Simplifying Multiloop Integrands and Ultraviolet Divergencesof Gauge Theory and Gravity Amplitudes

# Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes

Z. Bern, J. J. M. Carrasco, L. J. Dixon, H. Johansson and R. Roiban Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095, USA
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USATheory Group, Physics Department, CERN, CH–1211 Geneva 23, Switzerland
Stanford Institute for Theoretical Physics and Department of Physics,
Stanford University, Stanford, CA 94305, USA
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA
Institut de Physique Théorique, CEA–Saclay, F–91191 Gif-sur-Yvette cedex, France
Department of Physics, Pennsylvania State University, University Park, PA 16802, USA
###### Abstract

We use the duality between color and kinematics to simplify the construction of the complete four-loop four-point amplitude of super-Yang-Mills theory, including the nonplanar contributions. The duality completely determines the amplitude’s integrand in terms of just two planar graphs. The existence of a manifestly dual gauge-theory amplitude trivializes the construction of the corresponding supergravity integrand, whose graph numerators are double copies (squares) of the super-Yang-Mills numerators. The success of this procedure provides further nontrivial evidence that the duality and double-copy properties hold at loop level. The new form of the four-loop four-point supergravity amplitude makes manifest the same ultraviolet power counting as the corresponding super-Yang-Mills amplitude. We determine the amplitude’s ultraviolet pole in the critical dimension of , the same dimension as for super-Yang-Mills theory. Strikingly, exactly the same combination of vacuum integrals (after simplification) describes the ultraviolet divergence of supergravity as the subleading-in- single-trace divergence in super-Yang-Mills theory.

###### pacs:
04.65.+e, 11.15.Bt, 11.30.Pb, 11.55.Bq

UCLA/11/TEP/110 SU-ITP-11/34 Saclay–IPhT–T11/175
SLAC–PUB–14529 NSF–KITP–11–236 CERN–PH–TH/2011/190

## I Introduction

The past few years have brought remarkable advances in understanding scattering amplitudes in the maximally supersymmetric super-Yang-Mills (sYM) theory N4YM () in the planar limit of a large number of colors. It may soon be possible to completely determine all planar scattering amplitudes in this theory, for all values of the coupling, going far beyond the (now thoroughly understood) cases of four and five external gluons BDS (). Much of this progress has been surveyed recently SolveAmplReview (). Planar scattering amplitudes exhibit a new symmetry known as dual conformal symmetry DualConformal (); BCDKS (), which severely restricts their structure. Together with supersymmetry and (position space) conformal symmetry, dual conformal invariance gives rise to a Yangian Yangian ()—an algebraic structure common in integrable models. Indeed, it is widely believed that several aspects of the planar sector of sYM theory are controlled by an integrable model (see e.g. ref. IntegrableReview ()).

In contrast, much less is known about the nonplanar sector—the subject of the present paper. Consider sYM theory for the gauge group . In the limit , the nonplanar, or subleading-color, contributions are suppressed by powers of . Once one takes into account these corrections, for finite , the scattering amplitudes no longer appear to possess dual conformal symmetry, nor do they demonstrate any obvious integrability properties.

Understanding the subleading-color terms is critical to a complete description of the behavior of gauge theories. For example, many types of color correlations are suppressed in the large- limit. Furthermore, the information provided by the full-color expression for sYM amplitudes, expressed in terms of their loop-momentum integrands, can be used to construct corresponding amplitudes BDDPR (); GravityThree (); GravityFour (); Neq44np () in supergravity CremmerJulia (). From each set of amplitudes one can extract information about ultraviolet divergences in the respective theory.

The ultraviolet (UV) properties of supergravity have been the focus of intense investigation. There have been several recent reviews of the situation GravityUVReview (). Long ago, an supersymmetric local counterterm at three loops in was proposed DeserKayStelle (); Ferrara1977mv (); Deser1978br (); Howe1980th (); Kallosh1980fi (). An explicit computation of the three-loop four-graviton amplitude first revealed that the counterterm has a vanishing coefficient GravityThree (). Subsequently it was realized EKR4 () that this counterterm is forbidden in by the duality symmetry CremmerJulia (). Other analyses have extended the finiteness constraints from and linearized supersymmetry, such that the first potential divergence in is now at seven loops BHN (); BHS2010 (); Beisertetal (); BHSV (). Finiteness until this loop order happens to agree with an earlier naive power-counting, based on the assumption of an off-shell superspace Siegel7l (). A potential seven-loop divergence is also suggested by other approaches, including an analysis of string theory dualities GRV2010 (), a first-quantized world-line approach FirstQuantized (), and light-cone supergraphs KalloshRamond (). However, it has also been argued that the theory may remain finite beyond seven loops FiniteArgue ().

In this paper, we will show how a conjectured duality between color and kinematics BCJ (); BCJLoop () provides a powerful method for computing subleading-color terms in sYM amplitudes, in a way that makes the construction of the corresponding supergravity amplitudes extremely simple. Also, the result is expressed in a form that makes manifest the true ultraviolet behavior of the amplitude (when continued to higher space-time dimension ). Thus this method provides unprecedented access to the precise coefficients of potential counterterms in supergravity, as well as in its higher-dimensional versions. It may eventually offer a means for settling the question of whether additional UV cancellations exist in supergravity, beyond the known or expected ones. Perhaps even more importantly, the method gives a means for constructing complete amplitudes, allowing for detailed studies of their symmetries and properties.

A key point is that when the color-kinematics duality holds manifestly, it locks the nonplanar contributions to the planar ones. The nonplanar contributions are essential for evaluating gravity amplitudes, because in gravity theories no separation exists between planar and nonplanar contributions. This duality allows one to efficiently export information from the planar sector, e.g. that provided by dual conformal symmetry, to the much more intricate nonplanar sector.

A second key point is the claim BCJLoop () that if a duality-respecting representation of sYM amplitudes can be found, then the loop-momentum integrands of the corresponding supergravity amplitudes can be obtained simply by taking the graphs of sYM theory, dropping the color factors and squaring their kinematic numerators. This double-copy property is a loop-level generalization of the corresponding tree-level property BCJ (), equivalent to the Kawai-Lewellen-Tye (KLT) relations between gravity and gauge-theory amplitudes KLT (). Using the color-kinematics duality, followed by the double-copy property, advances in constructing integrands for the planar sector of gauge theory can be carried over to the nonplanar sector, and then on to gravity. The color-kinematic duality and the gravity double-copy property do not appear to require supersymmetry, although amplitudes in supersymmetric theories are generally much simpler to work with than non-supersymmetric amplitudes. Another important aspect of the duality is that it appears to hold in any dimension, thus making it compatible with dimensional regularization.

