On the Structure of Supersymmetric Sums in Multi-Loop Unitarity Cuts

On the Structure of Supersymmetric Sums in Multi-Loop Unitarity Cuts

Z. Bern, J. J. M. Carrasco, H. Ita, H. Johansson and R. Roiban Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547, USA
Department of Physics, Pennsylvania State University, University Park, PA 16802, USA
March, 2009

In this paper we describe algebraic and diagrammatic methods, related to the MHV generating function method, for evaluating and exposing the structure of supersymmetric sums over the states crossing generalized unitarity cuts of multi-loop amplitudes in four dimensions. We focus mainly on cuts of maximally supersymmetric Yang-Mills amplitudes. We provide various concrete examples, some of which are directly relevant for the calculation of four-loop amplitudes. Additionally, we discuss some cases with less than maximal supersymmetry. The results of these constructions carry over to generalized cuts of multi-loop supergravity amplitudes through use of the Kawai-Lewellen-Tye relations between gravity and gauge-theory tree amplitudes.

04.65.+e, 11.15.Bt, 11.30.Pb, 11.55.Bq          


I Introduction

Multi-loop scattering amplitudes in maximally supersymmetric gauge and gravity theories have received considerable attention in recent years for their roles BCDKS (); ABDK (); BDS (); AM () in helping to confirm and utilize Maldacena’s AdS/CFT correspondence Maldacena () and in probing the ultraviolet structure of supergravity theories Finite (); GravityThree (); CompactThree ().

In particular, multi-loop calculations offer important insight into the possibility that planar super-Yang-Mills scattering amplitudes can be resummed to all loop orders ABDK (); BDS (); AM (). In ref. ABDK () a loop iterative structure was suggested, leading to the detailed BDS conjecture BDS () for planar maximally-helicity-violating (MHV) amplitudes to all loop orders. Alday and Maldacena realized that certain planar scattering amplitudes at strong coupling may be evaluated as the regularized area of minimal surfaces in AdSS with special boundary conditions, and for four-point amplitudes they confirmed the BDS prediction. Direct evidence suggests that the all-order resummation holds as well for five-point amplitudes TwoLoopFive (). The structure of the four- and five-point planar amplitudes is now understood as a consequence DHKS () of a new symmetry dubbed “dual conformal invariance” DualConformal (); BCDKS (); FiveLoop (), with further generalizations at tree level DrummondYangian () and at infinite ’t Hooft coupling BerkovitsMaldacena (). However, beyond five points, the BDS conjecture requires modification AM2 (); Lipatov (); TwoLoopSixPt (). High-loop calculations in super-Yang-Mills theory should also play a useful role in clarifying the structure of subleading color contributions to the soft anomalous dimension matrix of gauge theories SoftMatrixPapers (), once the evaluation of the required nonplanar integrals becomes feasible at three loops and beyond.

In a parallel development, studies of multi-loop amplitudes in supergravity  CremmerJuliaScherk () have suggested that this theory may be ultraviolet finite in four dimensions Finite (); GravityThree (); CompactThree (), challenging the conventional understanding of the ultraviolet properties of gravity theories. For a class of terms accessible by isolating one-loop subamplitudes via generalized unitarity GeneralizedUnitarity (); TwoLoopSplit (); BCFGeneralized (), the one-loop “no-triangle” property (OneloopMHVGravity (); NoTriangle (); UnexpectedCancel (); AHCKGravity ()) shows that at least a subset of these cancellations persist to all loop orders Finite (). The direct calculation of the three-loop four-point amplitude of supergravity exposes cancellations beyond those needed for ultraviolet finiteness in in all terms contributing to the amplitude GravityThree (); CompactThree (). Interestingly, theory and string theory have also been used to argue either for the finiteness of supergravity DualityArguments (), or that divergences are delayed through at least nine loops Berkovits (); GreenII (), though issues with decoupling towers of massive states GOS () may alter these conclusions. A recent direct field theory study proposes that a divergence may first appear at the five loop order in , though this can be softer if additional unaccounted symmetries are present HoweStelleRecent (). If a perturbatively ultraviolet-finite point-like theory of quantum gravity could be constructed, the underlying mechanism responsible for the required cancellations is expected to have a fundamental impact on our understanding of gravity.

The recent studies of multi-loop amplitudes rely on the modern unitarity method UnitarityMethod (); Fusing () as well as various refinements GeneralizedUnitarity (); TwoLoopSplit (); BCFGeneralized (); FiveLoop (); FreddyMaximal (). In this approach multi-loop amplitudes are constructed directly from on-shell tree amplitudes. This formalism takes advantage of the fact that tree-level amplitudes are much simpler than individual Feynman diagrams, as well as makes use of various properties that hold only on shell. In particular, it provides a means of using an on-shell superspace—which is much simpler than its off-shell cousins—in the construction of loops amplitudes.

Summing over the physical states of propagating fields is one essential ingredient in higher-loop calculations. In particular, the modern unitarity method uses these sums over physical on-shell states in the reconstruction of any loop amplitude in terms of covariant integrals with internal off-shell lines. In supersymmetric theories the on-shell states can be organized in supermultiplets dictated by the supersymmetry. Systematic approaches to evaluate such supersymmetric sums—or supersums—have recently been discussed in refs. FreedmanGenerating (); FreedmanUnitarity (); KorchemskyOneLoop (); AHCKGravity (). As the calculations reach to ever higher loop orders these sums become more intricate. It is therefore helpful to expose their structure and simplify their evaluation as much as possible. In this paper we describe algebraic and diagrammatic methods which are helpful in this direction. These methods are the ones used in the course of computing and confirming the four-loop four-point amplitude of maximally supersymmetric Yang-Mills theory, including nonplanar contributions. The main aspects of the construction of this amplitude, as well as the explicit results, will be presented elsewhere FourLoopNonPlanar (). (The planar contributions are given in ref. BCDKS ().)

