Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
We present an ansatz for the planar five-loop four-point amplitude in maximally supersymmetric Yang-Mills theory in terms of loop integrals. This ansatz exploits the recently observed correspondence between integrals with simple conformal properties and those found in the four-point amplitudes of the theory through four loops. We explain how to identify all such integrals systematically. We make use of generalized unitarity in both four and dimensions to determine the coefficients of each of these integrals in the amplitude. Maximal cuts, in which we cut all propagators of a given integral, are an especially effective means for determining these coefficients. The set of integrals and coefficients determined here will be useful for computing the five-loop cusp anomalous dimension of the theory which is of interest for non-trivial checks of the AdS/CFT duality conjecture. It will also be useful for checking a conjecture that the amplitudes have an iterative structure allowing for their all-loop resummation, whose link to a recent string-side computation by Alday and Maldacena opens a new venue for quantitative AdS/CFT comparisons.
UCLA/07/TEP/04 ZU-TH 12/07 Saclay/SPhT–T07/050
Maximally supersymmetric Yang-Mills theory (MSYM) is an important arena for exploring the properties of gauge theories. The Maldacena weak-strong duality Maldacena () between MSYM and string theory in AdS provides an explicit realization of ’t Hooft’s old dream tHooft () of expressing the strongly coupled limit of a gauge theory in terms of a string theory. In addition, the higher-loop planar space-time scattering amplitudes of MSYM appear to have a remarkably simple and novel iterative structure ABDK (); BDS (). This structure allows higher-loop amplitudes to be expressed in terms of lower-loop amplitudes. This simplicity may well be connected to the observed integrability of the theory in the planar limit Integrability (); EdenStaudacher (); BES ().
The iterative structure of the planar amplitudes was first proposed in ref. ABDK (), and confirmed for the two-loop four-point amplitude. An independent two-loop check was given in ref. HiddenBeauty (). In ref. BDS (), this proposal was fleshed out for all planar maximally helicity violating (MHV) amplitudes, expressed via a specific all-loop exponentiation formula which was confirmed for three-loop four-point amplitudes. The iteration formula has also been shown to hold for two-loop five-point amplitudes FivePtTwoLoop (). At four loops the amplitudes are known in terms of a set of integrals, but the integrals themselves have not yet been evaluated fully BCDKS (); CSVFourLoop (). In this paper, we provide the corresponding integral representation of the five-loop four-point planar amplitude, for use in future studies of its properties. In a very recent paper, Alday and Maldacena AldayMaldacena () have shown how to perform a string-side computation of the same gluon amplitudes in the strong-coupling limit. This opens new and exciting possibilities of quantitative checks of the AdS/CFT correspondence, going beyond anomalous dimensions to detailed dependence on kinematics.
In addition to exhibiting an iterative structure, the scattering amplitudes provide new and nontrivial information on the AdS/CFT correspondence. Using considerations of integrability, an integral equation for the cusp (soft) anomalous dimension — valid to all loop orders — was written down by Eden and Staudacher EdenStaudacher (). This equation agreed with the first three loop orders KLOV (); BDS (), but its reliance on various assumptions cast doubt on whether it would hold beyond this. We now know that this original proposal requires modification because of the recent calculation of the four-loop cusp anomalous dimension from the infrared singular terms of a four-loop MSYM amplitude BCDKS (); CSVFourLoop (). A remarkable new integral equation proposed by Beisert, Eden and Staudacher (BES) BES () is in agreement with this calculation. Surprisingly, the first four loop orders of the planar cusp anomalous dimension contain sufficient information to test the AdS/CFT correspondence to the level of a few percent BCDKS (), using the interpolating function technique introduced by Kotikov, Lipatov, and Velizhanin KLV () as well as Padé approximants. This provides an independent guess of the entire perturbative series BCDKS (), matching the one generated by the BES integral equation. Detailed studies of the BES equation Klebanov () confirm that it has the proper behavior StrongCouplingLeading () at strong coupling, giving a high degree of confidence in it. Nonetheless, further checks on the perturbative side would be quite valuable. The first such check requires a computation of the five-loop anomalous dimension, which — following the approach taken at four loops — requires an expression for the five-loop four-point amplitude.
The unitarity method NeqFourOneLoop (); Fusing (); DDimUnitarity (); OneLoopReview (); DimShift (); GeneralizedUnitarity () provides a powerful method for computing gauge and gravity loop amplitudes and has played a central role in obtaining MSYM loop amplitudes NeqFourOneLoop (); Fusing (); BRY (); BDDPR (); ABDK (); BDS (); BRYProceedings (); FivePtTwoLoop (); BCDKS (). An important recent improvement BCFUnitarity () is the use of complex momenta WittenTopologicalString () within the framework of generalized unitarity GeneralizedUnitarity (). In particular, this allows one to define a non-zero massless three-point amplitude, which vanishes for real momenta. At one loop this enables an easy algebraic determination of the coefficient of any box integral appearing in the theory, because the cut conditions freeze the loop integrals BCFUnitarity (). Some of these ideas have also been applied at two loops BuchbinderCachazo (). In this paper we apply these ideas to develop a maximal-cutting method for efficiently determining coefficients of higher-loop integrals.
In the MSYM theory, a special set of cuts — the iterated two-particle cuts — give rise to the “rung rule” for systematically obtaining higher-loop integral representations of planar four-point amplitudes BRY (); BDDPR (). At two and three loops this rule generates all contributions appearing in the amplitudes. However, starting at four loops, new integrals arise which are not generated by the rung rule. In ref. BCDKS (), these were computed explicitly using generalized unitarity by relying on a set of mild assumptions.
These integrals, along with two others that do not appear in the amplitude, are predicted by a procedure relying on a beautiful observation due to Drummond, Henn, Smirnov and Sokatchev (DHSS) DHSS (). These authors noticed that, through three loops, the massless integrals appearing in the planar four-point amplitude are in direct correspondence with conformally invariant integrals. This correspondence comes from replacing dimensional regularization with an off-shell infrared regularization in four dimensions. We shall call the dimensional regularized version of the conformal integrals pseudo-conformal, since dimensional regularization breaks conformal invariance. DHSS also gave simple rules for generating all such integrals via “dual diagrams”. The direct evaluation of generalized unitarity cuts confirmed BCDKS () at four loops, that only such pseudo-conformal integrals appear in the planar amplitude. In the present work, we will assume that this is also true at higher loops, beginning with five loops. This provides a basis set of integrals for the planar (leading-color) contributions to the five-loop amplitude. We then use the unitarity method to determine the coefficients of these integrals in the planar five-loop four-point amplitude as well as to provide consistency checks on the absence of other integrals.