In this paper we will exploit the color-kinematic duality to construct the complete four-loop four-point amplitudes of sYM theory and supergravity. Both amplitudes were constructed previously by us GravityFour (); Neq44np (); however, the present construction is considerably more efficient, and makes various properties of the amplitudes more manifest. The color-kinematic duality relations allow us to express the four-loop loop-momentum integrands, for 83 different cubic (trivalent) graphs, as functionals of the integrands of just two planar graphs. For the one-, two- and three-loop four-point BCJLoop (); JJHenrikReview (), and the one- and two-loop five-point cases loop5ptBCJ (), the duality is even more restrictive: a single planar graph suffices to determine all the others. As it is becoming increasingly simple to construct planar amplitudes, a particularly attractive aspect of using the color-kinematic duality is that it determines nonplanar contributions from planar ones. Perhaps even more remarkable, in terms of measuring the redundancy found in local gauge-theory scattering amplitudes, we shall find that the entire non-trivial dynamical information in the four-loop four-point amplitude is contained in a single nonplanar graph; all other graphs are related to this one by the duality.

While a general proof of the duality conjecture is yet to be given beyond tree level, the four-loop construction we offer in this paper provides further evidence in favor of it, in the form of a highly nontrivial example. In this work, we have confirmed the duality-based construction by verifying that the integrand matches a complete (spanning) set of generalized unitarity cuts.

Based on the double-copy structure of supergravity amplitudes, we will give a new representation of the four-loop four-point supergravity amplitude. This construction provides a direct multiloop confirmation of the double-copy property, because we verify the generalized unitarity cuts for the new form of the supergravity amplitude, against the cuts of the known expression GravityFour (), originally constructed using the KLT relations. We also explore the ultraviolet properties of the amplitude in dimensions. An essential feature of the new representation is that the UV behavior is manifest: Individual integrals diverge logarithmically in precisely the same critical dimension as their sum. This property did not hold for the previous form of the amplitude GravityFour (). The critical dimension is also the same as that for the planar and single-trace sectors of sYM theory. In a previous paper GravityFour (), we showed that the supergravity amplitude is finite for , which is also the bound obeyed by sYM theory. However, the previous form of the amplitude did not display this bound manifestly. To see the cancellation of potential UV divergences, the integrals had to be expanded a few orders in powers of the external momenta. The lack of manifest UV behavior in that representation made it difficult to carry out the required integration in and to determine whether the amplitude actually does diverge in this dimension. With the new form, this task is greatly simplified, allowing us to carry it out here.

Due to the double-copy construction, the numerators of the integrands for the supergravity amplitude are perfect squares. However, they multiply propagator denominators that do not have definite signs. Therefore, individual integrals contributing to the amplitude can have different signs. To probe whether or not the four-loop amplitude diverges in , it is necessary to actually evaluate the UV divergences in the loop integrals in this dimension. Using the double-copy form of the four-loop four-point amplitude we do so, finding that the finiteness bound is in fact saturated at at four loops, which matches the behavior of sYM theory. Moreover, we calculate the precise coefficient of the supergravity divergence. We find that it exactly matches the coefficient of the divergence of the -suppressed single-trace term in the four-loop four-point amplitude of sYM theory, up to an overall rational factor. Although this property is most striking at four loops, only emerging after a number of simplifications, it is consistent with lower-loop behavior. Presumably this consistent connection is a clue for unraveling the general UV properties of supergravity.

Regularization is a crucial point in the construction of loop-level amplitudes in massless theories, because such amplitudes are usually either infrared or UV divergent. The issue of regularization has been studied in some detail in the context of unitarity cuts in ref. SixD (), where the six-dimensional helicity formalism SixDHel () was suggested as a general means for implementing either dimensional regularization DimReg () or a massive infrared regulator equivalent to the one in ref. HiggsReg (). In the present paper we take advantage of an earlier construction of the four-loop four-point amplitude of sYM theory, which provides expressions with demonstrated validity for  Neq44np (); SixD (). In this paper, we compare the -dimensional unitarity cuts of the new results with the cuts of the earlier results. We find exact agreement, confirming the new representations.

This paper is organized as follows. In section II we explain our strategy for constructing multiloop integrands, illustrating it with the three-loop four-point sYM amplitude. In section III (and appendix B) we present the new forms of the four-loop integrands of sYM theory and supergravity. We also outline their construction. In section IV, we obtain the explicit value of the UV divergence of the supergravity amplitude in and discuss its properties. We also determine the UV divergence of the color double-trace terms in the four-loop sYM amplitude in . (We had found earlier Neq44np () that the double-trace divergence canceled in the next possible lower dimension, .) We give our conclusions and outlook in section V. Several appendices are included. The first one gives functional defining relations between the numerators in the four-loop four-point amplitude, which are derived from the Jacobi relations after imposing some auxiliary conditions valid for sYM theory. Appendix B presents the analytic expressions for the numerators. Appendix C gives the values of the vacuum integrals entering the UV divergence for the supergravity amplitude in the critical dimension, as well as expressions for the vacuum integrals’ numerators. Explicit expressions for the color factors for each contribution to the full four-loop amplitude may be found online Online (), where we also provide plain-text, computer-readable versions of the numerator factors and the kinematic dual Jacobi relations that they obey.

## Ii Constructing multiloop integrands

The unitarity method UnitarityMethod (); BCFUnitarity (); FiveLoop (); Neq44np () has become a general-purpose tool for constructing multiloop amplitudes in gauge and gravity theories. In this section we demonstrate how one can dramatically reduce the complexity of unitarity-based calculations for gauge theories by assuming the conjectured duality between color and kinematics BCJ (); BCJLoop (). This duality reduces the construction of an amplitude at the integrand level to the determination of the numerator factors for a small set of graphs, which we call “master graphs”. For the four-loop four-point sYM amplitude, it suffices to use just two planar master graphs. Alternatively, a single nonplanar master graph is sufficient.

Given the duality-satisfying form of the sYM amplitude, the supergravity amplitude can be written down immediately by squaring the sYM numerator factors. We confirm the correctness of the derived gauge and gravity amplitudes using a spanning set of generalized unitarity cuts, showing that they agree with our previous forms GravityFour (); Neq44np () on these cuts.

