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Multi-Leg One-Loop Gravity Amplitudes from Gauge Theory

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Z. Bern, L. Dixon, M. Perelstein, J.S. Rozowsky

High Energy Physics - Theory · Submitted 15 Nov 1998 · 62 references

Pith's verdict: Careful, honest technical work: the n ≤ 6 amplitudes are solid and cross-checked, but the 'infinite sequences' claim rests on an ansatz whose N=8 collinear check is numerical only through n=7.

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CitationJ.S. Rozowsky, L. Dixon, M. Perelstein, Z. Bern. "Multi-Leg One-Loop Gravity Amplitudes from Gauge Theory." arXiv:hep-th/9811140 (1998). https://pith.science/paper/H4VSON6O
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The paper

The paper claims

The paper argues that one-loop graviton scattering amplitudes can be obtained, for two whole helicity families and any number of external legs, by recycling gauge-theory calculations through the Kawai-Lewellen-Tye squaring relations rather

The reading

The authors want to show that quantum gravity amplitudes with an arbitrary number of external gravitons are knowable in closed form, at least in two special helicity sectors, without evaluating a single gravity Feynman diagram. The engine is the Kawai-Lewellen-Tye (KLT) relation, which in the field-theory limit expresses a tree-level gravity amplitude as a sum of products of two gauge-theory amplitudes; applied inside unitarity cuts, it converts a hard gravity loop calculation into a bookkeeping exercise over gluon trees. Working in $D = 4 - 2ε$ dimensions so that the cuts fix the full amplitude rather than just its discontinuities, the authors compute the all-plus (self-dual) $n$-graviton amplitudes exactly for $n = 4, 5, 6$, and from them, via a gravitational 'dimension-shifting' relation, the maximally helicity-violating amplitudes of $N=8$ supergravity. Beyond six legs they do not compute; they write down closed-form expressions for all $n$ and argue that the soft and collinear limits of gravity, which they derive from gauge theory, pin those expressions down. If the argument holds, two infinite sequences of one-loop graviton amplitudes are known explicitly, and the same dimension-shifting trick that relates all-plus to MHV in gauge theory carries over to gravity with the loop dimension shifted by eight instead of four.

What carries the argument

Three objects carry the argument. First, the KLT relations, which write a tree-level $n$-graviton amplitude as a permutation sum over products of two colour-ordered $n$-gluon amplitudes weighted by kinematic factors $f, ħf$ built from Mandelstam invariants; this is what lets every gravity tree amplitude appearing on either side of a cut be replaced by gauge theory. Second, the unitarity cut in arbitrary dimension: because the cuts are known to all orders in $ε$, the loop amplitude is reconstructed without the subtraction ambiguities that plague four-dimensional dispersion relations, and the all-plus case is especially clean since a supersymmetry Ward identity lets any massless particle in the loop be replaced by a real scalar. Third, the 'half-soft' functions $h(a, M, b)$ — off-shell extensions of the tree amplitudes, defined recursively through symmetric coefficients $φ(i_1, …, i_m)$ or equivalently through non-standard Young tableaux — which obey a soft limit resembling the gravitational one and whose collinear behaviour, after cancellation between paired terms, reproduces the universal gravity splitting amplitude $[a b]/⟨a b⟩$. These $h$ functions are the building blocks of both all-$n$ ansätze, and they themselves satisfy squaring relations of the form $h(a,{1,2},b) = s_{12} g(a,{1,2},b) g(a,{2,1},b)$ with the gauge-theory functions $g$.