We make use of an additional observation: the cutting equations hold at the level of the integrands, prior to carrying out any loop integrals. Indeed they hold independently for each of the multiple solutions of the cutting equations. These properties are especially powerful when combined with the basis of pseudo-conformal integrals. The problem is then reduced to an algebraic problem of determining the coefficient of each integral. Remarkably, it turns out that after dividing by the tree amplitude, the coefficients are pure numbers taking on the values or . This property is already known to hold for the four-point amplitude through four loops BCDKS (), and here we confirm it through five loops.
At one loop, complete dimensionally regularized amplitudes in the MSYM theory can be constructed using only four-dimensional helicity amplitudes, greatly simplifying their construction NeqFourOneLoop (); Fusing (). Unfortunately, no such theorem exists at higher loops. Any rigorous construction of amplitudes requires that -dimensional momenta be used in the cuts. It is worth noting that if our assumption of a pseudo-conformal basis of integrals is correct, with dimension independent coefficients, then unitarity in four dimensions does suffice to determine these amplitudes in all dimensions. Our partial checks of -dimensional cuts provide non-trivial evidence that this assumption is correct. This is rather remarkable because away from four dimensions there is a priori no reason why a simple analytic continuation of the dimension of the integrals should give correct results.
Our expression for the five-loop four-point planar MSYM amplitude in terms of integral functions should be useful in a number of studies. The infrared singularities present in the amplitude encode the so-called cusp or soft anomalous dimension. At five loops these singularities begin at (where , as usual, is the dimensional regularization parameter: ). Evaluation of the infrared singular terms through would allow the extraction of the five-loop cusp anomalous dimension, as has been done at three and four loops BDS (); BCDKS (). An evaluation though order would allow the extraction of a second anomalous dimension connected to a form factor. An evaluation through would allow a five-loop check of the iterative structure of the amplitudes, providing strong evidence that it continues to all loop orders. Although evaluating loop integrals is rather challenging, there has been rapid progress using Mellin-Barnes representations VolodyaIntegrals () and in automating the required analytic continuations MB (), allowing for explicit computations through four loops. Recently, there has also been progress in isolating the subsets of terms which determine the anomalous dimension CSVFourLoop (), greatly simplifying its calculation. Further development will presumably be needed to apply these advances to the five-loop amplitude.
Another important reason for studying MSYM amplitudes is their intimate connection to supergravity amplitudes. The identification of additional cancellations in this theory GravityCancel () suggests that in four dimensions it may be ultraviolet finite to all loop orders GravityFinite (); ThreeLoopNEqEight (). (See also refs. KITPTalk ().) String dualities also hint at UV finiteness for supergravity StringFinite (), although this is weakened by issues with towers of light non-perturbative states from branes wrapped on the compact dimensions OoguriPrivate (). Remarkably, computational advances for gauge theory amplitudes can be imported BDDPR () directly into calculations of gravity amplitudes, by combining the unitarity method with the Kawai-Lewellen-Tye KLT () tree-level relations between gauge and gravity theories. This allows cuts of gravity loop amplitudes to be expressed as double copies of cuts of corresponding gauge theory amplitudes BDDPR (). This strategy has recently been applied to obtain the three-loop four-point amplitude of supergravity, starting from corresponding MSYM amplitudes ThreeLoopNEqEight (). That computation shows that at least through three loops, MSYM and supergravity share the same ultraviolet power-counting. Because they share the same critical dimension for ultraviolet finiteness, both are ultraviolet finite in four dimensions. The five-loop planar super-Yang-Mill amplitudes obtained here will be an important input for obtaining the corresponding five-loop supergravity amplitudes. (The supergravity calculation also requires the non-planar contributions.)
The paper is organized as follows. In section II, we briefly summarize known properties of the amplitudes and define the notation used in the remainder of the paper. In section III, we review the observations of refs. DHSS (); BCDKS (), on the exclusive appearance of pseudo-conformal integrals in planar four-point amplitudes. Because candidate integrals proliferate as the number of loops increases, we give a systematic procedure in section IV for constructing these integrals. The results of this procedure at five loops are given in section V, along with the coefficients of the integrals determined via unitarity. Our ansatz for the amplitude is also presented in this section. We then briefly review generalized unitarity in section VI. A description of the cuts used to determine the integral coefficients is given in sections VII and VIII, along with a description of the method of maximal cuts introduced in this paper. Our conclusions and some comments on the outlook are given in section IX.
Ii Notation and Review of MSYM Amplitudes
We use a standard color decomposition TreeReview (); OneLoopReview () for the MSYM amplitudes in order to disentangle color from kinematics. In this paper we focus on the leading-color planar contributions, which have a color structure similar to those of tree amplitudes, up to overall factors of the number of colors, . The color-decomposed form of planar contributions to the -loop gauge-theory -point amplitudes is,
where is an -loop color-ordered partial amplitude. We have followed the normalization conventions of ref. BDS (). Here is Euler’s constant, and the sum runs over non-cyclic permutations, , of the external legs. In this expression we have suppressed labels of momenta and helicities, leaving only the indices identifying the external legs. Our convention is that all legs are outgoing. This decomposition holds for all particles in the gauge super-multiplet.
We also define a loop amplitude normalized by the tree amplitude,
Supersymmetry Ward identities SWI () guarantee that, after dividing out by the tree amplitudes, MHV amplitudes are identical for any helicity configuration DimShift (). (The complete set of these tree amplitudes are tabulated in Appendix E of ref. BDDPR ().) Because four-point amplitudes are always maximally helicity violating, this holds for all four-point amplitudes. Because it is independent of the position of the two negative helicity legs, has complete cyclic and reflection symmetry. A practical consequence of this is that once a coefficient of a given integral is determined, the coefficient of integrals related by cyclic or reflection symmetry follow trivially.
where is a massless Weyl spinor with momentum and positive or negative chirality. Our conventions follow the QCD literature, with for real momenta so that,
We also define,
For complex momenta these two spinors are independent, though they are dependent for real momenta.