### ii.1 Duality between color and kinematics

In general, a massless -point -loop gauge-theory amplitude in space-time dimensions, with all particles in the adjoint representation, may be written as

 A(L)m = iLgm−2+2L∑i∈Γ∫L∏l=1dDpl(2π)D1SiniCi∏αip2αi, (1)

where is the gauge coupling constant. The sum runs over the complete set of -point -loop graphs with only cubic (trivalent) vertices, including all permutations of external legs. In each term, the product in the denominator runs over all propagators of the corresponding cubic graph. The integrations are over the independent loop momenta . The coefficients are the color factors obtained from the gauge-group structure constants by dressing every three-vertex in the graph with a factor

 ~fabc=i√2fabc=Tr([Ta,Tb]Tc), (2)

where the hermitian generators of the gauge group are normalized via . The coefficients are kinematic numerator factors depending on momenta, polarization vectors and spinors. For supersymmetric amplitudes in an on-shell superspace, the numerators will also contain Grassmann parameters. The symmetry factors of each graph remove any overcount introduced by summing over all permutations of external legs, as well as any internal automorphisms—symmetries of a graph with external legs fixed. This form of the amplitude may be obtained from representations involving higher-point contact interactions (as long as they are built out of ’s). One reexpresses all contact terms as the product between a propagator and its inverse; i.e. one inserts , and assigns the inverse propagator to be part of a numerator factor . The duality conjecture BCJ (); BCJLoop () states that there should exist a representation of the amplitude where the numerator factors satisfy equations in one-to-one correspondence with the Jacobi identities of the color factors. Explicitly, it requires that

 Ci=Cj+Ck⇒ni=nj+nk, (3)

where the first relation holds, thanks to the usual color Jacobi identity, for any triplet of graphs which are identical within the gray region in fig. 1. Moreover, the numerator factors carry the same antisymmetry properties as color factors, i.e. if a color factor changes sign under the interchange of two legs, then so does the corresponding kinematic numerator factor:

 Ci→−Ci⇒ni→−ni. (4)

These relations are conjectured to hold to all multiplicities, to all loop orders in a weak-coupling expansion, and in any dimension in both supersymmetric and non-supersymmetric Yang-Mills theory. Such relations were noticed long ago for four-point tree amplitudes Halzen (). Beyond the four-point level, the relations are rather nontrivial and only work after appropriate rearrangements of the amplitudes.

While the sign of the complete numerator factor of a graph is unambiguous, the sign of each factor, as well as the signs in eq. (3), may be changed as a consequence of the relations (4) by simply interchanging two lines at any vertex. This ambiguity reflects the different possible sign conventions in Jacobi identities. In this paper, the sign of each color factor (and implicitly of ) is fixed by the corresponding graph figure: for each integral, is built out of the structure constants corresponding to each trivalent vertex, with legs ordered clockwise in the plane of the figure.

The kinematic version of the Jacobi identity (3) is the key equation for the duality. At loop level, this equation relates graph numerators at the integrand level, as illustrated in fig. 1. Therefore it is important to properly line up both external and internal momenta. There is one such equation for every propagator of every graph. Of course, many of the equations are simply related to each other by automorphic symmetries of graphs, and the fact that any given equation can be obtained starting from each of the three contributing graphs. Simultaneous consideration of all relations gives a system of linear functional equations that the amplitude’s numerators should obey. As we will see, only a tiny subset of the possible equations needs to be used when solving the system. Once a tentative solution is found, one must verify that the full set of equations is satisfied, in order to have a duality-satisfying representation of the amplitude. The existence of at least one solution consistent with the unitarity cuts is the critical assumption of the conjecture. Indeed, as we will see, the system of equations at four loops is quite nontrivial and the emergence of a solution is striking.

There is by now substantial evidence in favor of the duality, especially at tree level (OtherTreeBCJ (); virtuousTrees (); Square (); Oconnell (), where explicit representations of the numerators in terms of partial amplitudes are known for any number of external legs ExplicitForms (). A consequence of this duality is the existence of nontrivial relations between the color-ordered partial tree amplitudes of gauge theory BCJ (), which have been proven both from field theory Feng () and string theory Bjerrum1 () perspectives. These relations were important in the recent construction of all open-string tree amplitudes MSSallStringAmpl (). A partial Lagrangian understanding of the duality has also been given Square (). An alternative trace-based presentation of the duality relation (3), which emphasizes its group-theoretic structure, was described recently Trace ().

While less is known at loop level, several nontrivial tests have been carried out. In particular, it has been confirmed that the duality holds for the three-loop four-point amplitude of sYM theory BCJLoop (). (The one- and two-loop four-point amplitudes GSB (); BRY (); BDDPR () in this theory also manifestly satisfy the duality). Similarly, the duality-satisfying five-point one-, two- and three-loop amplitudes of sYM theory have recently been constructed loop5ptBCJ (). The color-kinematic duality is also known to hold BCJLoop () for the two-loop four-point identical-helicity amplitude of pure Yang-Mills theory AllPlus2 (). At present there is no proof that the system of equations generated by the duality (3) always has a solution consistent with the unitarity cuts of a given theory, so it needs to be checked case by case. In section III we will find a solution for the four-loop four-point amplitude of sYM theory.

Perhaps more surprising than the duality itself is a consequent relation between gauge and gravity amplitudes. Once the gauge-theory amplitudes are arranged into a form satisfying eq. (3), the numerator factors of the corresponding -loop gravity amplitudes, , can be obtained simply by multiplying together two copies of gauge-theory numerator factors BCJ (); BCJLoop (),

 (5)

where is the gravitational coupling. The represent numerator factors of a second gauge-theory amplitude and the sum runs over the same set of graphs as in eq. (1). At least one family of numerators ( or ) must satisfy the duality (3). The construction (5) is expected to hold in a large class of gravity theories, including all theories that are the low-energy limits of string theories. At tree level, this double-copy property encodes the KLT relations between gravity and gauge-theory amplitudes KLT (). For supergravity both and are numerators of sYM theory.

The double-copy formula (5) has been proven Square () for both pure gravity and for supergravity tree amplitudes, under the assumption that the duality (3) holds in the corresponding gauge theories, pure Yang-Mills and sYM theory, respectively. The nontrivial part of the loop-level conjecture is the existence of a representation of gauge-theory amplitudes that satisfies the duality constraints. The double-copy property was explicitly checked for the three-loop four-point amplitude of supergravity in ref. BCJLoop () by comparing eq. (5) against a spanning set of unitarity cuts of the previously-calculated amplitude GravityThree (); CompactThree (). Here we perform a similar check for the four-loop four-point amplitude of supergravity, using the maximal cut method FiveLoop (); CompactThree (). The duality and double-copy property have also been confirmed in one- and two-loop five-point amplitudes in supergravity loop5ptBCJ (). For less-than-maximal supergravities, the double-copy property has been checked explicitly for the one-loop four- and five-graviton amplitudes of supergravity N46Sugra () by showing it matches known results DunbarEttle (). In the two-loop four-graviton amplitudes in these theories, it has been verified to be consistent with the known infrared divergences and other properties N46Sugra2 (). The double-copy property also leads to some interesting relations between certain supergravity and subleading-color sYM amplitudes SchnitzerBCJ ().

