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High Energy Physics - Theory · arXiv:hep-th/0510253 v2 · submitted 31 Oct 2005 · 74 references

From Trees to Loops and Back

Andreas Brandhuber, Bill Spence, Gabriele Travaglini

PDF · arXiv · The abstract, sentence by sentence · Cite

The paper says

“We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams. We first present a general proof of the covariance of one-loop non-MHV amplitudes obtained from MHV diagrams.”

more of the abstract

This proof relies only on the local character in Minkowski space of MHV vertices and on an application of the Feynman Tree Theorem. We then show that the discontinuities of one-loop scattering amplitudes computed with MHV diagrams are precisely the same as those computed with standard methods. Furthermore, we analyse collinear limits and soft limits of generic non-MHV amplitudes in supersymmetric Yang-Mills theories with one-loop MHV diagrams. In particular, we find a simple explicit derivation of the universal one-loop splitting functions in supersymmetric Yang-Mills theories to all orders in the dimensional regularisation parameter, which is in complete agreement with known results. Finally, we present concrete and illustrative applications of Feynman's Tree Theorem to one-loop MHV diagrams as well as to one-loop Feynman diagrams.

from the abstract

Pith says

Machine review · deepseek-v4.1-flash · 13 Nov 2026 · what was read was not recorded · not checked by a person

Reviewed in 2026 with later literature in view.

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Holds up, with conditions. The Feynman Tree Theorem covariance argument is the real contribution; the equivalence claim is honestly framed as an argument, and the one-particle-cut completion step is the softest joint. This line is wrong

2 major objections · 7 minor

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The paper says, at §3.1, Figs. 3 and 6, and §3.3

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Pith says: Objection 1 of 2 · major

The completion of the one-particle cut diagrams by the 'missing' diagrams of Figs. 3 and 6 is the load-bearing step for the covariance claim of abstract:s1, and it is argued rather than derived. Two justifications are offered: (i) the diagrams in which two adjacent opposite-helicity legs from the same MHV vertex are sewn together vanish after summing over particle species …

more of objection 1

… in a supersymmetric theory; (ii) in the anti-collinear limit they reduce to a splitting function times a sum of tree MHV diagrams and are therefore separately η-independent. Justification (i) is a standard supersymmetric collinear-vanishing identity, but the text does not display the algebra: an explicit line showing that the coefficient of 1/⟨l1 l2⟩ in the sewn diagram vanishes identically after the multiplet sum would make the step checkable rather than asserted. Justification (ii) applies a collinear-factorisation formula at an exact collinear point, which is heuristic. Section 3.3 then extends the completion to arbitrary v with 'it is easy to see'; either an induction on v should be supplied or the phrase 'general proof' in abstract:s1 should be softened to reflect that the general case is argued from the explicitly treated examples. If either justification fails for some helicity configuration, the one-cut diagrams are not covariant and the FTT-based covariance proof does not close, which matters because §4 uses covariance to exclude unphysical η-dependent singularities.

The paper says, at §4, discontinuity matching for non-MHV amplitudes

The review did not quote this passage.

Pith says: Objection 2 of 2 · major

The claim of abstract:s3, that the discontinuities computed with MHV diagrams are precisely those of the standard calculation, is argued in §4 by fixing the two propagators of a given cut and inspecting all MHV diagrams that share that cut.

more of objection 2

The explicit demonstration is given for the MHV case, where the two-particle cut of each diagram is mapped onto a unitarity cut by citation to [4]; for generic non-MHV amplitudes the identification of which MHV diagrams contribute to a given cut, and that their sum equals the two-particle unitarity cut, is asserted by analogy with the pattern of §3.2–3.3. A single worked NMHV example, following the cut diagrams of Fig. 7 through to the unitarity cut, would make the general statement checkable. I note that the authors themselves acknowledge at the end of §4 that multi-particle factorisation is not proven and is the final piece needed for a complete equivalence proof; that acknowledgement is appropriate and I do not present it as a new finding, but it does mean the equivalence argument of §4 is conditional on a step the paper does not supply.

7 minor objections

The paper says, at §4.1.3, Eqs. (4.10)–(4.15)

Pith says: minor objection 1 of 7

The cancellation of bubble and triangle contributions when the four terms of Eq. (4.10) are summed, which justifies replacing R by the effective Reff of Eq. (4.15), is stated rather than shown. One explicit line summing the first term of Eq. (4.14) over the four R's would help the reader verify the cancellation.

The paper says, at §4.1.3, Eq. (4.22)

Pith says: minor objection 2 of 7

The identification of the prefactor multiplying Eq. (4.21) as Splittree times Atree is stated without the intermediate step in which the sum of tree MHV diagrams is recognised as the (n−1)-point amplitude. A sentence indicating how the tree amplitude is reconstructed would improve readability.

show 5 more

The paper says, at Footnote 16

Pith says: minor objection 3 of 7

The η-independence of the all-orders two-mass easy box function is supported by 'extensive numerical checks' rather than an analytic argument. This is acceptable as a check, but the text should say clearly whether an analytic proof is available elsewhere or whether the numerical verification is the only support.

The paper says, at §5.2.1, Eqs. (5.44)–(5.47)

Pith says: minor objection 4 of 7

The passage from the measure in Eq. (5.44) to the final form in Eq. (5.47), including the sign conventions for ε1 and ε2 and the change of variable from z to P²_{L;z}, involves a little algebra that is compressed. A short intermediate display would make the derivation easier to follow.

The paper says, at Figure 10

Pith says: minor objection 5 of 7

The caption defines the box by its two massless legs but does not state explicitly which invariants s, t, P² and Q² correspond to which pair of external legs in the figure. Making this correspondence explicit in the caption would help.