In ref. ABDK (), a conjecture was presented that MSYM amplitudes possess an iterative structure, based on an observed iteration of two-loop four-point amplitudes. In ref. BDS (), this was fleshed out for MHV amplitudes into an explicit exponentiation ansatz to all loop orders. Through five loops, the expansion of the exponential gives the iteration relations,
where is a three term series in ,
and are numbers independent of the kinematics and of the number of external legs . Similarly, the are also pure numbers. The constant is proportional to the -loop cusp (soft) anomalous dimension ,
After subtracting the known infrared singularities MagneaSterman () the iteration relation takes on a rather simple exponential form,
where is the -point one-loop finite remainder, is the complete cusp anomalous dimension, and depends on the coupling but not on the external momenta. Very recently Alday and Maldacena have matched this expression at strong coupling for using string theory AldayMaldacena ().
The iteration conjecture has so far been confirmed for two- and three-loop four-point amplitudes ABDK (); BDS () as well for two-loop five-point amplitudes FivePtTwoLoop (). For the four-loop four-point amplitude, the integrand is known BCDKS () and has been shown to generate the correct form of the infrared singularities though . This has been used to extract the four-loop contribution to the cusp anomalous dimension BCDKS (); CSVFourLoop () numerically.
Iii Pseudo-Conformal Integrals
Conformal properties offer a simple way to identify integrals that can appear in planar MSYM amplitudes DHSS (); BCDKS (). This observation allows us to easily identify a basis of integrals, whose coefficients can be determined via the unitarity method. This greatly simplifies the cut analysis because we need to determine only these coefficients to obtain the planar amplitudes.
Although the underlying theory is conformally invariant, there is as yet no proof that only integrals dictated by conformal invariance can appear. One obvious complication to providing such a proof is the infrared divergence of the amplitudes, and the subsequent need to regulate the integrals (via dimensional regularization), which breaks the conformal invariance. As mentioned in the introduction, we therefore call the integrals corresponding to conformally-invariant ones “pseudo-conformal”. We shall describe in this section how to implement this correspondence. In principle, it is possible that individual integrals appearing in the amplitude would have no special conformal properties, yet the complete amplitude would retain simple conformal properties because of cancellations between integrals. Through four loops BCDKS (), however, only pseudo-conformal integrals appear in the four-point amplitude. This provides compelling evidence that this is a general property of MSYM four-point amplitudes.111For higher-point amplitudes, the situation is more complicated, as can be confirmed by checking the conformal properties of known results FivePtTwoLoop () for the five-point amplitudes at two loops.
We therefore assume that only pseudo-conformal integrals appear in the five-loop planar amplitudes. One way of proving the correctness of this assumption would be to compute a sufficient number of cuts in dimensions to determine the amplitude completely. In this paper, we discuss only a partial confirmation.
Dimensional regulation of the infrared singularities breaks the conformal symmetry. For the purposes of exposing the conformal symmetry, we instead regulate the infrared divergences by taking the external momenta off shell and letting the dimension be . We will obtain a pseudo-conformal integral from a suitable conformal integral by reversing this change of regulator.
The authors of ref. DHSS () provide a simple way of making manifest the conformal properties of planar integrals via “dual diagrams”. The dual diagrams provide a direct method of identifying all conformally invariant loop integrals.222The dual diagrams are related to the dual graphs used in graph theory Nakanishi (). In general, conformal properties are not obvious in the momentum-space representation of loop integrals, but with a simple change of variables encoded by the dual diagrams we can make these properties manifest.
Let us give an illustrative example. Consider the two-loop double-box integral of fig. 1(a),
where and .
After replacing the regulator as mentioned above, the conformal symmetry can then be exposed via the change of variables,
where . The new variables automatically satisfy momentum conservation
After substituting the new variables into eq. (13) and taking , the double-box integral takes on a very symmetric form,
The conformal-invariance properties follow from examining its behavior under inversion, ,
Under this inversion the double-box (16) is invariant because each external point appears equally many times in the numerator as in the denominator, while the internal points appears exactly four times in the denominator, precisely canceling the behavior of the integration measure. The variables are useful because inversion respects momentum conservation, which is not true for an inversion of the original momentum variables.
More generally, following refs. DHSS (); BCDKS (), we keep track of conformal weights using dual diagrams. To obtain the dual representation, start with the momentum representation shown in fig. 1(a) and place internal points inside each loop, as well as external points between each pair of external momenta, as shown in fig. 1(b). Following ref. DHSS () we mark the internal integration points by solid dots at the center of each loop but in most cases leave the external points unmarked. Solid lines represent an inverse power of , corresponding to the dual propagator , which crosses exactly one Feynman propagator whose momentum is equal to . Dashed lines represent a positive power of an , such as the numerator factors of and . The represented by a dashed line correspond the sum of the momenta of the ordinary propagator lines they cross. (The dashed lines can be deformed to cross different propagators, but momentum conservation ensures that this does not affect the value of the dual invariants .) The dual diagram constructed in the way in fig. 1(b), is in direct correspondence to the dual integral (16).
We can further restrict the possible set of conformal integrals by requiring that they have only logarithmic behavior in the on-shell limit. That is, we are not interested in conformal integrals which vanish or diverge with a power-law behavior in any , because these do not correspond to massless integrals in dimensional regularization. For example, numerator factors such as are not allowed. Similarly, factors such as are excluded because their power singularities are too severe for the required logarithmic behavior of infrared singularities. We then obtain a pseudo-conformal integral, as discussed earlier, by replacing the off-shell regulator with the usual dimensional one.
It is straightforward to generalize this graphical mapping to any loop order, allowing for a relatively simple bookkeeping of the change of variables between the momenta and the dual variables. The map in eq. (14) exemplifies the convention we use for external momenta for all diagrams in this paper.
The conformal weights are easy to read off directly from the dual diagrams. For a dual diagram to be conformally invariant it must satisfy the following: The number of solid lines minus the number of dashed lines entering a point must be zero for external points, and four for internal points. The conformal weight of four for internal points cancels the conformal weight of the integration measure given in eq. (17). As observed in ref. BCDKS (), a consequence of requiring integrals to be conformal is that integrals with triangle or bubble subdiagrams are not allowed.
One class of pseudo-conformal integrals may be understood in terms of the “rung-rule” of ref. BRY (). This rule instructs its user to generate contributions to an -loop amplitude from a known -loop amplitude by inserting a new leg between each possible pair of internal legs, as shown in fig. 2. From this set, all diagrams with either triangle or bubble subdiagrams are removed. The new loop momentum is integrated over, after including an additional factor of in the numerator, where and are the momenta flowing through the indicated lines. (With the conventions used here it is convenient to remove a factor of from the numerator factor, compared to ref. BRY ().) Each distinct contribution should be counted only once, even if it can be generated in multiple ways. Contributions arising from identical diagrams (that is, having identical propagators) but with distinct numerators count as distinct contributions. The diagrams obtained by iterating this procedure are sometimes called Mondrian diagrams, because of their similarity to Mondrian’s art.