### ii.2 Calculational setup

We now demonstrate how the conjectured duality between color and kinematics streamlines the construction of integrands of multiloop gauge-theory amplitudes. We first give an overview of the procedure and illustrate it with the three-loop four-point amplitude, before turning to the four-loop case in the following section.

To start the construction we enumerate the graphs with only cubic vertices that can appear in a particular amplitude. Although this step can be carried out in many different ways, we describe one that conveniently also generates the needed duality relations: We assume we have a given set of known cubic graphs (e.g. at four loops we can start with the planar cubic graphs ones given in ref. BCDKS ()). Any missing graphs can then be generated by applying the Jacobi relations (3) to the set of known graphs. New graphs generated in this way are then added to the list of known ones. This process continues recursively, until no further graphs or relations are found. At the end of the process all cubic graphs related via the duality are known, and all duality relations (3) have been written down.

The next step is to solve the relations thus generated. This is the most complicated part of the construction. We can, however, simplify the step by dividing it up into two separate parts, the first of which is straightforward. First use a subset of the duality relations to express all numerator factors in terms of the numerators of a judiciously chosen small set of graphs, which we call “master graphs”. We identify master graphs by systematically eliminating numerator factors from the duality relations, via a functional analog of the standard row reduction of systems of linear equations. This problem is analogous to the reduction of loop integrals to a set of master integrals using the Laporta algorithm Laporta (). In both cases, there is freedom to change the order in which the linear equations are solved. Here, there is a freedom in the choice of master graphs, which is equivalent to a choice of path in solving the system of duality relations. In all cases we have examined, it is convenient to choose the master graphs to be planar (although such a restriction does not necessarily yield the smallest set). This choice has the advantage that the planar contributions are relatively simple and well studied in the literature. In particular, the planar contributions to the four-loop four-point amplitude have a fairly simple form BCDKS (). For the three-loop four-point sYM amplitude we only need a single master graph BCJLoop (); JJHenrikReview (). In section III, we will find that at four loops we can express all numerators in terms of the numerators of only two planar master graphs (or a single nonplanar master graph).

After the reduction of the system of duality constraints, our task is to find explicit expressions for the master numerators. As with any functional equations, a good strategy is to write down Ansätze for the master numerators. The Ansätze are then constrained using input from unitarity cuts, as well as symmetry requirements on both the master numerators and on the numerators derived from them through the duality relations.

In addition to the duality relations (3) and unitarity cuts, we may add extra constraints on numerator factors, motivated by our prejudices about the structure and properties of the amplitude. Although not necessary, such constraints, when well chosen, can greatly facilitate the construction. To find the four-loop four-point sYM amplitude we use the following auxiliary constraints, which are known to be valid for the duality-satisfying numerators at three loops BCJLoop ():

1. One-loop tadpole, bubble and triangle subgraphs do not appear in any graph.

2. A one-loop -gon subgraph carries no more than powers of loop momentum for that loop.

3. After extracting an overall factor of , the numerators are polynomials in -dimensional Lorentz products of the independent loop and external momenta.

4. Numerators carry the same relabeling symmetries as the graphs.

We will assume that these observations carry over to the four-loop four-point amplitude. If one of these auxiliary conditions had turned out to be too restrictive, it would have led to an inconsistency with either the unitarity cuts or the duality relations. We would then have removed conditions one by one until a consistent solution were found. As we shall see in section III, these auxiliary constraints are quite helpful for quickly finding a duality-satisfying representation for the four-loop four-point amplitude. A surprisingly small subset of generalized unitarity cuts is then sufficient to completely determine this amplitude.

The specific auxiliary constraints that should be imposed depend on the problem at hand. The third constraint is clearly specific to the four-point amplitude, and should be modified for higher-point amplitudes because they have a more complicated structure. For the five-point case, however, a simple generalization has been found loop5ptBCJ (), involving pre-factors that are proportional virtuousTrees () to linear combinations of five-point tree-amplitudes. For amplitudes in less supersymmetric theories, the first and second conditions should be relaxed (in addition to the third one), because one-loop triangle and bubble subgraphs are known to appear in such theories.

### ii.3 Three-loop warmup

To illustrate the above procedure in some detail, we reconstruct the well-studied three-loop four-point amplitude of sYM theory. This amplitude was originally constructed in refs. GravityThree (); CompactThree (). A form compatible with the duality (3) was then found BCJLoop (). Here we describe how to streamline the construction of the latter form, before following a similar procedure at four loops.

A straightforward enumeration shows that there are 17 distinct cubic graphs with three loops and four external legs, which do not have one-loop triangle or bubble (or tadpole) subgraphs. Only 12 contribute to the amplitude, as shown in ref. BCJLoop (). These 12 nonvanishing graphs, shown in fig. 2, are sufficient for explaining the construction. (Had we kept all 17 graphs, the construction would be only slightly more involved, with the result that the numerators of the additional five graphs vanish.)

Each numerator depends on three independent external momenta, labeled by , and on (at most) three independent loop momenta, labeled by , as well as on the external states. The Mandelstam variables are , and . We denote the color-ordered tree-level amplitude by . The four-point amplitudes of sYM theory are special. Supersymmetry Ward identities fix the external state dependence, and imply that an overall prefactor of , or equivalently the crossing-symmetric prefactor  GravityThree (); CompactThree (), can be extracted from every numerator factor , leaving behind new numerator factors that depend only on the momenta,

 n(x) = stAtree4(1,2,3,4)N(x), N(x) ≡ N(x)(k1,k2,k3,l5,l6,l7). (6)

Here refers to the label for each graph in fig. 2. This result has been argued to be valid in any dimension Neq44np (), justifying the third assumption above for these amplitudes. The homogeneity of the Jacobi relations implies that they hold for just as for . The crossing symmetry of implies that the symmetry properties of are the same as those of .