The paper says, at §4, n = 4 boundary case

Pith says: minor objection 6 of 7

The statement that for n = 4 'the only non-vanishing one-loop amplitudes in supersymmetric Yang-Mills are the MHV amplitudes' should be qualified to the gluon amplitudes considered in the paper, since the argument does not address external fermion or scalar states.

The paper says, at Abstract and §3.3

Pith says: minor objection 7 of 7

The phrase 'general proof of the covariance' in abstract:s1 slightly outruns what §3.3 demonstrates, since the arbitrary-v case is asserted with 'it is easy to see'. Aligning the abstract with the level of detail actually supplied would be more accurate.

What would settle it

Take a specific non-MHV configuration, e.g. the 6-point NMHV amplitude with v = 3, and isolate a missing diagram: an MHV vertex carrying adjacent legs l1 = l and l2 = -l (l² = 0) together with the appropriate external legs. Compute the sum over the N=4 multiplet of this vertex and extract the coefficient of 1/⟨l1 l2⟩. If the coefficient is not identically zero, the missing diagrams do not vanish, the 1-cut diagrams are not separately covariant, and the covariance proof has a genuine gap. A secondary check: verify that the same sum, when the two legs are not exactly back-to-back, reproduces the known tree collinear splitting function times the reduced amplitude, confirming argument (ii).

from the skeptic's reading

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  4. Referee minor revision: 2 major objections, 7 minor 2 min 7 s · 34,966 tokens read, 6,195 written

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The review in full

The review in brief, machine-written

The thing worth your time here is Section 3: a proof that one-loop MHV-diagram amplitudes are independent of the reference spinor η, using the Feynman Tree Theorem to break the loop into cuts and showing each cut is separately covariant. That is a genuinely new structural argument, and it is the piece that lets Section 4 exclude unphysical η-dependent poles and discontinuities. Section 5.2.2, where they re-derive the one-loop measure of their earlier paper [4] directly from FTT, is a clean independent check of their own machinery. The all-orders-in-ε two-mass easy box in the Appendix is a useful reference result, and they credit [74] for the four-dimensional form. The splitting function (Eq. 4.24) is a re-derivation of [37,38], and they say so plainly — no overclaiming there. The bubble example in 5.1 checks out term by term, including the one-cut/two-cut cancellation in (5.13)-(5.14). The paper is technically careful and the algebra is explicit where it matters. On the stress-test concern: it lands, but softly. The p ≥ 2 cuts are trivially covariant, as the note says. The p = 1 case needs the 'missing' diagrams of Figs. 3 and 6 to complete the cut into a full tree amplitude. Argument (i) — that adjacent opposite-helicity legs sewn together vanish after summing over the multiplet — is a standard SUSY collinear identity, not something they invented, though they do not derive it. Argument (ii), the anti-collinear reduction to a splitting function times a tree amplitude, is heuristic. And §3.3 generalises to arbitrary v with 'it is easy to see'. That is the weakest sentence in the paper. It is not fatal: the two arguments are plausible and the SUSY one is on solid ground, but a referee should ask them to spell out the general-v case rather than assert it. The other gap — multi-particle factorisation — they state themselves in Section 4 and call it the final missing piece. Credit for that; it is why the abstract says 'we argue' rather than 'we prove'. So: the abstract's 'general proof' (abstract:s1) is a touch stronger than what the body delivers, but only a touch, and the body is honest about it. Who should read it: anyone working on MHV diagrams at loop level, or on FTT as a computational tool. The FTT machinery is more broadly useful than the MHV application. I would cite it for the covariance argument and the box appendix. Send it to referees — this deserves a careful read, not a desk decision.

The strongest objection

The covariance proof (Section 3) rests on showing that each p-particle Feynman cut of the one-loop MHV diagrams is separately η-independent. For p ≥ 2 this is immediate: the cut factorises into on-shell tree amplitudes, each a complete MHV-diagram sum and hence covariant by [2]. The p = 1 case is different. Cutting one propagator produces a connected tree diagram with v vertices and n+2 legs in which the two cut legs sit at different vertices. To recover a complete tree amplitude one must add the 'missing' diagrams of Figs. 3 and 6, where the two cut legs attach to the same vertex. The paper offers two justifications: (i) these diagrams vanish after summing over the N=4 multiplet because the adjacent (+,-) pair is exactly collinear; (ii) in the anti-collinear limit they reduce to a splitting function times a tree amplitude and are therefore separately η-independent. Neither is proved in the text. Argument (i) is a standard SUSY collinear-vanishing identity but is asserted, not derived; argument (ii) applies a collinear-factorisation formula at an exact collinear point and is heuristic. Section 3.3 then generalises the completion to arbitrary v with 'it is easy to see'. If either argument fails for some helicity configuration—e.g. if the coefficient of 1/⟨l1 l2⟩ in the missing diagram does not vanish identically—the 1-cut diagrams are not covariant, and the FTT-based covariance proof does not close. This matters because Section 4 explicitly uses covariance to exclude unphysical η-dependent poles and discontinuities; without it the equivalence argument loses a leg.