The rung rule may be understood as a consequence of the conformal properties of the integrals. As illustrated in fig. 3, if the starting integral is pseudo-conformal, inserting a rung splits the inner loop into two side-by-side loops, and the conformal weight of the central dots in the figure is unchanged. However, the upper and lower loop need an additional dashed line connecting their central dots to maintain their conformal weight. This dashed line corresponds exactly to the factor of required by the rung rule in fig. 2.
The rung-rule, unfortunately, does not generate the complete set of planar integrals BCDKS (). However, at least through four loops any diagram that it generates is obtained with the correct sign. Here we confirm this observation at five loops, using the unitarity method. To obtain the remaining, non-rung-rule contributions to the five-loop amplitude, we start with the other pseudo-conformal integrals and determine their coefficients using the unitarity method.
Iv Generating the Planar Pseudo-Conformal Diagrams
The proliferation of candidate pseudo-conformal integrals with increasing number of loops encourages the development of a systematic construction procedure. One approach is suggested by examining the -particle cuts of an -loop amplitude. Using such a cut we can decompose the loop integrals into products of tree diagrams which then simplifies the bookkeeping.333At six loops and beyond it turns out that there are pseudo-conformal integrals which do not have an -particle cut. These can however be obtained from a “parent diagram” containing an -particle cut, by canceling one of the propagators. Our procedure will be:
Construct the set of all possible amputated tree configurations on each side of the cut.
Identify all possible loop integrals by sewing each configuration from the left side of the cut with each configuration on the right side of the cut.
Identify all possible overall factors in each integral which make it conformal.
Because the conformal properties are most obvious in the dual representation described in section III, we have found it convenient to work with dual coordinates and translate back to the momentum representation at the end.
iv.1 Constructing all dual tree diagrams
It is useful to consider first the reverse procedure of splitting an -loop integral into two trees via an -particle cut. As an illustration, consider the four-loop “window” integral as cut in fig. 4. This integral contributes to the four-loop four-point amplitude BCDKS (). As shown in the figure, a five-particle cut separates the integral into a product of two tree diagrams. In dual space, we also split integrals into tree diagrams with a five-particle cut. Figure 5 depicts the separation of the corresponding dual diagram into tree diagrams, which we carry out in two steps. In the first step of the figure we divide the dual diagram along the cut marked with arrows, keeping the cut line on both sides. In the next step we drop the lines with arrows, giving us dual diagrams corresponding to amputated (i.e. with external propagators removed) tree-level momentum-space diagrams. In this representation each dual line crosses an internal propagator of the momentum-space diagrams. The diagrams will always have a fixed cyclic ordering. The dual-diagram points also respect this cyclic ordering. In this example, for the tree amplitude on the left, the points are ordered , while the points for the tree diagram on the right are ordered .
Our systematic construction of all dual loop diagrams simply reverses the process in this example. At loops we start with two ordered lists of points: for the left tree and for the right tree. These lists correspond to -particle cuts in the channel; the corresponding construction in the channel is easily obtained by relabeling the final result. The assignment of labels to points is determined by the external momenta, and the remainder follow from the cyclic ordering.
We obtain all possible pairs of dual tree diagrams by connecting non-adjacent points with propagators in all possible ways such that the lines do not cross. Dual diagrams where nearest-neighbor points are connected are not included as they correspond to momentum-space diagrams whose external propagators have not been truncated.
After identifying the possible dual tree diagrams we restore the dual lines representing the cut, by retracing our steps in the example shown in fig. 5. That is, in both sets of tree diagrams we draw lines with arrows connecting to , etc., to , which is then connected to . Once this is done the pairs of tree diagrams can be glued together along the lines with arrows, which then gives us the -loop dual diagrams that we wish to construct. At this stage we can remove diagrams trivially related by cyclic or flip symmetry.
iv.2 Finding the pseudo-conformal integrals
Once we have a set of candidate loop-level dual diagrams, we must find the numerator factors necessary to make the corresponding integrals conformal. This can be accomplished as follows DHSS (); BCDKS ():
If one of the internal points appears in less than four dual propagators, discard the diagram as it cannot be made conformal.
To determine possible numerator factors one first identifies all external points from the set appearing in one or more dual propagators, and all internal points in the set appearing in five or more dual propagators. All such points require numerator factors to cancel the extra conformal weight. That is, the number of times a given external appears in the dual propagators minus the number of times it appears in the numerator should be zero. Similarly, for an internal point , the number of times it appears in the dual propagators minus the number of times it appears in the numerator should be four. To find the conformally invariant integrals we sweep through products of all candidate numerators to identify the ones where the conformal invariance constraints are satisfied. (In principle, there might also be an overall resulting factor of or , but this does not occur at five loops, nor do we expect such contributions to enter the amplitudes with non-zero coefficients at any loop order.)
If the previous step yields a previously-identified pseudo-conformal integral, go on to the next case. Such repeated integrals can arise when a numerator factor cancels a propagator or when diagrams are related by symmetries.
Once we have the set of conformal dual diagrams we can convert these back to momentum space with a change of variables,
where the are the momenta of the lines in -particle cut used in the construction. Since our construction was only a bookkeeping device for finding pseudo-conformal integrals, at the end there is no on-shell restriction on the .
In the next section we apply this procedure to construct a basis of all pseudo-conformal integrals appearing in the five-loop planar MSYM amplitudes.
V The five-loop planar pseudo-conformal integrals
v.1 The five-loop pseudo-conformal integral basis
Following the procedure described in the previous section we find a total of 59 independent pseudo-conformal integrals potentially present in the five-loop four-point planar amplitude (not counting those related by permutations of external legs). They are shown in figs. 7, 8, 9 and 10. The ‘parent’ integrals, shown in fig. 7, have only cubic vertices. The remaining integrals have both cubic and quartic vertices. They may be obtained by omitting propagators and modifying numerator factors present in the parent integrals, As we shall show in the following section using unitarity cuts, the integrals in figs. 7 and 8 appear in the amplitude (2) with relative coefficients of , which we have absorbed into the definitions of the numerator factors in the figures. The remaining ones shown in figs. 9 and 10 do not appear at all. We do not have an explanation for the remarkable simplicity of the coefficients of the integrals, but presumably it is tied to the superconformal invariance of the theory.