Extended reading notes

Core claim

The central claim is that two infinite sequences of one-loop $n$-graviton amplitudes are determined: the all-plus helicity amplitudes of gravity with any minimally coupled massless matter content (which correspond to self-dual field configurations and vanish in supersymmetric theories), and the MHV amplitudes of $N=8$ supergravity. For $n ≤ 6$ the all-plus amplitudes are obtained exactly from their unitarity cuts in arbitrary dimension, giving explicit results in eqs. (4.12), (4.20) and (4.25), with coefficients built from products of gauge-theory box coefficients in a way that mirrors the tree-level KLT relations. The gravitational dimension-shifting relation, eq. (3.16), then yields the $n ≤ 6$ $N=8$ MHV amplitudes from the all-plus ones by removing eight powers of the $( -2ε)$-dimensional loop momentum and multiplying by $⟨i j⟩^8/2$; the authors derive this relation at the level of cuts by combining KLT with the known gauge-theory version, eq. (3.15), which shifts the loop dimension by four. For $n > 6$ the paper offers ansätze, eqs. (6.13) and (6.18), built from 'half-soft' functions $h(a, M, b)$ that reproduce the correct soft and collinear behaviour, and reports numerical checks of the collinear limits through $n = 7$ and of the infrared divergences through $n = 8$.

Load-bearing premise

The all-$n$ claim rests on the assumption that an amplitude with the correct soft behaviour and the correct collinear phase singularity is the amplitude, and the authors state plainly in section 6 that they have no proof the $n > 6$ expressions are the unique functions with those limits.

Editorial extensions

If this is right

  • One-loop graviton amplitudes in the all-plus and $N=8$ MHV helicity sectors are available in closed form for any number of external legs, removing the need for leg-by-leg Feynman-diagram computation in those sectors.
  • The one-loop all-plus amplitudes constitute the complete perturbative solution of the Chalmers-Siegel self-dual gravity action, since that action has no tree-level on-shell amplitudes and no contributions beyond one loop.
  • The gravitational dimension-shifting relation provides a working dictionary: an $N=8$ MHV amplitude at loop dimension $D+8$ is the corresponding all-plus gravity amplitude at $D$, up to a spinor factor, mirroring the four-unit shift in gauge theory.
  • The $N=8$ MHV amplitudes contain only scalar box integrals with at most two external masses, fewer integral types than naive power counting from squaring $N=4$ super-Yang-Mills would allow, indicating cancellations beyond those expected from supersymmetry alone.
  • The tree-level soft and collinear splitting functions of gravity are shown to receive no loop corrections at any order, unlike their gauge-theory counterparts, because the dimensionful coupling suppresses the would-be corrections.

Reading between the lines

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

  • If the soft-and-collinear uniqueness premise holds, the same bootstrap strategy should extend to other helicity sectors of $N=8$ supergravity at one loop, and possibly to the two-loop MHV amplitudes where only two-particle cuts are currently under control.
  • The absence of triangle and bubble integrals in the $N=8$ MHV amplitudes, despite power counting that permits them, suggests an organisational principle at the level of the integrand rather than the integrated result; a string-based or twistor-style derivation would be the natural place to look for it.
  • The $h$ functions' squaring relation to the gauge-theory $g$ functions hints that the one-loop coefficient functions, and not just the tree amplitudes, admit a KLT-like factorisation; if so, the loop-level gravity-gauge correspondence could be stated as an algebraic identity on integrands rather than as an equality of integrated amplitudes.
  • Because the all-plus amplitudes are rational functions with singularities only at soft and collinear configurations, they are a natural testing ground for whether the modern combinatorial machinery used for scattering amplitudes reproduces the older unitarity-cut answers.
Original abstract

By exploiting relations between gravity and gauge theories, we present two infinite sequences of one-loop n-graviton scattering amplitudes: the `maximally helicity-violating' amplitudes in N=8 supergravity, and the `all-plus' helicity amplitudes in gravity with any minimally coupled massless matter content. The all-plus amplitudes correspond to self-dual field configurations and vanish in supersymmetric theories. We make use of the tree-level Kawai-Lewellen-Tye (KLT) relations between open and closed string theory amplitudes, which in the low-energy limit imply relations between gravity and gauge theory tree amplitudes. For n < 7, we determine the all-plus amplitudes explicitly from their unitarity cuts. The KLT relations, applied to the cuts, allow us to extend to gravity a previously found `dimension-shifting' relation between (the cuts of) the all-plus amplitudes in gauge theory and the maximally helicity-violating amplitudes in N=4 super-Yang-Mills theory. The gravitational version of the relation lets us determine the n < 7 N=8 supergravity amplitudes from the all-plus gravity amplitudes. We infer the two series of amplitudes for all n from their soft and collinear properties, which can also be derived from gauge theory using the KLT relations.