The paper's central claim is that one-loop SUSY Yang-Mills amplitudes can be computed equivalently with MHV or Feynman diagrams. The argument has four pillars: covariance (Section 3), matching discontinuities (Section 4), matching collinear and soft poles (Sections 4.1-4.2), and multi-particle factorisation. The last is explicitly acknowledged as missing (Section 4: 'we do not have a proof of the correct multi-particle factorisation at this point; this would be the final piece needed to construct a complete proof of equivalence'). I do not present that as a new finding; the authors state it plainly, and the reader already credits it. The remaining unacknowledged soft spot is the completion of one-particle cuts by the missing diagrams. The reader identifies this correctly. The paper's two justifications are plausible and grounded in known SUSY and collinear-factorisation results, so the concern is not fatal; but neither is proved in the text, and the arbitrary-v generalisation is asserted. Because Section 4 leans on covariance to rule out η-dependent artefacts, a failure of this step would weaken the equivalence argument more than a local technicality. The rest of the paper checks out: the propagator decomposition (Eqs. 2.7-2.8) is standard, the bubble example (Section 5.1) is correct including the one-cut/two-cut cancellation (Eqs. 5.13-5.14), the collinear algebra leading to Eq. 4.24 is explicit and lands on the published splitting function of [37,38], and the re-derivation of the measure in Section 5.2.2 is an independent check of [4]. The verdict CONDITIONAL is appropriate: the paper argues rather than proves its headline, and says so. My concern does not move the verdict.

The referee's report

If the argument holds, the MHV expansion is a valid computational route at one loop and not merely a tree-level curiosity, which is a substantial claim for the field. The manuscript has several concrete strengths that are checkable rather than asserted. The all-orders-in-ε derivation of the one-loop splitting function (Eq. 4.24, via the all-orders two-mass easy box of Eq. 4.18) reproduces the independent published results of [37,38], which is a nontrivial benchmark. The FTT re-derivation of the one-loop measure (Eqs. 5.50–5.59, culminating in Eq. 5.47) is a clean and reproducible calculation. The bubble example of §5.1 is worked term by term, including the cancellation between the one-cut and two-cut pieces (Eqs. 5.13–5.14), and the discussion in §5.1.3 correctly distinguishes Feynman cuts from conventional unitarity cuts. The paper is also candid about its scope: it states that it argues the equivalence, and it flags the multi-particle factorisation gap explicitly at the end of §4.

Pith's reading of the paper (a machine's summary, not the authors')

The paper sets out to show that a diagrammatic method built from maximally helicity violating (MHV) tree amplitudes is not just a tree-level trick: at one loop in supersymmetric Yang-Mills, it gives the same scattering amplitudes as ordinary Feynman diagrams. The authors first prove that the one-loop MHV result does not depend on the arbitrary null reference momentum used to continue MHV amplitudes off shell, then argue that its discontinuities and generalised cuts match standard unitarity cuts. They also derive the universal one-loop splitting functions to all orders in the dimensional-regularisation parameter $epsilon$ and find agreement with known results. The authors flag multi-particle factorisation as the remaining piece needed for a complete equivalence proof.

What carries the argument. The Feynman Tree Theorem rewrites a one-loop integral as a sum over cuts by decomposing each Feynman propagator into a retarded propagator plus an on-shell delta function; the retarded-only loop vanishes by causality and locality, leaving the physical loop as a sum of one-, two-, and higher-particle cuts. In this paper the theorem is applied to MHV diagrams rather than Feynman diagrams, using the fact that an MHV tree amplitude is local in Minkowski space: it localises on a twistor line, which corresponds to a spacetime point. That locality is what makes the causality argument go through and lets the authors show that each cut sector is independent of the MHV reference spinor.

Core claim. The central claim is that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams. The load-bearing step is a proof of covariance: the sum of one-loop MHV diagrams is independent of the reference spinor $eta$ because each $p$-particle cut produced by the Feynman Tree Theorem separately assembles into a covariant tree amplitude or product of tree amplitudes. Matching discontinuities and generalised cuts then fixes the analytic structure, and explicit collinear and soft limits reproduce the known universal one-loop splitting and soft functions, including the all-order-in-$epsilon$ expression for $r^{[1]}_1(z)$.

The premise it rests on. The argument's load-bearing premise is that the 'missing' one-particle cut diagrams can be added back and either vanish after summing over particle species or reduce to a splitting function times a covariant tree amplitude, making each cut sector separately independent of the MHV reference spinor.

If the paper is right (Pith's inferences, not the authors')

If correct, MHV diagrams become a practical one-loop calculational method for generic supersymmetric Yang-Mills amplitudes, not only for MHV helicity configurations.

The Feynman Tree Theorem supplies a general route from on-shell tree data to loop amplitudes, which the authors suggest extends to massive particles and higher loops.

The all-order-in-$epsilon$ splitting functions can be derived diagrammatically, giving a cross-check on infrared structure and dimensional regularisation.

The cut-constructible part of non-supersymmetric amplitudes can also be computed with MHV diagrams, according to the authors' extension of the argument.

The equivalence would reduce reliance on the Feynman-diagram expansion for these amplitudes and support MHV methods beyond one loop.

The covariance proof suggests a general template: any local vertex expansion whose tree amplitudes are covariant may yield covariant loops if the Tree Theorem cuts assemble into tree amplitudes, which could apply to other twistor-inspired or recursion-based formalisms.

If the missing-diagram completion can be made rigorous for arbitrary multiplicity, the per-cut covariance argument would become a general theorem about reference-spinor independence in MHV loop calculations.

The all-order $epsilon$ agreement hints that the MHV diagram method may capture the full infrared and collinear structure even where rational terms are absent; a natural test is to compare finite rational remainders in non-supersymmetric theories.

A practical extension would be to automate the cut assembly for higher-point non-MHV amplitudes and compare against numerical unitarity, though the paper does not attempt this.