We draw the diagrams in momentum space, but also include the relevant for tracking numerator factors. The numerators are written out as Mandelstam variables and or as dual invariants, . As discussed in section III a dual invariant is equal to where is the total momentum flowing through a line spanned between points and . For example, in fig. 6, .
Some of the integrals have identical sets of propagators but differing numerator factors. These sibling integrals would be identical were one to omit the numerators. Examples are and or and in fig. 7. The numerator factors often have different symmetries than the propagators in any given integral. The different numerator factors in sibling integrals will also typically have different symmetries. For example, integral is completely symmetric under a cyclic permutation of its arguments, (corresponding to a rotation of the diagram in fig. 7) and under flips, (corresponding to reflection of the diagram). Its sibling , in contrast, has only one symmetry, (corresponding to a rotation of the diagram by radians). Accordingly, appears only once in the amplitude, but appears four times. This makes it inconvenient to combine them into a single integral.
All planar five-loop pseudo-conformal integrals with the exception of have at most four-point vertices. ( has quintic vertices where the external legs attach. Internal quintic vertices do not occur in conformal integrals until seven loops.)
v.2 The five-loop four-point amplitude
In sections VII and VIII, we evaluate a sufficient number of cuts in order to determine the numerical prefactors of each pseudo-conformal integral as it appears in the amplitude (2). We find that the complete five-loop four-point MSYM planar amplitude is,
where the integrals are shown in figs. 7 and 8, and is defined in eq. (2). There are a total of 193 integrals in the sum. As the integrals depend only on the kinematic invariants and , instead of having leg labels, each integral can appear only as or as . In eq. (19), we have suppressed the arguments “” and combined identical terms, leaving a symmetry factor in front. The relative signs between integrals, determined from the unitarity cuts in sections VII and VIII, have been incorporated in the numerator factors in figs. 7 and 8, though we have chosen to leave an overall sign outside the integrals. The normalization factor of follows the conventions of refs. BDS (); BCDKS () and accounts for the factor of in eq. (1). The integrals in eq. (19) are therefore normalized as:
To understand the relative signs of the diagrams we classify terms into those derived from the rung rule and the rest. Any diagram generated by the rung rule in fig. 2 simply inherits the sign of the lower loop diagram from which it was derived. This gives the correct numerator factor, including the sign, for all contributing integrals containing only cubic vertices, except for – the only diagram in fig. 7 having a non-rung rule numerator. Integrals with quartic vertices, such as and , are given by the rung rule applied to known BCDKS () four-loop diagrams. Using two-particle cuts, other examples of integrals whose prefactors are easy to understand are and . The latter two have vanishing coefficient because their four-loop parent diagrams also have vanishing coefficients. Integral can be understood in terms of a rung inserted between an external leg and an internal line.
Another class of prefactors and signs can be understood from a “substitution rule”. Consider diagram in fig. 7. As shown in fig. 11, this diagram inherits its prefactor and sign by replacing the four-point vertex by a one-loop box integral. (The one-loop box enters with a relative plus sign.) The numerator factor of the substituted box is simply the factor needed to make the box conformal. The negative sign is inherited from the sign of the four-loop diagram. Also other signs can be understood from this substitution rule. For example, the sign on and the zero coefficient on follow from similar substitutions on four loop conformal diagrams. This rule can more generally be understood as a substitution of the normalized four-point function into a four-point vertex, which can obtained using generalized cuts.
With the rung rule, two-particle cuts, and substitution rule we may understand the signs of all diagrams that appear in the amplitude, except for , and . As of yet we have not found a rule giving the sign of these diagrams, other than resorting to computations of cuts.
At four loops BCDKS (), integrals not containing at least one factor of and also one factor of are absent from the amplitude. This is again true at five loops. Factorization arguments using complex momenta can give a suggestive explanation of this property. As we already noted, supersymmetry identities ensure that after dividing by the tree amplitude, the non-vanishing MSYM four-point loop amplitudes are identical for all external helicity and particle configurations. Consider then the helicity configuration , with all external legs gluons. The tree amplitude
factorizes in both the and channels into products of three-point vertices. The absence of compensating factors of and would imply factorization into one-particle irreducible loop three vertices. Such vertices have not appeared in the factorization of any previous MSYM amplitude, and so it is not surprising that the offending integrals, shown in fig. 10, do not contribute here either.
We may understand the remaining vanishing coefficients using the known harmonic-superspace power counting of MSYM HoweStelleNew (). It is compatible in with six powers of loop momenta canceling from integral numerators. The missing engineering dimensions of the amplitude are then supplied by external momenta, requiring at least three powers of either or . This result agrees with the arguments of refs. BRY (); BDDPR (), which provide a bound on dimensions for which the -loop amplitude is ultraviolet finite,
It is natural to assume that this power counting holds independently for each integral. This rules out integrals which do not have at least three powers of or . This includes all of those in fig. 9 and those with either a single power of or in fig. 10.
v.3 All-loop structure
Inspecting the contributions of the integrals in the basis to the five-loop four-point amplitude reveals the following features:
All pseudo-conformal integrals containing a factor of or (and possibly additional powers of or ) enter the amplitude with relative weight of or .
Any pseudo-conformal integral without a factor of or has a vanishing coefficient.
All integrals that could be obtained from the rung rule or from two-particle cuts inherit their weights from the lower-loop integrals used to construct them.
All contributing integrals satisfy the ultraviolet finiteness bound (22).
These observations seems to suggest that, in general, the set of pseudo-conformal integrals with nonvanishing coefficients are the ones with a prefactor divisible by either or . However, at six loops a new structure appears where a pseudo-conformal integral is a simple product of lower-loop integrals, as displayed in fig. 12. We have checked that the conformal integral in fig. 12 does not contribute to the amplitude, although its prefactor is divisible by . While this particular integral does not appear in the amplitude, its existence suggests that at higher loops there will be additional classes of pseudo-conformal integrals with vanishing coefficients.
Vi Generalized Unitarity
The unitarity method NeqFourOneLoop (); Fusing (); DDimUnitarity (); OneLoopReview (); DimShift (); GeneralizedUnitarity (); BCFUnitarity () has proven an effective means for computing scattering amplitudes in gauge and gravity theories. So-called generalized unitarity is particularly powerful for computing amplitudes GeneralizedUnitarity (); BCFUnitarity (), as it allows an -loop amplitude to be built directly from products of tree amplitudes. When combined with complex momenta WittenTopologicalString (); BCFUnitarity (); BCFW (), it allows the use of maximal cuts, in which all propagators in an integral are cut. (The term “generalized unitarity”, corresponding to leading discontinuities of diagrams, dates back to ref. EarlyGeneralizedUnitarity ().)