The compiled record

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

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,837 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.48 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.21 · compares with prior results 0.66 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

By exploiting relations between gravity and gauge theories, we present two infinite sequences of one-loop n-graviton scattering amplitudes: the `maximally helicity-violating' amplitudes in N=8 supergravity, and the `all-plus' helicity amplitudes in gravity with any minimally coupled massless matter content. The all-plus amplitudes correspond to self-dual field configurations and vanish in supersymmetric theories. We make use of the tree-level Kawai-Lewellen-Tye (KLT) relations between open and closed string theory amplitudes, which in the low-energy limit imply relations between gravity and gauge theory tree amplitudes. For n < 7, we determine the all-plus amplitudes explicitly from their unitarity cuts. The KLT relations, applied to the cuts, allow us to extend to gravity a previously found `dimension-shifting' relation between (the cuts of) the all-plus amplitudes in gauge theory and the maximally helicity-violating amplitudes in N=4 super-Yang-Mills theory. The gravitational version of the relation lets us determine the n < 7 N=8 supergravity amplitudes from the all-plus gravity amplitudes. We infer the two series of amplitudes for all n from their soft and collinear properties, which can also be derived from gauge theory using the KLT relations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 By exploiting relations between gravity and gauge theories, we present two infinite sequences of one-loop n-graviton scattering amplitudes: the `maximally helicity-violating' amplitudes in N=8 supergravity, and the `all-plus' helicity amplitudes in gravity with any minimally coupled massless matter content. contribution 0.74 0.95 0.86 asserted 1.00
s1 The all-plus amplitudes correspond to self-dual field configurations and vanish in supersymmetric theories. interpretation 0.60 0.13 0.05 asserted 1.00
s2 We make use of the tree-level Kawai-Lewellen-Tye (KLT) relations between open and closed string theory amplitudes, which in the low-energy limit imply relations between gravity and gauge theory tree amplitudes. method 0.68 0.15 0.04 asserted 1.00
s3 For n < 7, we determine the all-plus amplitudes explicitly from their unitarity cuts. method 0.63 0.94 0.06 asserted 1.00
s4 The KLT relations, applied to the cuts, allow us to extend to gravity a previously found `dimension-shifting' relation between (the cuts of) the all-plus amplitudes in gauge theory and the maximally helicity-violating amplitudes in N=4 super-Yang-Mills theory. method 0.78 0.79 0.10 asserted 1.00
s5 The gravitational version of the relation lets us determine the n < 7 N=8 supergravity amplitudes from the all-plus gravity amplitudes. method 0.62 0.88 0.14 asserted 1.00
s6 We infer the two series of amplitudes for all n from their soft and collinear properties, which can also be derived from gauge theory using the KLT relations. method 0.62 0.82 0.32 asserted 0.93