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The paper

Abstract

We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams. We first present a general proof of the covariance of one-loop non-MHV amplitudes obtained from MHV diagrams. This proof relies only on the local character in Minkowski space of MHV vertices and on an application of the Feynman Tree Theorem. We then show that the discontinuities of one-loop scattering amplitudes computed with MHV diagrams are precisely the same as those computed with standard methods. Furthermore, we analyse collinear limits and soft limits of generic non-MHV amplitudes in supersymmetric Yang-Mills theories with one-loop MHV diagrams. In particular, we find a simple explicit derivation of the universal one-loop splitting functions in supersymmetric Yang-Mills theories to all orders in the dimensional regularisation parameter, which is in complete agreement with known results. Finally, we present concrete and illustrative applications of Feynman's Tree Theorem to one-loop MHV diagrams as well as to one-loop Feynman diagrams.

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. No sentence in this record was written by a model.

abstract compiled · 2026-09-23

The authors' main claim, as the classifier picked it out 1.00

“We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams.”

How the main claim is stated: asserted 0.83

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.95 · mentions code or data availability 0.02

The abstract, sentence by sentence

This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.

SentenceRoleThe authors' own workStates the main conclusionStrength
We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams. result 0.54 0.36 0.91 asserted 0.83
We first present a general proof of the covariance of one-loop non-MHV amplitudes obtained from MHV diagrams. contribution 0.95 0.94 0.08 asserted 0.99
This proof relies only on the local character in Minkowski space of MHV vertices and on an application of the Feynman Tree Theorem. method 0.99 0.71 0.04 asserted 1.00
We then show that the discontinuities of one-loop scattering amplitudes computed with MHV diagrams are precisely the same as those computed with standard methods. result 0.98 0.93 0.22 asserted 1.00
Furthermore, we analyse collinear limits and soft limits of generic non-MHV amplitudes in supersymmetric Yang-Mills theories with one-loop MHV diagrams. contribution 0.99 0.95 0.04 asserted 1.00
In particular, we find a simple explicit derivation of the universal one-loop splitting functions in supersymmetric Yang-Mills theories to all orders in the dimensional regularisation parameter, which is in complete agreement with known results. result 0.98 0.95 0.10 asserted 1.00
Finally, we present concrete and illustrative applications of Feynman's Tree Theorem to one-loop MHV diagrams as well as to one-loop Feynman diagrams. contribution 0.97 0.94 0.06 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,674 input tokens, $0.0002 reported.