We begin our discussion with a brief review, including earlier applications of maximal cuts to the computation of two-loop amplitudes BCFUnitarity (); BuchbinderCachazo (). We record a number of observations useful for computation at higher loops. In section VI.1 we modify the maximal-cut procedure and use it to determine the coefficients of all pseudo-conformal integrals appearing in the MSYM five-loop four-point amplitudes efficiently and systematically.
vi.1 Maximal cuts
Cut calculations can be simplified by increasing the number of cut legs. This isolates a smaller number of integrals, making it simpler to determine the values of their coefficients. This technique is especially powerful for computing one-loop MSYM amplitudes, because only box integrals can appear NeqFourOneLoop (). As observed by Britto, Cachazo and Feng BCFUnitarity (), taking a quadruple cut, where all four propagators in a box integral are cut, freezes the four-dimensional loop integration. This allows its kinematic coefficient to be determined algebraically, with no integration (or integral reduction) required. The use of complex momenta, as suggested by twistor space theories WittenTopologicalString (), makes it possible to define massless three vertices and thereby to use quadruple cuts to determine the coefficients of all box integrals including those with massless external legs.
For three massless momenta , and one has the following consistency requirement
For real momenta in Minkowski signature, and (see eq. (5)) are complex conjugates of each other (up to a sign determined by incoming or outgoing nature of the corresponding particle). Hence if vanishes then must also vanish. This constraint holds for all three legs , leaving no non-vanishing quantities out of which to build a three vertex. If the momenta are taken to be complex, however, the two spinors and are independent. This gives two independent solutions to eq. (23),
with the other spinor product non-vanishing in each case. In the three-gluon case, there are overall two possible solutions: all s proportional, and hence all vanishing, or all proportional, and hence all vanishing. This means that exactly one of,
does not vanish. Similar statements hold for amplitudes involving fermions or scalars: one of the two independent helicity configurations will not vanish. The non-vanishing amplitudes involving a fermion pair are,
where the subscript denotes a fermionic leg. Similarly the non-vanishing scalar amplitudes are,
where the subscript denotes a scalar leg. (For complex scalars, the two helicities correspond to particle and antiparticle.) We have chosen the signs in these amplitudes to be consistent with the supersymmetry Ward identities SWI (), as given in ref. Fusing (), and with the parity conjugation rules of ref. QQGGG ().
The method of quadruple cuts has been generalized by Buchbinder and Cachazo BuchbinderCachazo () to two loops using hepta- and octa-cuts. Although the two-loop four-point double-box integral only has seven propagators it secretly enforces an additional, eighth constraint, yielding an octa-cut which localizes the integration completely. But as the authors of ref. BuchbinderCachazo () point out this last cut condition is not really necessary for the evaluation of the two-loop four-point amplitude: the integrand was already independent of loop momenta after imposing the constraints from the seven on-shell propagators in the hepta-cut. We will return to this point below.
Let us then focus on the constraints imposed by the delta functions corresponding to the propagators alone. In the hepta-cut construction of ref. BuchbinderCachazo (), various classes of solutions are allowed by the seven delta function constraints arising from localizing the propagators. In the context of generalized unitarity these delta functions correspond to solving the on-shell cut conditions . As discussed above the solution to these conditions is always complex when three-point vertices are present. These cut conditions have a discrete set of solutions because of the two-fold choice in eq. (24) at each three-point vertex. Each of these solutions depends on continuous parameters, corresponding to the degrees of freedom not frozen by the cut conditions. The discrete choice coincides with the choice of three-point amplitude at each vertex as given in eqs. (25) and (26) (or similar vertices for fermionic and scalar lines), which suggests a convenient way to represent the possible solutions using additional labels at the vertices of the cut diagrams. For the four-point hepta-cut two inequivalent arrangements of three-point vertices are shown in fig. 13. External legs represent outgoing external momenta while internal lines represent cut propagators and thus on-shell loop momenta. The signs inside the blobs in the diagrams indicate the corresponding choice for the three-point vertex, and implicitly, that the spinors of the opposite helicity are proportional. A ‘ ’ vertex will have all spinors proportional to each other, so that the vertex is built out of spinors of the attached legs, while the roles of the two kinds of spinors are interchanged for a ‘ ’ vertex. We can sum over all possible solutions to obtain the multiply-cut integrand, as was done in ref. BuchbinderCachazo (). For the sevenfold cut of the double-box diagram, there are six distinct solutions of the two types shown in fig. 13.
Once we choose external helicities, as in fig. 14, the blobs then dictate the possible assignments of helicities for internal lines. The rules for finding the complete set of kinematic solutions associated with a given assignment of plus and minus labels to a diagram are as follows:
A ‘ ’ label means the three spinors corresponding to the lines attached to the blob are proportional to each other. Similarly, a ‘ ’ label denotes having the three spinors proportional to each other.
If one of the lines attached to a vertex is an external line then the spinors are proportional to an external spinor, either or .
If a ‘ ’ vertex is directly connected to another ‘ ’ vertex then all s of the lines attached to both vertices are proportional to each other. A similar statement holds for the s of two connected ‘ ’ vertices.
If there is a chain of vertices of the same sign connecting any two external lines then the diagram vanishes, because one cannot solve the on-shell and momentum conservation constraints for the diagram. A solution would require that two external spinors of the same type are proportional to each other; this cannot be true in general, because they are independent.
Applying these rules to the four-point double-box does indeed give the six allowed solutions of the two types shown in fig. 13. The remaining solutions are related to the depicted ones by flip symmetries. The complete set of solutions to the cut constraints is solely determined by the topology of a given diagram. Each solution determines a pattern of‘ ’ and ‘ ’ vertices in the diagram. For each solution to the cut constraints, one of the two types of three-point amplitudes vanishes at each vertex. This pattern (along with the topology) will in general restrict the helicity assignments along internal lines, and may also restrict the particle types allowed in different internal lines.
The strongest constraint that can arise in a kinematic solution is the restriction to a single allowed helicity configuration for the internal lines. We will refer to this configuration as a “singlet”. In this case only gluons can propagate inside the diagram, as in fig. 14(a). Fermions or scalars are not allowed because the only potentially non-vanishing vertices are of the wrong type and vanish for the given solution. The second-strongest constraint allows two helicity configurations. In such configurations the particle content is purely gluonic except for one loop in which any particle type can propagate, as shown in fig. 14(b). (A fermionic loop always allows two helicity assignments, corresponding to interchanging fermion and antifermion, and the same is true for complex scalar loops.)