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

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

Figures
Figure 1. Kinematics of the two-mass box integrals I aK1bK2 4 that enter n-point MHV amplitudes in both N = 4 super-Yang-Mills theory and N = 8 supergravity. Here a and b label the external massless legs for the integral, which coincide with two external momenta for the amplitude, ka and kb. The massive legs carry momenta K1 and K2, which are sums of the remaining external momenta. The four Lorentz invariants are the masses K2 1 and K2 2 , and the Mandelstam invariants S1a = (K1 + ka) 2 and S1b = (K1 + kb) 2 . over all distinct non-adjacent pairs of massless legs a, b. The coefficients of the…
Figure 2. Relations between infinite sequences of one-loop amplitudes in four different theories. The vertical arrows correspond to the ‘dimension-shifting’ relations of ref. [15], within (super-) Yang-Mills theory and (super-) gravity. (These remain a conjecture for n ≥ 7 legs.) The horizontal arrows correspond to the gauge-gravity relations which follow from the KLT equations. Using eq. (3.16) it is then sufficient to calculate the all-plus graviton amplitudes as a function of ǫ (i.e., for arbitrary D = 4 − 2ǫ), in order to obtain the N = 8 supergravity MHV amplitudes. 4.3 All-Plus…
Figure 3. As two momenta become collinear, the gravity S-matrix element develops a phase singularity which can be detected by rotating the two momenta about the axis formed by their sum. For example, consider the two factors, (a) [1 2] h1 2i , (b) [1 3] h1 3i . (5.6) If we take ~k1 to be nearly collinear with ~k2 and rotate ~k1 and ~k2 around the vector P~ = ~k1 + ~k2 the factor (b) undergoes only a slight numerical variation. On the other hand, from eq. (2.7), the factor (a) undergoes a large phase variation, proportional to the angle of rotation. Thus a Fourier analysis 25
Figure 4. The class of tree diagrams in a gravity theory that can have a phase singularity factorizes in the collinear limit a k b. The appearance of the same three-vertex for any number of external legs implies the universality of the tree-level splitting amplitudes. The splitting amplitudes are given by evaluating the three-vertex in the collinear limit in a helicity basis. Similarly, the validity of the soft factor (2.21) for an arbitrary number of external legs also follows from the factorization properties of Feynman diagrams. In this case the tree diagrams that contribute to the soft…
Figure 5. The class of tree diagrams in a gravity theory that contribute in the soft limit, where leg s is soft. The soft functions are found by summing over all three-vertices containing a soft leg. One may also prove the universality of the tree-level splitting amplitudes using the n-point version of the KLT relations given in appendix A. Alternatively, one may obtain the tree-level soft and collinear splitting functions from string theory by extending the gauge theory discussion given 3 In the helicity formalism [29], a reference momentum entering the polarization vector or tensor could…
Figure 6. One-loop factorizing corrections to (a) the collinear splitting amplitudes and (b) the soft functions. Now consider the non-factorizing contributions. In one-loop gauge theory amplitudes the infrared divergences, for e.g. a pure gluon amplitude, are of the form −A tree n Xn i6=j h 1 ǫ 2 − ln(−sij ) ǫ i . (5.7) The mismatch between the infrared divergence of the n- and (n − 1)-point one-loop amplitudes on the left- and right-hand-sides of the collinear limit (3.3) implies that there must be a non-trivial contribution to the one-loop gauge splitting amplitude. This may be cont
Figure 7. One- and two-loop examples of diagrams that can produce a pole in k1 from the loop integration in the gauge theory case, but not in the gravity case. For the case of gravity, the graviton vertex attached to leg 1 (and marked by a cross in the figure) contains one extra power of k1 or L1, as compared to the gauge theory vertex, and the contribution is therefore suppressed compared to gauge theory. Since one can obtain at most a single power of 1/k1 in the gauge theory case, the gravity case cannot have a pole in k1 and is therefore suppressed compared to the tree gravity soft S…
Figure 8. The configurations of external legs that are summed over in eq. (6.13). The fact that the amplitudes (6.13) have the correct soft limits is a consequence of the soft properties of the h functions, eq. (6.2). As kn → 0, the term labeled by (a, M, b, N) in Mn−1 gets contributions from two terms in Mn, those labeled by (a, M + n, b, N) and (a, M, b, N + n). Each 32
Figure 9. Sewing of a double-off-shell current S and a single-off-shell current J to obtain the all-plus loop amplitude Mn(1+, 2 +, . . . , n+). The particle circulating in the loop is a scalar, while the remaining lines are gravitons. The nth leg must belong to J. Thus we have to retain the µ-dependence in the last step of the recursion relation for Sq. The µ 2 terms in eq. (B.15) have an additional propagator, so they correspond to a triangle integral in D = 6 − 2ǫ, multiplied by a factor of −ǫ. Because such a triangle integral has an ultraviolet 1/ǫ pole, we get a finite contribution as ǫ →

Figures reproduced from arXiv:hep-th/9811140 by the authors.