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Figures (17)
Figure 1 of the paper
Figure 1. The MHV diagrams contributing to a one-loop MHV scattering amplitude. The blobs represent MHV vertices, which should then be dressed with external particles in all possible ways compatible with cyclic ordering, and in such a way that the two vertices have the MHV helicity configuration. Let us start illustrating this strategy by considering the simplest case, namely that of an MHV scattering amplitude at one loop. 3.1 One-loop MHV amplitude The one-loop MHV diagrams contributing to an n-point MHV amplitude are pre￾sented in
Figure 2 of the paper
Figure 2. One-particle and two-particle MHV diagrams contributing to the one-loop MHV scattering amplitude. For the MHV amplitude we have to consider only one type of MHV diagram, represented in
Figure 3 of the paper
Figure 3. In this Figure we represent “missing diagrams”, mentioned in the text. It remains to justify the omission of these missing diagrams. We will present two alternative and somewhat complementary explanations. The first one relies on supersymmetry. The diagrams where two adjacent and opposite helicity legs from the same MHV vertex are sewn together vanish when summed over particle species in a supersymmetric theory. Individual diagrams before 10
Figure 4 of the paper
Figure 4. These are the MHV diagrams contributing to a one-loop NMHV scattering amplitude. 3.2 The NMHV amplitude at one loop We now move on and consider the MHV diagrams contributing to a one-loop NMHV amplitude. These diagrams are drawn in
Figure 5 of the paper
Figure 5. This Figure shows the one-particle cut diagrams generated by cutting open one loop propagator in the diagrams of
Figure 6 of the paper
Figure 6. Here we draw the missing diagrams corresponding to a NMHV scattering amplitude at one loop. These are one-particle diagrams which would never be gen￾erated by cutting open one loop propagator in the one-loop MHV diagrams of
Figure 7 of the paper
Figure 7. Two-particle cuts of an NMHV amplitude. 3. Three-particle cuts produce an NMHV amplitude and two MHV amplitudes; 4. Finally, four-particle cuts give rise to four disconnected MHV amplitudes which are trivially covariant. Again, each set of p-particle cut diagrams is covariant. As a consequence of the Feynman Tree Theorem, the sum of these sets reproduces the full physical amplitude, which is therefore covariant. A final comment before closing this section – so far we have not been working in any particular Yang-Mills theory; in particular, we have not required it to be…
Figure 8 of the paper
Figure 8. MHV diagrams contributing to a N2MHV scattering amplitude at one loop. the spinor associated to an off-shell momentum P in the MHV diagram prescription, and the label i refers to a generic external particle, with momentum λiλ˜ i . However, the sum of all tree-level MHV diagrams is covariant (i.e. η-independent), hence these unphysical poles must cancel out in the sum A (n) MHV [2, 13]. It then follows that the difference A (n) F − A(n) MHV must be a polynomial in the momenta of the particles. But scattering amplitudes of n gluons have dimension 4 − n; so for n > 4 the polynomial
Figure 9 of the paper
Figure 9. A schematic example of a one-loop MHV diagram contributing to a generic non-MHV one-loop amplitude where sab is a singular channel. In the collinear limit a k b, diagrams of this type generate the second term on the right hand side of (4.1). loop integrand involves the function [4, 5] Rˆ := ha − 1 aihl2 l1i ha − 1 l1ih−l1 ai hb b + 1ihl1 l2i hb l2ih−l2 b + 1i , (4.8) where the spinors l1 and l2 are those associated to the off-shell loop momenta L1 and L2 according to the prescription of [2],13 that is Li;α,α˙ = liα˜liα˙ + zi ηαη˜α˙ , i = 1, 2 . (4.9) Notice that momentum…
Figure 10 of the paper
Figure 10. The two-mass easy box function F 2me, whose all-order in ǫ expression is given in (4.18) and further studied in the Appendix. The vertical (horizontal) cuts correspond to the s-channel (t-channel) cuts, and the upper left (lower right) corner cuts to the P 2 -channel (Q2 -channel) cuts. The four terms in (4.10) give rise to four different boxes in the sab-cut, which are then integrated with the appropriate dispersive measure in the sab-channel (see section 5.2 and [4] for further details). Thus, what we are seeing here is that the one-loop collinear singularity arises precisely…
Figure 11 of the paper
Figure 11. A schematic example of a one-loop MHV diagram contributing to the non-singular channel diagrams for the −− → − collinear limit, giving rise to the contribution Splittree + (a −, b−) A 1−loop n−1 (1, . . . ,(a + b) −, . . . , n). We now move on to consider the diagrams which are at the origin of the term Split1−loop + (a −, b−) Atree n−1 (1, . . . ,(a+b) −, . . . , n). These diagrams, represented in
Figure 12 of the paper
Figure 12. A schematic example of a one-loop MHV diagram contributing to the singular channel diagrams for the −− → − collinear limit, giving rise to the term Split1−loop + (a −, b−) Atree n−1 (1, . . . ,(a + b) −, . . . , n). The calculation of the diagram in question proceeds along lines very similar to that followed for the ++ → + collinear limit, so we will not spell it out. We would only like to point out one aspect of it which we believe is worth mentioning. In performing the algebra, one encounters ratios such as (kak)/(kbk), where k is the null momentum associated to the internal leg
Figure 13 of the paper
Figure 13. A schematic example of a one-loop MHV diagram contributing to the one-loop soft function Soft1−loop(a +, s+, b+). where Soft1−loop(a, s, b) is a one-loop soft function [54–59]. We will now obtain the all orders in ǫ expression for the one-loop soft function found in [38, 39], in the case where the helicities of the gluons a, s and b are all positive. MHV 4pt MHV s + b + MHV MHV MHV MHV a +
Figure 14 of the paper
Figure 14. The second class of MHV diagrams contributing to Soft1−loop(a +, s+, b+). The relevant one-loop MHV diagrams are drawn in Figures 13-15. After perform￾ing the spinor algebra as in section 4.1, one can see that the term singular in the soft limit gives Soft1−loop(a, s, b) = Softtree(a, s, b) (BI + BII + BIII) , (4.31) where BI , BII and BIII are three dispersion integrals in the s, t and P 2 cut of a one￾mass box function with adjacent massless legs a, s and b. For this box, the parameter a defined in (4.19) is equal to a = −sab/(sasssb), with sij := (ki + kj ) 2 . Using the form (
Figure 15 of the paper
Figure 15. The third class of one-loop MHV diagrams contributing to Soft1−loop(a +, s+, b+). we get Soft1−loop(a, s, b) = Softtree(a, s, b)  − cΓ ǫ 2 πǫ sin(πǫ)   − sab sasssb µ 2 ǫ . (4.33) This is in agreement with the results of [38, 39]. 5 Further applications of the Feynman Tree The￾orem We have seen in previous sections that the Feynman Tree Theorem (2.19) leads us to consider various multiple cuts of a loop diagram. As reviewed in section 2, these cuts assign a specific sign to the energy of the cut line; using the “retarded” version of the theorem, the sign of the ti
Figure 16 of the paper
Figure 16. A bubble diagram, which we firstly evaluate directly, and then by making use of the Feynman Tree Theorem. The two results are in perfect agreement. The bubble diagram we consider is represented in
Figure 17 of the paper
Figure 17. MHV diagrams contributing to a one-loop MHV scattering amplitude. In the text we show that these diagrams can equivalently be computed either by a direct calculation, as in [4], or by resorting to the Feynman Tree Theorem. We can now convert the integration over z1 and z2 in (5.29) to an integration over z defined in (5.28), and z ′ defined by z ′ := z1 + z2 . (5.30) Indeed [4], neither the measure nor the integrand depend on z ′ , which can therefore be integrated out. To do this, we observe that dz1 z1 + iε1 dz2 z2 + iε2 → 2 dz dz′ (z ′ + z + iε1)(z ′ − z + iε2) . (5.3

Figures are the authors', reproduced from the paper.

