7.2)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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":"Appendix II, Eqs. (II.10)-(II.15)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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":"Section 5, below Eq. (5.11)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Section 4, Eq. (4.3)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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":"Appendix I, Eqs. (I.5)-(I.8)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Section 4, opening paragraph","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The result is correct and has held up, and the n=6 string-based check plus the later literature give me confidence in it. My one substantive request is that Section 6 either enumerate the remaining box topologies or soften the word 'prove' in the abstract, since as written the uniqueness argument rests on a single worked example plus 'one can continue in this way.' That is a short, local fix and should not delay publication. The citation to ref. [42] as 'in preparation' for the quark splitting amplitudes is worth watching but is cross-checked against the published Giele-Glover work, so I would not hold the paper on it."},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"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.","headline":"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.","tokens_in":37811,"tokens_out":1559,"would_cite":true,"duration_ms":26567,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["11.15.Bt","11.30.Pb","12.38.Bx"],"model":"selfhosted/deepseek-v4.1-flash","headline":"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.","keywords":["N=4 super-Yang-Mills","one-loop amplitudes","unitarity cuts","collinear limits","maximally helicity-violating amplitudes","Parke-Taylor amplitudes","scalar box integrals","splitting amplitudes"],"falsifier":"A direct string-based or superspace computation of a seven- or eight-point MHV gluon amplitude in $N=4$ super-Yang-Mills that disagrees with eqs. (4.3)-(4.8), or a demonstration that the $N=4$ integrand retains a term of loop-momentum degree $m-3$ that survives reduction as a triangle integral, would break the uniqueness argument.","tokens_in":38130,"feed_emoji":"⚛️","tokens_out":2605,"duration_ms":50300,"temperature":0.7,"pith_summary":"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.","feed_headline":"Cuts and collinear limits pin down n-gluon loop amplitudes","feed_subtitle":"Unitarity fixes the branch cuts, collinear behavior fixes the rest, and the closed form holds for any number of gluons.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Parke-Taylor tree-level MHV formula that the entire cut calculation is built on.","marker":"[2]"},{"why":"Provides the one-loop five-gluon helicity amplitudes used to jump-start the collinear bootstrap and to extract the loop splitting functions.","marker":"[13]"},{"why":"The string-based method that yields the one-loop amplitudes used as checks and as the source of the subleading-color color decomposition.","marker":"[14]"},{"why":"The Cutkosky rules, the unitarity technique applied at the amplitude level throughout section 5.","marker":"[17]"},{"why":"Passarino-Veltman reduction, used to reduce the cut hexagon and tensor integrals to scalar boxes.","marker":"[24]"},{"why":"Defines and tabulates the scalar box integrals in terms of which the amplitude is expressed in eq. (4.19) and appendix I.","marker":"[26]"},{"why":"Supersymmetry Ward identities that force identical-helicity amplitudes to vanish and relate gluon amplitudes to gluino and scalar ones.","marker":"[5]"},{"why":"The loop-momentum power-counting argument that in $N=4$ the numerator degree is four lower than the gluonic case, leaving only scalar boxes.","marker":"[19]"},{"why":"Dilogarithm identities (including the Abel-identity rearrangement) needed to show the ansatz has the correct collinear limit.","marker":"[46]"}],"fun_headline_variants":["Unitarity cuts fix every coefficient in N=4 gluon loops","Collinear limits and cuts give all-n N=4 MHV amplitudes","A scalar box sum proves the N=4 gluon loop ansatz","The six-point string calculation checks the all-n N=4 formula","The proof evaluates each unitarity cut and fixes the coefficients"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Unitarity cuts fix every coefficient in N=4 gluon loops","Collinear limits and cuts give all-n N=4 MHV amplitudes","A scalar box sum proves the N=4 gluon loop ansatz","The six-point string calculation checks the all-n N=4 formula","The proof evaluates each unitarity cut and fixes the coefficients"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":1500,"prompt_tokens":914,"completion_tokens":586,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":507}},"tokens_in":914,"tokens_out":586,"duration_ms":3955,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":507,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-29T19:40:06.989842+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"falsifier":"A direct string-based or superspace computation of a seven- or eight-point MHV gluon amplitude in $N=4$ super-Yang-Mills that disagrees with eqs. (4.3)-(4.8), or a demonstration that the $N=4$ integrand retains a term of loop-momentum degree $m-3$ that survives reduction as a triangle integral, would break the uniqueness argument.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Parke-Taylor tree-level MHV formula that the entire cut calculation is built on."},{"cited_title":"Phys. Rev. Lett. 70:2 677","cited_arxiv_id":null,"evidence_quote":"Provides the one-loop five-gluon helicity amplitudes used to jump-start the collinear bootstrap and to extract the loop splitting functions."},{"cited_title":"Nucl. Phys. B379:451","cited_arxiv_id":null,"evidence_quote":"The string-based method that yields the one-loop amplitudes used as checks and as the source of the subleading-color color decomposition."},{"cited_title":"Nucl. Phys. 13:181 (1959); S. Mandelstam, Phys. Rev. 112:1344 (1958), 115:1741 (1959) ; R.E. Cutkosky, J. Math. Phys. 1:429","cited_arxiv_id":null,"evidence_quote":"The Cutkosky rules, the unitarity technique applied at the amplitude level throughout section 5."},{"cited_title":"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","cited_arxiv_id":null,"evidence_quote":"Passarino-Veltman reduction, used to reduce the cut hexagon and tensor integrals to scalar boxes."},{"cited_title":"Phys. Lett. B302:299 (1993); erratum B318:649 (1993); Nucl. Phys. B412:751","cited_arxiv_id":null,"evidence_quote":"Defines and tabulates the scalar box integrals in terms of which the amplitude is expressed in eq. (4.19) and appendix I."},{"cited_title":"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","cited_arxiv_id":null,"evidence_quote":"Supersymmetry Ward identities that force identical-helicity amplitudes to vanish and relate gluon amplitudes to gluino and scalar ones."},{"cited_title":"String-Based Perturbative Methods for Gauge Theories","cited_arxiv_id":"hep-ph/9304249","evidence_quote":"The loop-momentum power-counting argument that in $N=4$ the numerator degree is four lower than the gluonic case, leaving only scalar boxes."},{"cited_title":"Lewin, Dilogarithms and Associated Functions (Macdonald, 1958)","cited_arxiv_id":null,"evidence_quote":"Dilogarithm identities (including the Abel-identity rearrangement) needed to show the ansatz has the correct collinear limit."}],"review_version":1}