Next, we write down a subset of the duality relations that allows us to identify the master graphs JJHenrikReview (). For the three-loop four-point amplitude, one such restricted set of duality relations is:

 N(a) = N(b)(k1,k2,k3,l5,l6,l7), N(b) = N(d)(k1,k2,k3,l5,l6,l7), N(c) = N(a)(k1,k2,k3,l5,l6,l7), N(d) = N(h)(k3,k1,k2,l7,l6,k1,3−l5+l6−l7)+N(h)(k3,k2,k1,l7,l6,k2,3+l5−l7), N(f) = N(e)(k1,k2,k3,l5,l6,l7), N(g) = N(e)(k1,k2,k3,l5,l6,l7), N(h) = −N(g)(k1,k2,k3,l5,l6,k1,2−l5−l7)−N(i)(k4,k3,k2,l6−l5,l5−l6+l7−k1,2,l6), N(i) = N(e)(k1,k2,k3,l5,l7,l6)−N(e)(k3,k2,k1,−k4−l5−l6,−l6−l7,l6), N(j) = N(e)(k1,k2,k3,l5,l6,l7)−N(e)(k2,k1,k3,l5,l6,l7), N(k) = N(f)(k1,k2,k3,l5,l6,l7)−N(f)(k2,k1,k3,l5,l6,l7), N(l) = N(g)(k1,k2,k3,l5,l6,l7)−N(g)(k2,k1,k3,l5,l6,l7), (7)

where . For convenience we have suppressed the canonical arguments of the numerators on the left-hand side of the equations (7), as we will often do in the remainder of the paper. Each relation specifying an is generated by considering the dual Jacobi relations focusing around the lightly colored (pink) line labeled in fig. 2. In general, the duality condition relates triplets of numerators; sometimes, however, one or two of the numerators vanish because the associated graph has a one-loop triangle subgraph forbidden by our auxiliary constraints. Specifically, for five of the above equations, the duality sets pairs of numerators equal; this occurs because the third term in the triplet of numerators of eq. (3) vanishes due to the presence of a triangle subgraph. The above system can be used to express any numerator factor in terms of combinations of the numerator , with various different arguments. Thus, graph (e) can be taken as the sole master graph. This is a convenient choice, but not the only possible one; for example, either graph (f) or (g) can also be used as a single master graph. None of the remaining nine graphs, however, can act alone as a master graph.

One valid numerator factor (consistent with unitarity cuts) for graph (e) is the “rung-rule” numerator BRY (),

 N(e)rr=s(l5+k4)2. (8)

With this numerator, the graph possesses dual conformal symmetry. However, it turns out that this numerator is incompatible with the duality between color and kinematics (3).

We are therefore looking for a modification of consistent with both the maximal cut of the graph and with the duality constraints (7). We start by requiring that the maximal cut of graph (e) is correct, and that the auxiliary constraints in section II.2 are satisfied. That is, the numerator has mass dimension four and possesses the symmetry of the graph; no loop momentum for any box subgraph in (e) appears in it (ruling out and ); and is at most quadratic in the pentagon loop momenta . (This last condition is looser than the second auxiliary constraint in section II.2; we will tighten it shortly.) The symmetry condition implies that should be invariant under

 {k1↔k2,k3↔k4,l5→k1+k2−l5}. (9)

The most general polynomial consistent with these constraints is of the form,

 N(e)=s(l5+k4)2+(αs+βt)l25+(γs+δt)(l5−k1)2+(αs+βt)(l5−k1−k2)2, (10)

where the four parameters are to be determined by further constraints. All added terms are proportional to inverse propagators and therefore vanish on the maximal cut. Thus, since eq. (8) is consistent with the maximal cuts so is eq. (10).

According to the second auxiliary constraint in section II.2, the numerator of a pentagon subgraph should be at most linear in the corresponding loop momentum, not quadratic as assumed above. Therefore the coefficient of in eq. (10) should vanish, yielding the relations and . Consequently, the Ansatz for is reduced to

 N(e)=s(τ45+τ15)+(αs+βt)(s+τ15−τ25), (11)

where we use the notation,

 (12)

Now there are just two undetermined parameters, and .

To determine one of the remaining parameters we use the properties of graph (j), and the expression for its numerator in terms of the numerator of graph (e), which is given by the 9th duality constraint in eq. (7). Inserting eq. (11) into this relation leads to

 N(j)=s(1+2α−β)(τ15−τ25)+βs(t−u). (13)

Because the smallest loop in graph (j) carrying is a box subgraph, our auxiliary constraints require that this momentum be absent from . Setting the first term in eq. (13) to zero implies that , which in turn leads to

 N(e) = s(τ45+τ15)+(α(t−u)+t)(s+τ15−τ25), (14) N(j) = (1+2α)(t−u)s, (15)

leaving undetermined a single parameter .

There are a variety of ways to determine the final parameter. For example, one can use planar cuts to enforce that the planar part of the amplitude is correctly reproduced. A particularly instructive method is to use the duality relations to obtain the numerator for planar graph (a) in terms of master numerator . The numerator is quite simple once we impose the condition that a one-loop box subgraph cannot carry loop momentum. Since three independent one-loop subgraphs of graph (a) are boxes, the numerator cannot depend on any loop momenta. Indeed, the iterated two-particle cuts, or equivalently the rung insertion rule BRY (), immediately fix this contribution to be

 N(a)=s2. (16)

Solving the duality relations (7) to express in terms of we find,

 N(a) = N(e)(k1,k2,k4,−k3+l5−l6+l7,l5−l6,−l5) (17) +N(e)(k2,k1,k4,−k3−l5+l7,−l5,l5−l6) −N(e)(k4,k1,k2,l6−l7,l6,l5−l6)−N(e)(k4,k2,k1,l6−l7,l6,−l5) −N(e)(k3,k1,k2,l7,l6,l5−l6)−N(e)(k3,k2,k1,l7,l6,−l5).

Plugging in the value of the numerator factor in eq. (14), we obtain

 N(a)=s2+(1+3α)((τ16−τ46)s−2(τ17+τ37)s+(τ16−2τ17−τ26+2τ27)t+4ut). (18)

Demanding that this expression matches the numerator factor given in eq. (16), or alternatively that it is independent of loop momenta, fixes the final parameter to be . This constraint completely determines the numerator of graph (e) to be

 N(e)=s(τ45+τ15)+13(t−s)(s+τ15−τ25), (19)

matching the result of ref. BCJLoop ().

Remarkably, numerator (e) in eq. (19) generates all other numerators , via eq. (7), giving us the entire integrand at three loops. For all graphs, the resulting numerators reproduce the expressions quoted in ref. BCJLoop (), and the resulting amplitude matches previous expressions GravityThree (); CompactThree () on all -dimensional unitarity cuts. As already noted, it is highly nontrivial to have a consistent solution where all duality relations hold, all numerators have the graph symmetries and all unitarity cuts are correct. Squaring these numerators , using eq. (5), immediately yields the numerators for the three-loop four-point supergravity amplitude. This form has also been confirmed against previous expressions GravityThree (); CompactThree () on a spanning set of -dimensional unitarity cuts BCJLoop ().