References (74)
  1. [4] A. Brandhuber, B. Spence and G. Travaglini, One-loop gauge theory amplitudes in N = 4 super Yang-Mills from MHV vertices, Nucl. Phys. B 706, 150 (2005), hep-th/0407214 on PithPith's reviewarXiv
  2. [74] G. Duplancic and B. Nizic, Dimensionally regulated one-loop box scalar integrals with massless internal lines, Eur. Phys. J. C20 (2001) 357-370, hep-ph/0006249 on PithPith's reviewarXiv
  3. [1] E. Witten, Perturbative gauge theory as a string theory in twistor spac e, Com- mun. Math. Phys. 252, 189 (2004), hep-th/0312171 on PithPith's reviewarXiv
  4. [2] F. Cachazo, P. Svrˇ cek and E. Witten, MHV vertices and tree amplitudes in gauge theory, JHEP 0409, 006 (2004), hep-th/0403047 on PithPith's reviewarXiv
  5. [3] R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, The Analytic S-Matrix, Cambridge University Press, 1966 on Pith
  6. [5] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217, hep-ph/9403226 on PithPith's reviewarXiv
  7. [6] C. J. Zhu, The googly amplitudes in gauge theory, JHEP 0404, 032 (2004), hep-th/0403115 on PithPith's reviewarXiv
  8. [7] G. Georgiou and V. V. Khoze, Tree amplitudes in gauge theory as scalar MHV diagrams, JHEP 0405, 070 (2004), hep-th/0404072 on PithPith's reviewarXiv
  9. [8] J. B. Wu and C. J. Zhu, MHV vertices and scattering amplitudes in gauge theory, JHEP 0407, 032 (2004) hep-th/0406085 on PithPith's reviewarXiv
  10. [9] J. B. Wu and C. J. Zhu, MHV vertices and fermionic scattering amplitudes in gauge theory with quarks and gluinos, JHEP 0409, 063 (2004) hep-th/0406146 on PithPith's reviewarXiv
  11. [10] I. Bena, Z. Bern and D. A. Kosower, Twistor-space recursive formulation of gauge theory amplitudes, Phys. Rev. D 71, 045008 (2005), hep-th/0406133 on PithPith's reviewarXiv
  12. [11] G. Georgiou, E. W. N. Glover and V. V. Khoze, Non-MHV tree amplitudes in gauge theory, JHEP 0407, 048 (2004) hep-th/0407027 on PithPith's reviewarXiv
  13. [12] K. J. Ozeren and W. J. Stirling, MHV techniques for QED processes, JHEP 0511 (2005) 016, hep-th/0509063 on PithPith's reviewarXiv
  14. [13] R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level re- cursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602, hep-th/0501052 on PithPith's reviewarXiv
  15. [14] R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715, 499 (2005) hep-th/0412308. 46 on PithPith's reviewarXiv
  16. [15] Z. Bern, L. J. Dixon and D. A. Kosower, On-shell recurrence relations for one- loop QCD amplitudes, Phys. Rev. D 71 (2005) 105013, hep-th/0501240 on PithPith's reviewarXiv
  17. [16] Z. Bern, L. J. Dixon and D. A. Kosower, The last of the finite loop amplitudes in QCD, Phys. Rev. D 72 (2005) 125003, hep-ph/0505055 on PithPith's reviewarXiv
  18. [17] Z. Bern, L. J. Dixon and D. A. Kosower, Bootstrapping Multi-Parton Loop Am- plitudes in QCD, Phys. Rev. D 73 (2006) 065013, hep-ph/0507005 on PithPith's reviewarXiv
  19. [18] Z. Bern, N. E. J. Bjerrum-Bohr, D. C. Dunbar and H. Ita, Recursive Cal- culation of One-Loop QCD Integral Coefficients, JHEP 0511 (2005) 027, hep-ph/0507019 on PithPith's reviewarXiv
  20. [19] S. D. Badger, E. W. N. Glover, V. V. Khoze and P. Svrˇ cek, Recursion rela- tions for gauge theory amplitudes with massive particles, JHEP 0507 (2005) 025, hep-th/0504159 on PithPith's reviewarXiv
show 54 more
  1. [20] S. D. Badger, E. W. N. Glover and V. V. Khoze, Recursion relations for gauge theory amplitudes with massive vector bosons and fermions, JHEP 0601 (2006) 066, hep-th/0507161 on PithPith's reviewarXiv
  2. [21] L. J. Dixon, E. W. N. Glover and V. V. Khoze, MHV rules for Higgs plus multi- gluon amplitudes, JHEP 0412 (2004) 015, hep-th/0411092 on PithPith's reviewarXiv
  3. [22] K. Risager, A direct proof of the CSW rules, JHEP 0512 (2005) 003, hep-th/0508206 on PithPith's reviewarXiv
  4. [23] J. Bedford, A. Brandhuber, B. Spence and G. Travaglini, A recursion relation for gravity amplitudes, Nucl. Phys. B 721 (2005) 98 hep-th/0502146 on PithPith's reviewarXiv
  5. [24] F. Cachazo and P. Svrˇ cek, Tree level recursion relations in general relativity, hep-th/0502160 on PithPith's reviewarXiv
  6. [25] N. E. J. Bjerrum-Bohr, D. C. Dunbar, H. Ita, W. B. Perkins an d K. Risager, MHV-vertices for gravity amplitudes, JHEP 0601 (2006) 009, hep-th/0509016 on PithPith's reviewarXiv
  7. [26] J. Bedford, A. Brandhuber, B. Spence and G. Travaglini, A twistor approach to one-loop amplitudes in N = 1 supersymmetric Yang-Mills theo ry, Nucl. Phys. B 706, 100 (2005), hep-th/0410280 on PithPith's reviewarXiv
  8. [27] C. Quigley and M. Rozali, One-loop MHV amplitudes in supersymmetric gauge theories, JHEP 0501 (2005) 053 hep-th/0410278 on PithPith's reviewarXiv
  9. [28] J. Bedford, A. Brandhuber, B. Spence and G. Travaglini, Non-supersymmetric loop amplitudes and MHV vertices, Nucl. Phys. B 712, 59 (2005) hep-th/0412108. 47 on PithPith's reviewarXiv
  10. [29] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59, hep-ph/9409265 on PithPith's reviewarXiv
  11. [31] R. P. Feynman, Quantum Theory Of Gravitation, Acta Phys. Polon. 24 (1963) 697 on Pith
  12. [32] R. P. Feynman, Closed Loop And Tree Diagrams,in J. R. Klauder, Magic Without Magic, San Francisco 1972, 355-375; in Brown, L. M. (ed.): Selected papers of Richard Feynman, 867-887 on Pith
  13. [33] R. P. Feynman, Problems In Quantizing The Gravitational Field, And The Mas s- less Yang-Mills Field, In J. R. Klauder, Magic Without Magic , San Francisco 1972, 377-408; in Brown, L. M. (ed.): Selected papers of Richard Feynman , 888- 919 on Pith
  14. [34] F. Cachazo, P. Svrˇ cek and E. Witten, Twistor space structure of one-loop am- plitudes in gauge theory, JHEP 0410 (2004) 074, hep-th/0406177 on PithPith's reviewarXiv
  15. [35] Z. Bern and G. Chalmers, Factorization in one loop gauge theory, Nucl. Phys. B 447, 465 (1995), hep-ph/9503236 on PithPith's reviewarXiv
  16. [36] D. A. Kosower, All-order collinear behavior in gauge theories, Nucl. Phys. B 552 (1999) 319, hep-ph/9901201 on PithPith's reviewarXiv
  17. [37] D. A. Kosower and P. Uwer, One-loop splitting amplitudes in gauge theory, Nucl. Phys. B 563 (1999) 477, hep-ph/9903515 on PithPith's reviewarXiv
  18. [38] 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 ord er, Phys. Rev. D 60 (1999) 116001, hep-ph/9903516 on PithPith's reviewarXiv
  19. [39] Z. Bern, V. Del Duca and C. R. Schmidt, The infrared behavior of one-loop gluon amplitudes at next-to-next-to-leading order, Phys. Lett. B 445 (1998) 168, hep-ph/9810409 on PithPith's reviewarXiv