Solving for the spinors in any diagram is then straightforward. Consider the singlet case, diagram (a) in fig. 14. The on-shell conditions together with momentum conservation at each vertex give a set of equations that must be satisfied,
The solution to these equations is,
where is an arbitrary parameter, corresponding to the remaining degree of freedom in the integration not frozen by the hepta-cut. Since a bispinor is invariant under a rescaling of the spinors, , the above solution can be written in many other forms. In addition, there is a choice as to where to include the remaining degree of freedom. While individual three-point amplitudes are not invariant under this transformation, the product of amplitudes forming the cut is invariant.
vi.2 Solving for integral coefficients using maximal cuts
We now consider how to solve for the coefficient of an integral using the maximal cuts. At two loops, there is only a single conformal integral, the double box. It can appear, of course, in both - and -channel configurations, but the hepta-cut shown in figs. 13 and 14 selects only the -channel double box. Our candidate expression for the amplitude is then,
where is the pseudo-conformal two-loop double-box integral in eq. (13) and is a coefficient that we need to solve for. This integral contains a factor of in the numerator, which as we shall see is necessary for satisfying the cut conditions.
Imposing the sevenfold cut condition, we obtain,
where is the three-point amplitude corresponding to one of the six three vertices and the sum over helicities runs over all possible helicity and particle configurations. (We have taken the loop integrals to be four-dimensional for the purposes of our discussion here, but in any explicit evaluation of the integrals they should be continued to dimensions to regulate the infrared singularities.)
As discussed above, the delta-function constraints are solved by a discrete set of solutions, so we obtain
where runs over the different kinematic solutions, and are the values of the independent loop momenta expressed in terms of the external momenta, and the remaining degree of freedom is . For each discrete solution , only a subset of helicity and particle configurations denoted by gives a non-vanishing contribution. The Jacobian from the change of variables is .
Buchbinder and Cachazo BuchbinderCachazo () noted that the integrand is constant after imposing the seven cut conditions arising directly from cutting propagators, without need to impose the eighth cut condition. Another curiosity they noted is that all six discrete kinematic solutions for the hepta-cut give the same answer for the amplitude. This was true for kinematic solutions that permitted only gluons in the two loops as well as for those which also permitted fermions and scalars. This simplicity is related to the absence of terms which integrate to zero upon performing the loop integral.444This property is special to four-point amplitudes and is already violated at one loop for five-point amplitudes.
We will exploit this observation, and assume its generalization to higher loops. It allows us to match integrands, and indeed to pick individual solutions to match the left-hand and right-hand side of eq. (35), determining the overall coefficient. That is, we assume that there is a single overall coefficient to solve for in front of each integral, instead of a different contribution for each solution. This also avoids any need for integral reductions or analysis of the integrals, and translates integrands into algebraic coefficients of integrals. Our knowledge of an integral basis — given by the pseudo-conformal integrals — is not essential but greatly simplifies the extraction of these coefficients. This equality of contributions from different solutions is likely special to MSYM at four points or perhaps to conformal supersymmetric gauge theories more generally. In general, there is no reason to expect solutions which allow different particle types to circulate to yield equal answers.
The assumption can be checked directly, of course, by comparing different solutions. While we have not checked it exhaustively, it does pass the large number of such comparisons that we have carried out. The use of the assumption and the maximal-cut procedure described here also leads to a determination of coefficients at three and four loops in agreement with known answers BDS (); BCDKS (). Furthermore, a violation would likely lead to inconsistent determinations of integral coefficients at five loops; we find no such inconsistency. We can also rely on cross checks from non-maximal cuts.
(The reader may be puzzled by the appearance of complex solutions in what was originally an integral over real loop momenta. This is not special to the amplitudes under consideration here. In extracting the cut by replacing propagators with delta functions, one must sum over complex solutions as well as real ones BCFUnitarity (). This was necessary in other circumstances such as evaluating the connected prescription for tree-level gauge-theory amplitudes RSV () in twistor string theory WittenTopologicalString (). It can also be understood by reinterpreting Vergu () the original integral as a fourfold contour integral in each component of the loop momentum, and replacing the propagators by products of an expression of the form times Jacobians; Cauchy’s theorem makes it act like a delta function, but allowing complex solutions. The details are not important to us here because we are only determining coefficients and not evaluating any integrals.)
Choosing one of the kinematic solutions then gives,
where runs over the helicity configurations and particle content with non-vanishing contributions for the given solution.
Obviously, it is advantageous to choose the simplest solution, where the kinematics restricts us to the fewest possible particle types circulating in the loop. The best choice is a singlet where only gluons contribute. Using the kinematic solution (32), corresponding to the singlet solution in fig. 14(a), we have,
with no sum over intermediate helicities. In the singlet case there is only one term in the helicity sum and all six three-point amplitudes are purely gluonic. Here, represents one of the three-gluon tree amplitudes in eqs. (25) and (26), with the plus and minus labels on these amplitudes matching the labels of the vertices in the figure. We have confirmed that eq. (37) holds for any value of the arbitrary parameter (other than , where the right-hand-side of eq. (37) is ill defined) in the kinematic solution (32). This equation then determines , so that the pseudo-conformal double box integral in fig. 1 appears in the two-loop amplitude with a coefficient of , in agreement with known results BRY (); BDDPR ().
vi.3 Generalized unitarity with real momenta at two loops
The kinematics in maximal cuts is highly constrained. It is therefore useful to have a way of checking results using less-restricted kinematics. Let us begin by considering generalized (non-maximal) cuts which are well defined for real momenta in four dimensions.
As an example of a four-dimensional generalized cut, consider the two-loop iterated two-particle cuts shown in fig. 15. This helicity configuration has the property that it is a singlet under supersymmetry transformations BDDPR (), the only contributions coming from gluon internal states. Hence we call it the “singlet” contribution. The remaining contributions, containing all other helicity and particle-type assignments, we will collectively call the “non-singlet” contributions. These latter contributions transform into each other under supersymmetry. (The action of supersymmetry on the amplitudes is described in appendix E of ref. BDDPR ().)
The singlet configuration is especially simple to evaluate because it involves only a single particle type, similar to the singlet configurations of the maximal cuts. At higher loops, the number of different particle and helicity configurations grows rapidly. If possible, it would be simpler to use only singlet configurations to confirm our ansatz for the amplitude, just as with maximal cuts. Unlike maximal cuts, however, the generalized cuts considered here do not enforce a particular choice of internal-line helicities or particle types. Nonetheless, in special cases, the singlet can be used to determine the coefficient of integrals in the amplitude.