References
How later papers used it

159 later papers cite it

Evidence: substantial. 156 of the citing papers share no author with it, and 33 of those carry a confident label from Pith's citation classifier.

Pith read 20 of the 295 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

Builds on it 1 paper
Compares with it 1 of 15 papers with this label
How the literature uses it

153 arXiv papers cite this in 290 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 · 188 sentences without a confident label

Builds on it 1 paper
Uses its method or tool 13 papers
  • To write the gravity kernel in a double-copied form, we first perform a basis expansion of the Lie polynomial $b(P)$ in ([ref]): [equation] As shown in [ref] , this relation can be inverted to obtain the kinematic numerator $n^p ( ℓ[1Q] )$ by means the KLT matrix $S(P|Q)_1$ [ref] : [equation] Substituting in ([ref]) we obtain a double-copy form for the gravity kernel, expressed in terms two copies of the color-stripped Yang-Mills kernel and the KLT matrix: [equation] To make the expression fully explicit, we recall from ([ref], [ref]) that [equation] and that the KLT matrix is given by [ref] [equation] where $θ(i_t, i_q)$ is 0 if $i_t$ sequentially comes before $i_q$ in ${j_1, …, j_k}$, and 1 otherwise.

    Color-kinematics and double-copy relations for selfdual solutions · Double-copy form · 0.81

  • We shall use a simple formula for general one-loop amplitude in terms of tree amplitudes [ref] follows from “no triangle hypothesis” [ref] .

    A Note on Single Soft Scalar Emission of $\mathcal{N}=8$ SUGRA and $E_{7(7)}$ Symmetry · Introduction · 0.81

  • This paper explores the KLT relation in these aspects by referring to the NSVW/BDPR tree formula [ref] to discuss the tree structure of the KLT relation.

    The KLT Relation from the Tree formula and Permutohedron · Introduction · 0.90

Uses its data or values 2 papers
Compares with it 14 papers
Cites it as support 1 paper
  • The structure of the soft (also given in [ref] ) and collinear limits give additional evidence for the extra simplicity of gravity compared to gauge theory: remarkably, the soft factors and collinear splitting functions are not renormalized beyond tree level, as a direct consequence of the dimensionful coupling constant in gravity!

    What is the Simplest Quantum Field Theory? · Absence of Rational Terms · 0.96

Cites it as background 56 papers
  • Once the large string tension limit (also called the field theory limit) is taken, the KLT relation turns into a relation between gravity and gauge theory tree level amplitudes [ref] .

    Soft and Collinear Limits in $\mathcal{N}=8$ Supergravity using Double Copy Formalism · Double copy and collinear limit · 0.86

  • On-shell recursion relation for tree-level gluon amplitudes [ref] has been shown to be important not only for real calculations, but also for theoretical understanding of many important properties such as the BCJ relaton [ref] [The BCJ relation has first been proved in string theory [ref] , and then in field theory [ref] .] and Kawai-Lewellen-Tye relation [ref] [See also recent review [ref] .].

    Note on Cyclic Sum and Combination Sum of Color-ordered Gluon Amplitudes · Introduction · 0.98

  • The field-theory limit of the celebrated Kawai-Lewellen-Tye (KLT) relations [ref] between tree amplitudes in open and closed string theory gives double copy relations linking gauge-theory amplitudes to gravitational ones [ref] .

    String amplitudes from field-theory amplitudes and vice versa · Introduction · 0.95

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