  20. [40] F. Cachazo, P. Svrˇ cek and E. Witten, Gauge theory amplitudes in twistor space and holomorphic anomaly, JHEP 0410 (2004) 077, hep-th/0409245 on PithPith's reviewarXiv
  21. [41] I. Bena, Z. Bern, D. A. Kosower and R. Roiban, Loops in twistor space, Phys. Rev. D 71 (2005) 106010, hep-th/0410054 on PithPith's reviewarXiv
  22. [42] F. Cachazo, Holomorphic anomaly of unitarity cuts and one-loop gauge th eory amplitudes, hep-th/0410077. 48 on PithPith's reviewarXiv
  23. [43] R. Britto, F. Cachazo and B. Feng, Computing one-loop amplitudes from the holomorphic anomaly of unitarity cuts, Phys. Rev. D 71 (2005) 025012, hep-th/0410179 on PithPith's reviewarXiv
  24. [44] Z. Bern, V. Del Duca, L. J. Dixon and D. A. Kosower, All non-maximally- helicity-violating one-loop seven-gluon amplitudes in N = 4 super-Yang-Mills theory, Phys. Rev. D 71 (2005) 045006, hep-th/0410224 on PithPith's reviewarXiv
  25. [45] S. J. Bidder, N. E. J. Bjerrum-Bohr, L. J. Dixon and D. C. Dunb ar, N = 1 supersymmetric one-loop amplitudes and the holomorphic an omaly of unitarity cuts, Phys. Lett. B 606 (2005) 189, hep-th/0410296 on PithPith's reviewarXiv
  26. [46] R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725, 275 (2005), hep-th/0412103 on PithPith's reviewarXiv
  27. [47] Z. Bern, D. Forde, D. A. Kosower and P. Mastrolia, Twistor-inspired con- struction of electroweak vector boson currents, Phys. Rev. D 72 (2005) 025006, hep-ph/0412167 on PithPith's reviewarXiv
  28. [48] Z. Bern, L. J. Dixon and D. A. Kosower, All next-to-maximally helicity-violating one-loop gluon amplitudes in N = 4 super-Yang-Mills theory, Phys. Rev. D 72, 045014 (2005), hep-th/0412210 on PithPith's reviewarXiv
  29. [49] S. J. Bidder, N. E. J. Bjerrum-Bohr, D. C. Dunbar and W. B. Pe rkins, One-loop gluon scattering amplitudes in theories with N < 4 supersymmetries, Phys. Lett. B 612 (2005) 75, hep-th/0502028 on PithPith's reviewarXiv
  30. [50] R. Britto, E. Buchbinder, F. Cachazo and B. Feng, One-loop amplitudes of gluons in SQCD, Phys. Rev. D 72, 065012 (2005), hep-ph/0503132 on PithPith's reviewarXiv
  31. [51] L. J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 on PithPith's reviewarXiv
  32. [52] T. G. Birthwright, E. W. N. Glover, V. V. Khoze and P. Marquard , Multi-gluon collinear limits from MHV diagrams, JHEP 0505 (2005) 013, hep-ph/0503063 on PithPith's reviewarXiv
  33. [53] T. G. Birthwright, E. W. N. Glover, V. V. Khoze and P. Marquard , Collinear limits in QCD from MHV rules, JHEP 0507 (2005) 068, hep-ph/0505219 on PithPith's reviewarXiv
  34. [54] Z. Kunszt, A. Signer and Z. Trocsanyi, One loop helicity amplitudes for all 2 → 2 processes in QCD and N=1 supersymmetric Yang-Mills theory , Nucl. Phys. B 411 (1994) 397, hep-ph/9305239 on PithPith's reviewarXiv
  35. [55] Z. Bern and D. A. Kosower, The Computation of loop amplitudes in gauge theo- ries, Nucl. Phys. B 379, 451 (1992) on Pith
  36. [56] R. K. Ellis and J. C. Sexton, QCD Radiative Corrections To Parton Parton Scattering, Nucl. Phys. B 269, 445 (1986). 49 on Pith
  37. [57] Z. Bern, L. J. Dixon and D. A. Kosower, One loop corrections to five gluon amplitudes, Phys. Rev. Lett. 70, 2677 (1993), hep-ph/9302280 on PithPith's reviewarXiv
  38. [58] Z. Kunszt, A. Signer and Z. Trocsanyi, One loop radiative corrections to the helicity amplitudes of QCD processes involving four quarks and one gluon, Phys. Lett. B 336, 529 (1994), hep-ph/9405386 on PithPith's reviewarXiv
  39. [59] Z. Bern, L. J. Dixon and D. A. Kosower, One loop corrections to two quark three gluon amplitudes, Nucl. Phys. B 437 (1995) 259, hep-ph/9409393 on PithPith's reviewarXiv
  40. [60] A. Brandhuber, S. McNamara, B. Spence and G. Travaglini, Loop ampli- tudes in pure Yang-Mills from generalised unitarity, JHEP 0510, 011 (2005) hep-th/0506068 on PithPith's reviewarXiv
  41. [61] Y. t. Huang, N = 4 SYM NMHV loop amplitude in superspace, Phys. Lett. B 631 (2005) 177, hep-th/0507117 on PithPith's reviewarXiv
  42. [62] K. Risager, S. J. Bidder and W. B. Perkins, One-loop NMHV amplitudes in- volving gluinos and scalars in N = 4 gauge theory, JHEP 0510 (2005) 003, hep-th/0507170 on PithPith's reviewarXiv
  43. [63] D. Forde and D. A. Kosower, All-multiplicity amplitudes with massive scalars, Phys. Rev. D 73 (2006) 065007, hep-th/0507292 on PithPith's reviewarXiv
  44. [64] D. Forde and D. A. Kosower, All-multiplicity one-loop corrections to MHV am- plitudes in QCD, Phys. Rev. D 73 (2006) 061701, hep-ph/0509358 on PithPith's reviewarXiv
  45. [65] C. Anastasiou, Z. Bern, L. J. Dixon and D. A. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91 (2003) 251602, hep-th/0309040 on PithPith's reviewarXiv
  46. [66] C. Anastasiou, Z. Bern, L. J. Dixon, and D. A. Kosower, Cross-order relations in N = 4 supersymmetric gauge theories, hep-th/0402053 on PithPith's reviewarXiv
  47. [67] Z. Bern, L. J. Dixon and V. A. Smirnov, Iteration of planar amplitudes in max- imally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001, hep-th/0505205 on PithPith's reviewarXiv
  48. [68] E. I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N = 4 super Yang-Mills, JHEP 0511 (2005) 036, hep-th/0506126 on PithPith's reviewarXiv
  49. [69] L. D. Landau, On Analytic Properties Of Vertex Parts In Quantum Field Theo ry, Nucl. Phys. 13 (1959) 181 on Pith
  50. [70] R. E. Cutkosky, Singularities And Discontinuities Of Feynman Amplitudes, J. Math. Phys. 1 (1960) 429 on Pith
  51. [71] C. Itzykson and J. B. Zuber, Quantum Field Theory, McGraw-Hill (1985). 50 on Pith
  52. [72] D. A. Kosower, Next-to-maximal helicity violating amplitudes in gauge th eory, Phys. Rev. D 71, 045007 (2005), hep-th/0406175 on PithPith's reviewarXiv
  53. [73] V. P. Nair, A Current Algebra For Some Gauge Theory Amplitudes, Phys. Lett. B 214 (1988) 215 on Pith
  54. [75] Z. Bern, L. J. Dixon and D. A. Kosower, Dimensionally regulated pentagon in- tegrals, Nucl. Phys. B 412 (1994) 751, hep-ph/9306240. 51 on PithPith's reviewarXiv
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  2. One-loop Gravitational Bremsstrahlung and Waveforms from a Heavy-Mass Effective Field Theory hep-th · Mar 2023
  3. The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space hep-th · Sep 2010
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  6. The Cosmological Tree Theorem hep-th · Aug 2023
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  16. An Introduction to On-shell Recursion Relations hep-th · Nov 2011
How the literature uses it