We shall use the same streamlined strategy to construct the four-loop four-point amplitude in section III. Before carrying out this construction, however, we need to address an important subtlety that appears in the construction of the three-loop amplitude and affects the four-loop construction as well.

### ii.4 Comment on one-particle-reducible graphs and snails

Beyond tree-level, the on-shell three-point amplitudes of sYM theory vanish. The appearance of one-particle reducible (1PR) graphs in the three-loop four-point amplitude may therefore seem surprising. Indeed, graphs (i), (j) and (k) of fig. 2 do not appear in the original representations of the same amplitude GravityThree (); CompactThree (). The existence of 1PR graphs may seem to imply that the three-point amplitude is non-vanishing. However, these graphs’ numerators are proportional to the Mandelstam invariant , which is also the inverse propagator for the sole line on which the graph is 1PR. Thus, the superficially 1PR graphs are in fact just one-particle-irreducible (1PI) contact graphs. Even though they are kinematically equivalent to 1PI graphs, the non-contact form of graphs (i), (j) and (k) in fig. 2 is needed to describe easily their color structure, and to allow the amplitude to obey the duality (3) between color and kinematics. As we shall see, this feature continues at four loops, where we encounter, not only graphs with three-point subgraphs, but also nontrivial two-point subgraphs. Some of these graphs contain four-loop two-point bubble subgraphs on external legs, and must be treated with particular care.

At first sight, it may appear surprising that two- and three-point subgraphs show up; indeed in sYM theory we expect the vanishing of on-shell two- and three-point loop amplitudes. This property has been known in string theory for some time Martinec (). By taking the low-energy limit, it should hold in field theory as well. A direct field theory argument for the vanishing of the on-shell two-point function can be made as follows: Quite generally, a (diagonal) on-shell two-point loop contribution represents a correction to the mass of the corresponding field. Gauge invariance forbids such a term from being generated in the gluon two-point function. The chirality of sYM gluinos forbids such mass terms from being generated by perturbative quantum effects for fermions as well. Thus, gluon and gluino two-point functions vanish on shell. Manifest off-shell supersymmetry, which can be maintained, then implies that the scalar field two-point function also vanishes on shell.

We can also argue that three-point amplitudes vanish on shell. Because supersymmetry relates all such amplitudes to each other, it suffices to focus on the scattering amplitude of two fermions and one scalar. Up to an -symmetry transformation, we may further assume that neither of the fermions is the superpartner of the gluon. Thus we consider only the interaction between matter multiplets. Conservation of the matter -symmetry subgroup requires that the three-field interaction is controlled by invariance, and thus is either holomorphic or antiholomorphic. Now, in the effective action language, the three-point amplitude originates either from terms in the superpotential or the Kähler potential. Due to the perturbative nonrenormalization of the superpotential SeibergNonrenormalization (), only the tree amplitude comes from the former. A nonvanishing loop amplitude can only originate from a correction to the Kähler potential. For this case, a nonvanishing full superspace integral and Lorentz invariance require that the product of three chiral superfields containing the relevant wave functions must be accompanied by at least two additional superderivatives. In turn, this implies that the product of one scalar and two fermion wave functions is always accompanied by an external momentum invariant, originating from the superspace integration measure. For massless fields, any such product vanishes on shell. Thus, all quantum corrections to three-point amplitudes in sYM theory vanish on shell, completing the argument.

While these arguments confirm the vanishing of two- and three-point amplitudes at one-loop and beyond, we emphasize that this does not mean that we cannot have graphs with two- and three-point subgraphs. However, when such graphs appear they should always carry factors that make their contributions vanish whenever legs are cut (placed on shell) to isolate two- and three- point subamplitudes. Indeed, we shall find that at four points, through four loops, all such graphs with three- or four-point subgraphs can be absorbed as contact terms in other graphs. This property is consistent with the fact that previous representations of the three- and four-loop amplitudes GravityThree (); CompactThree (); Neq44np () do not use any 1PR graphs with two- or three-point loop subgraphs.

At loops, the four-point amplitude in sYM theory is expected to have a representation with at most powers of the loop momentum in the numerator of each 1PI cubic graph BDDPR (); HoweStelleRevisited (). At three loops, this power counting allows for cancellation of one internal propagator, as in graphs (i), (j) and (k) of fig. 2. However, it precludes the existence of two-point graphs or propagator corrections (and tadpole graphs), which would require two inverse propagators or four powers of the loop momentum in the numerator. On the other hand, at four loops and beyond, such graphs can and indeed do appear. Propagator corrections can be of two types:

1. on internal legs, as shown in fig. 3(a), or

2. on external legs, as shown in fig. 3(c).

In both cases the graph’s numerators must contain momentum invariants that cancel out the unwanted poles, so that they are kinematically equivalent to the 1PI graphs shown in fig. 3(b) and fig. 3(d), respectively.

For case 1, this cancellation is straightforward because the momentum invariant is nonvanishing for generic on-shell kinematics. For case 2, the external leg corrections, the mechanism is more subtle. On the one hand, because the amplitudes have on-shell external legs, a propagator in fig. 3(c) diverges: . On the other hand, from the vanishing of the on-shell two-point function we expect that the numerator of fig. 3(c) is proportional to . In order to resolve this 0/0 ambiguity, we need to regulate the external leg by taking , and cancel factors of between numerator and denominator. This procedure yields the “snail graph”111With suitable imagination, the graph resembles a snail (as much as a penguin diagram resembles a penguin). in fig. 3(d), which is perfectly well behaved at the level of the integrand, even with all external momenta on-shell.

It is important to note that the snail graph in fig. 3(d) contains a scale-free integral, which vanishes by the usual rules of dimensional regularization. We cannot, however, simply ignore these contributions. In dimensional regularization, scale-free integrals evaluate to zero because of cancellations between infrared and UV singularities. Ignoring the snail graph contributions in sYM theory would lead to incorrect values for the UV divergences.222In QCD, propagator corrections on external legs can be ignored because the UV divergences are known a priori. It is therefore quite simple to restore the missing terms. In sYM theory, UV divergences in are unknown a priori. Since we are interested in this paper in the coefficient of the UV divergences, these snail graphs must be included.