For example, in the iterated two-particle cuts at two loops shown in fig. 15, the singlet contribution gives exactly the coefficient of the double box integral BRY (). With this external helicity configuration, this cut has no non-singlet contribution. The non-singlet contribution appears in the other channel and gives the identical result. In the three-particle cut, however, the singlet and non-singlet contributions appear in the same cut and are not identical. This cut does nonetheless have simple properties that we can exploit. The singlet contribution, depicted in fig. 16, has been previously evaluated in refs. BRY (); BDDPR (), with the result,
where . The tree amplitudes that appear are,
The terms in the singlet and non-singlet appear with opposite signs. In the sum over singlet and non-singlet contributions to the cut, the terms therefore cancel algebraically at the level of the integrand. (Alternatively, the difference between the singlet and non-singlet contributions integrates to zero.) From a practical standpoint, it is easier to compare cuts with target ansätze prior to integration and to use only the singlet, so the key observation is that we may use is that the non- term of the singlet is exactly half the total contribution to the cut.
We need to extend these observations to higher loops in order for them to be useful. We have confirmed that the same properties hold at three and four loops for any combination of two- and three-particle cuts composed only of four- and five-point tree amplitudes. (If six- or higher-point tree amplitudes are present in the cuts, non-MHV configurations with three or more negative and three or more positive helicities enter into the computation, which renders the structure of supersymmetric cancellations more elaborate and prevents us from using the singlet contribution alone to evaluate the cut.) We will assume that this observation continues to hold true at five loops. With this assumption we will be able to check (in section VIII) the coefficients of a variety of pseudo-conformal integrals, using only singlet cuts. This is a strong consistency check, because it relies not only on the coefficient under examination being correct, but also on the assumption remaining valid. (It seems extremely implausible that a breakdown of the assumption at five loops could be compensated by an incorrect coefficient.)
To do better we need to sum over all states crossing the cuts. Moreover, a proper treatment of the cuts requires that the cuts be evaluated using -dimensional states and momenta DDimUnitarity (); OneLoopReview (); DimShift (). This ensures that no contributions have been dropped, as can happen when four-dimensional momenta are used. At one loop, the improved power counting of supersymmetric theories allows one to prove a theorem that unitarity cuts with four-dimensional momenta are sufficient to determine dimensionally-regulated supersymmetric amplitudes (that is, “near” four dimensions) completely NeqFourOneLoop (); Fusing (). (The regulator must of course maintain manifest supersymmetry; as mentioned earlier, we use the four-dimensional helicity scheme (FDH) to do so. In this scheme, the helicity algebra is always four dimensional, but the momenta are continued to dimensions.) Unfortunately, no such theorem is as yet known beyond one loop. A subtlety in deriving such a theorem arises from infrared singularities: the singularities in one loop can effectively “probe” the contributions from another loop, the product giving a surviving contribution even as . When computing with -dimensional momenta, one can no longer use the spinor helicity representation SpinorHelicity (), which makes expressions for tree amplitudes used in the cuts more complicated. A good way to ameliorate this additional complexity is to consider instead in ten dimensions super-Yang-Mills dimensionally reduced to dimensions. The remaining states are completely equivalent to those of MSYM in the FDH scheme FDH (), except that the bookkeeping of contributions is much simpler.
At two loops all cuts of MSYM amplitudes were evaluated in dimensions TwoLoopGluons (), providing a complete proof of the planar and non-planar expressions for the MSYM amplitudes first obtained in ref. BRY () using four-dimensional momenta. At three loops, we have also re-evaluated the planar amplitude using -dimensional cuts. The four-loop planar amplitude has been evaluated using -dimensional cuts, assuming that the full result (as an abstract tensor in polarization vectors and momenta) is proportional to the tree amplitude, and making the reasonable assumption that no contributions can have a triangle subintegral. (Only the terms involving polarization vectors dotted into each other, after tensor reductions, were evaluated explicitly. Also, one can rule out all bubble and some triangle subintegrals using supersymmetry along with generalized unitarity.) In section VIII, we will make use of these results to provide non-trivial evidence in favor of the various assumptions we have used to obtain our ansatz for the five-loop four-point planar amplitude. A complete proof would require additional -dimensional cuts be evaluated in order to confirm the coefficient of every potential integral that might appear, including non-conformal ones.
Vii Maximal Cut Technique for Determining Integral Coefficients
vii.1 Overview of maximal cut method
In this section, we further develop the maximal-cut technique for higher loops using the observations of the previous section. This allows us to extend the two-loop maximal cuts of Buchbinder and Cachazo BuchbinderCachazo () to higher-loop orders. Unlike Buchbinder and Cachazo, we do not require the loop integration be frozen by the cut conditions. That is, we do not require the number of cut conditions to match the number of loop integrations. As discussed in the previous section, instead, we perform all evaluations of the cuts at the level of the integrand, prior to performing any loop integrations.
Moreover, we do not solve for all possible kinematic configurations satisfying the cuts. We instead focus on those solutions which allow the simplest determinations of the coefficients of the pseudo-conformal integrals as they appear in the amplitude. As discussed in section VI.1, the simplest kinematic solutions are the singlets, to which only gluons contribute. At five loops it turns out that only four pseudo-conformal integrals do not have singlet solutions. However, even in these cases one can choose kinematics which forces the fermion and scalar contributions into specific loops, again greatly simplifying the determination of the coefficients. In order to speed up the process of extracting the coefficients we solve the constraint equations numerically, although in some cases we find it useful to solve the constraints analytically.
At two-, three- and four-loops, where the complete results for the four-point planar amplitudes are known BRY (); BDS (); BCDKS (), we have confirmed that singlet maximal cuts correctly determine the coefficients of all integrals appearing in the amplitudes. This suggests that the same will be true at five loops. Again, if this were not true it would reveal itself as an inconsistency in the results. In particular, we would find that different kinematic solutions of the cut conditions would lead to inconsistent determinations of integral coefficients. We would also find inconsistencies with cuts with less restrictive kinematics.
A drawback of our maximal-cut method is its reliance on four-dimensional spinor helicity, which as mentioned in section VI.3 might drop contributions. Nevertheless, it does provide a relatively simple and systematic means to obtain an ansatz for the coefficients of integrals that appear in the amplitude.