33 arXiv papers cite this in 58 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 · 37 sentences without a confident label

Uses its method or tool 1 paper
Compares with it 2 papers
  • This is different from the usual analyses of MHV one-loop calculations in [ref] which use four dimensional MHV vertices.

    S-matrix equivalence restored · Conclusion and higher loops · 0.98

  • Before performing this expansion we note that now the integrals over $q_1$ and $q_2$ simply transform the HEFT amplitude with four scalars to impact parameter space, and hence can be stripped off, leaving us with [We note that this expression for the forward limit is similar to expressions found in e.g. [ref] and it would be interesting to investigate this further.] [equation] where we have reparameterised the momentum as $q_3=-ℓ_3=ℓ_1+q_1$ to make contact with our usual loop integration variables in [ref].

    One-loop Gravitational Bremsstrahlung and Waveforms from a Heavy-Mass Effective Field Theory · Details of the calculation · 0.86

Cites it as support 2 papers
  • Note that at tree level and for one loop MHV amplitudes this is not a conjecture, see respectively [ref] and [ref] .[The CSW construction was argued to hold for any helicity configuration in [ref] .

    Symmetries of the N=4 SYM S-matrix · Generalizations to All Loops and Helicities · 0.96

  • As at tree level, the proof [ref] that the momentum space rules are equivalent to the usual Feynman rules ensures that the complete integrand is independent of $|ι]$.

    MHV Diagrams in Momentum Twistor Space · General 1-loop integrands · 0.96

Cites it as background 15 papers

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