While enforcing the duality constraints (3) brings the phenomenon of snail graphs to the forefront, we emphasize that the potential appearance of such contributions to amplitudes is independent of the color-kinematic duality. Snail contributions can in principle occur within any representation; it is therefore important to always check the unitarity cuts for such contributions. Because snails are associated with external leg contributions, ordinary unitarity cuts fail to detect them, and generalized cuts are required. The momenta of the states crossing the cut must either be complex, or else have an indefinite sign of their energy. (We have examined such cuts, and have confirmed thereby that no snail contributions are present in the representation found in ref. Neq44np ().)

Although the snail contributions are important for sYM amplitudes, they will not infect the corresponding supergravity amplitudes. This may be understood heuristically as a consequence of the double-copy formula (5). In sYM theory, graphs of the form in fig. 3(c) carry a factor of 0 in their numerator to cancel the 1/0 from the propagator. In supergravity we get a second factor of from the second copy, making the numerator vanish faster than the denominator, and giving a vanishing snail contribution. Below we confirm this heuristic argument directly from unitarity cuts.

Finally, we remark that very similar considerations appear in the analysis of inverse derivative factors arising from the collision of vertex operators in the discussion of nonrenormalization conditions for amplitudes in superstring theory—see section 3.2 of ref. DoubleTraceNonrenormalization ().

## Iii The four-loop four-point integrand

We now turn to the construction of the four-loop four-point amplitude and follow the same strategy as described in the previous section for the corresponding three-loop amplitude.

### iii.1 Overview of the result

We will find that, in terms of the 85 distinct graphs of figs. 511, the four-loop sYM amplitude is given by

 A(4)4 = g10stAtree4∑S485∑i=1∫(8∏j=5dDlj(2π)D)1SiNi(kj,lj)Ci∏13αi=1p2αi, (20)

where are the four independent loop momenta and are the three independent external momenta. The are the momenta of the internal propagators (corresponding to the internal lines of each graph ), and are linear combinations of the independent loop momenta and the external momenta . In the case of 1PR graphs, some will depend only on the external momenta. As usual, is the -dimensional integration measure for the loop momentum. The numerator factors are polynomial in both internal and external momenta, and are given in appendix B. The color factors are collected online Online (), but they can also be read directly off the figures. The full amplitude is obtained by summing over the group of 24 permutations of the external leg labels. Overcounts are removed by the symmetry factors , which include both external symmetry factors (related to the overcount from the sum over ), as well as any internal symmetry factors associated with automorphisms of the graphs holding the external legs fixed. As at three loops, we extract the crossing-symmetric, -invariant prefactor , which contains all dependence on the external states. (Notice that we have used a slightly different notation for the independent loop momenta in eq. (20), compared with in eq. (1).)

Out of the 85 integrals in eq. (20), graphs 50 and 79 are somewhat peculiar: Their integrands are nonvanishing, but they integrate to zero. The vanishing of their integrals can be seen from symmetry considerations alone. For example, graph 50 has a symmetry exchanging legs 1 and 4, and legs 2 and 3, flipping the graph across a vertical midline. It is easy to check that the color graph picks up a minus sign under this operation; therefore the kinematic integrand must also be antisymmetric, causing the integral to vanish. In fact, as the duality between color and kinematics might suggest, the color factors and vanish after the internal color sum is carried out (for any gauge group ). However, both graphs give nonvanishing contributions to the supergravity amplitude. Therefore we retain them here. (While the vanishing gauge-theory integrals are odd under the above relabeling of the loop momenta, the double-copy property makes the gravity integrals even under the same relabeling.)

As we discussed in section II.4, graphs 83-85, (displayed in fig. 11), superficially appear as propagator corrections on external legs. These graphs give rise to the snail contributions described there, after an external propagator is canceled by a corresponding factor in the numerator.

Using the double-copy relation (5), the four-loop four-point supergravity amplitude is obtained simply by trading the color factor for in eq. (20). Employing the relation and changing the gauge coupling to the gravitational coupling, we have

 M(4)4=−(κ2)10stuMtree4∑S482∑i=1∫(8∏j=5dDlj(2π)D)1SiN2i(kj,lj)∏13αi=1p2αi, (21)

where are the gauge-theory numerator factors given in appendix B. In contrast to the sYM amplitudes, potential snail contributions from graphs 83-85 vanish identically, as expected from our heuristic argument in section II.4, and confirmed by an analysis of the unitarity cuts.

The symmetry factors appearing in eqs. (20) and (21) are given explicitly as:

 85∑i=11SiIi =14I1+14I2+116I3+14I4+18I5+12I6+12I7+I8+14I9+14I10+12I11+14I12+12I13 +12I14+14I15+I16+12I17+I18+I19+I20+I21+I22+I23+12I24+I25+I26 +12I27+14I28+I29+12I30+12I31+I32+I33+12I34+I35+I36+12I37+14I38 +12I39+14I40+12I41+I42+I43+12I44+14I45+12I46+18I47+12I48+12I49+18I50 +12I51+I52+14I53+12I54+12I55+12I56+12I57+12I58+12I59+12I60+14I61+12I62 +12I63+12I64+12I65+14I66+12I67+14I68+14I69+12I70+18I71+12I72+12I73+14I74 +14I75+12I76+12I77+14I78+18I79+116I80+18I81+116I82+14I83+12I84+14I85,

where should be interpreted only as placeholders for the graphs, including both the numerator or color factors, in either theory.

### iii.2 The calculation

As at three loops, the construction of the amplitude begins by writing down a sufficient number of duality constraints so that a set of master numerators can be identified. We have constructed a set of duality equations similar to the three-loop ones of eq. (7). In appendix A we collect a set of simplified equations derived from these duality constraints by imposing the four auxiliary constraints presented in section II.2. Because of these additional simplifications, the duality equations in the appendix are valid only for sYM theory. The duality equations allow us to express all non-snail numerators directly as linear combinations of the numerators and . The corresponding graphs are shown in fig. 4; we will choose them as the master graphs.

It is interesting to note that as an alternative we can use a single nonplanar master graph that does the same job, such as graph 33 (or equivalently 35 or 36, which have identical numerators up to a sign). However, we prefer to use planar graphs as master graphs because their numerators have a somewhat simpler structure. If we choose planar graphs as master graphs then the minimal number is two. In our treatment the snail contributions are only given partially in terms of the master numerators, because the latter are specified using on-shell external kinematics, whereas the numerators of the former require an off-shell regularization to be nonvanishing.

Our next task is to determine the master numerators. To this end we begin by constructing an Ansatz for the numerator factors and that satisfies the auxiliary constraints discussed in section II.2 and a restricted set of duality relations. We then constrain the Ansatz by demanding that other duality relations are satisfied, and that the numerator factors of the other integrals obey the auxiliary constraints. For both graphs, the numerator must be independent of loop momenta and because they are assigned to one-loop box subgraphs, whose momenta should not appear in their numerators. Similarly, momenta and are assigned to one-loop pentagon subgraphs, so and should be no more than linear in these momenta, according to our auxiliary constraints. Thus, each of the two master numerators should be a polynomial built from the monomials,

 M={s3,st2,s2t,t3,τi5s2,τi5t2,τi5st,τi6s2,τi6t2,τi6st,τi5τj6s,τi5τj6t,τ56s2,τ56t2,τ56st}, (23)

where labels the three independent external momenta, . In total this gives us a polynomial with 43 terms for each master graph. Labeling the monomials consecutively as , and including arbitrary coefficients, we have as our starting Ansatz,

 N18=43∑j=1ajMj,N28=43∑j=1bjMj. (24)

The 86 free coefficients and are to be determined from various consistency conditions obtained from the color-kinematic duality, graph symmetries and unitarity cuts. The number of free parameters that need to be determined in this construction is remarkably small, considering the expected analytic complexity of amplitudes at four loops.

Using the 86-parameter Ansatz and the solution to the restricted set of duality constraints listed in appendix A gives us expressions for the numerator factors of any of the 82 non-snail graphs appearing in the amplitude. (The snail graphs will be determined below in terms of the non-snail graphs using generalized unitarity cuts.) These expressions do not yet satisfy all duality constraints; thus far we have imposed only the relatively few relations in appendix A sufficient to determine all numerators in terms of the master numerators, but we have not yet accounted for the complete set of duality relations. To further constrain the master Ansatz we could require that all other dual Jacobi relations are satisfied; there are on the order of such functional relations (not all independent). An alternate strategy, which we follow here, is to first impose the consistency constraints on the numerators of the graphs derived from and through eqs. (101) and (102). After obtaining a complete solution for all 86 parameters appearing in the ansatz, we then verify that they indeed satisfy all remaining duality relations and unitarity cuts. An advantage of this strategy is that it allows us to illustrate the remarkably small number of unitarity cuts needed to find the complete amplitude, including nonplanar contributions.

As we shall see, to construct the complete amplitude we need only information about the unitarity cuts of the four-loop planar amplitude, obtained previously in refs. BCDKS (); Neq44np (). The list of constraints needed to fix all 86 parameters in the Ansatz, thus determining the amplitude, is remarkably short. It is sufficient to enforce:

1. the graph automorphism symmetries on numerators , and ;

2. the maximal cut of graph 12;

3. the next-to-maximal cut of graph 14, where is the off-shell leg. Graph 68 also contributes to this cut.

Strikingly, only two rather simple planar cuts are needed to fully determine the amplitude. Let us now discuss some details of fixing the parameters.

We start by analyzing the consequences of the symmetries of the master graph 28: This graph is invariant under two independent transformations:

 {k1↔k3,l5→k2−l5,l6→k4−l6,l7↔l8}, (25)

and

 {k2↔k4,l5↔l6,l7→k1−l7,l8→k3−l8}. (26)

Imposing the invariance of numerator under eqs. (25) and (26) reduces the number of its unknown coefficients from 43 to 14. The other master graph, graph 18, does not have any such automorphism relations; we are therefore left to determine a total of 57 parameters.

We then impose similar symmetry conditions on , which may be written in terms of and as

 N12 = −N18(k4,k3,k1,l6,−l5,−l6,l8)+N18(k4,k3,k2,l6,k2+l8,l5,l7) (27) +N18(k4,k3,k2,k3+l8,l5−l8,−l6,l8)−N28(k1,k2,k3,l5−l8,k3−l6+l8,−l6,l8) +N28(k2,k1,k3,−l5,0,−l6,l8)−N28(k4,k3,k1,l6−l8,k2−l5+l8,l8,l8) +N28(k4,k3,k2,k3,k1+l5,−k3+l6,l8),

by combining the 2nd, 6th, 14th and 21st relations in eq. (101) in appendix A. Invariance under the automorphisms of graph 12 fixes 37 parameters, leaving undetermined 20 parameters. Similarly, imposing the graph symmetry condition on reduces the total number of unknown parameters to 17. (These parameter counts are for the specific set of duality relations given in eq. (101). Using another set of relations would result in somewhat different parameter counts; however, the final solution would be the same.)

We could continue imposing more symmetry constraints on other numerators, but we already have a very small set of undetermined parameters. Ultimately, dynamical information provided by unitarity cuts should become necessary. Therefore we will now inspect some cuts. A good starting point is the maximal cut of graph 12. Its explicit value is easily obtained using the simple rung-rule numerator of that graph BRY (); BCDKS (),

 Nrr12=s2(l5+l6+k1+k4)2. (28)

The rung rule was originally designed to reproduce iterated two-particle cuts. Since maximal cuts can be obtained from iterated two-particle cuts by imposing additional cut conditions, the rung rule reproduces the maximal cuts as well. Our task is to match , as obtained from the duality relations, and in eq. (28) on the maximal cut kinematics that uniquely single out this graph, i.e. we impose on all 13 propagators of graph 12. Solving these conditions, we obtain

 l2i=0,τ15=τ25=τ36=0,τ16=−τ26,τ17=−τ57, (29)

Thus, on the maximal cut, becomes

 Nmax.cut12=s2(t−τ26−τ35+τ56). (30)

Requiring that the numerator obtained from the Ansatz (24) via eq. (27) matches on the maximal cut, the number of undetermined parameters is reduced from 17 to 8.

Finally, all remaining parameters can be determined by requiring the next-to-maximal cut of graph 14, where all propagators except for are placed on shell, to be satisfied. Graph 68 also contributes to this cut since it contains the same set of cut propagators. Relabeling graph 68 so it matches graph 14, and appropriately weighting the numerators by the remaining off-shell propagators, we find under the cut kinematics,

 N14+l25s(N68∣∣l5→k1−l5,l6→−l6l7→−l7,l8→−l8) = s(l5+k2+k3)4. (31)

The right-hand side of this equation is the numerator of graph 14, as constructed using the rung rule; in the rung-rule representation of the planar four-loop amplitude, graph 68 vanishes BCDKS (). This requirement fixes all remaining eight coefficients, giving us a unique expression for the master numerators,

 N18 = 14(6u2τ25+u(2s(5τ25+2τ26)−τ15(7τ16+6t)) (32) +t(τ15τ26−τ25(τ16+7τ26))+s(4τ15(t−τ26)+6τ36(τ35−τ45) −τ16(4t+5τ25)−τ46(5τ35+τ45))+2s2(t+τ26−τ35+τ36+