Supersymmetric cancellations were extensively discussed at one and two loops in refs. UnitarityMethod (); Fusing (); BRY (); BDDPR () using a component formalism that exploits supersymmetry Ward identities SWI (). These supersums were relatively simple, making it straightforward to sum over the contributions from the supermultiplet in components. The recent calculations of more complicated amplitudes in refs. TwoLoopFive (); FiveLoop (); GravityThree (); TwoLoopSixPt (); LeadingApplications (); CompactThree (), are performed in ways obscuring the systematics of the supersums. For example, as explained in ref. FiveLoop (), it is possible to avoid evaluating (sometimes complicated) supersums in maximally supersymmetric Yang-Mills theory via the method of maximal cuts, where kinematics can be chosen to restrict scalars and fermions to a small (even zero) number of loops. Remarkably, this trick is sufficient to construct ansätze for super-Yang-Mills amplitudes. However, any such ansatz needs to be confirmed by more direct evaluations incorporating all particles in the supermultiplet, to ensure that no terms are dropped. It is therefore necessary to compare the cuts of the ansatz with the cuts of the amplitude for more general kinematic configurations, allowing all states to cross the cuts. The calculation of supersums is a crucial ingredient in carrying out this comparison. Moreover, formal studies of the ultraviolet behavior of multi-loop amplitudes of supersymmetric theories, in particular of supergravity, are substantially aided by a formalism that exposes the supersymmetric cancellations.

Nair’s original construction of an on-shell superspace Nair () captured only MHV tree amplitudes in super-Yang-Mills theory; more recent developments extend this to any helicity and particle configuration. The approach of GGK (); FreedmanGenerating (); FreedmanUnitarity (); FreedmanProof () makes use of the MHV vertex expansion CSW () to extend this on-shell superspace to general amplitudes. Another strategy, discussed in refs. AHCKGravity (); RecentOnShellSuperSpace (), makes use of the Britto, Cachazo, Feng, and Witten (BCFW) on-shell recursion BCFW () to extend the MHV on-shell superspace to general helicity configurations. A new key ingredient of this approach is a shift involving anti-commuting parameters which may be thought of as the supersymmetric extension of the BCFW shift of space-time momenta. A recent paper uses shifts of anti-commuting parameters to construct a new super-MHV expansion Kiermaier (), which we do not use here. With the unitarity method UnitarityMethod (); Fusing (); BRY (); BDDPR (), superspace expressions for tree amplitudes can be extended to loop level. One-loop constructions along these lines were discussed in refs. FreedmanGenerating (); KorchemskyOneLoop (); AHCKGravity (), while various examples of supersums in higher-loop cuts, including four-loop ones, have already been presented in ref. FreedmanUnitarity ().

The MHV vertex expansion suggests an inductive structure for supersymmetric cancellations. Once these cancellations are exposed and understood for cuts with only MHV or tree amplitudes, more general cuts with non-MHV amplitudes follow rather simply FreedmanUnitarity (). Indeed, the prescription for evaluating these more general cuts involves summing over MHV contributions with shifts of certain on-shell intermediate momenta.

To evaluate the supersymmetric sums that appear in unitarity cuts we introduce complementary algebraic and diagrammatic approaches. The algebraic approach has the advantage of exposing supersymmetric cancellations, in many cases leading to simple expressions. It is a natural approach for formal proofs. In particular, it allows us to systematically expose supersymmetric cancellations—within the context of the unitarity method—sufficient for exhibiting the well known Mandelstam () all-loop ultraviolet finiteness of super-Yang-Mills theory. The diagrammatic approach gives us a means of pictorially tracking contributions, allowing us to write down the answer directly by drawing a set of simple diagrams. It also leads to a simple algorithms for writing down the results for any cut by sweeping over all possible helicity labels. Since it tracks contributions of individual states, it can be easily applied to a variety of cases with fewer supersymmetries. To illustrate these techniques we present various examples, including those relevant for evaluating the four-loop four-point amplitude of super-Yang-Mills theory FourLoopNonPlanar (). We will also show that these techniques are not restricted to four-point amplitudes by discussing some higher-point examples.

One potential difficulty with any four-dimensional approach is that unitarity cuts are properly evaluated in dimensions DDimUnitarity (); SelfDual (), since they rely on a form of dimensional regularization FDH () related to dimensional reduction Siegel (). Moreover, a frequent goal in multi-loop calculations is the determination of the critical dimension in which ultraviolet divergences first appear. Consequently, such calculations often need to be valid away from four dimensions. This requirement complicates the analysis significantly, because powerful four-dimensional helicity methods SpinorHelicity () can no longer be used. Any ansatz for an amplitude obtained with intrinsically four-dimensional methods, such as the ones of the present paper, needs to be confirmed through -dimensional calculation. Nevertheless, the analysis offers crucial guidance for the construction of -dimensional amplitudes. Additionally, methods appear to capture the complete result for four-point super-Yang-Mills amplitudes with fewer than five loops BRY (); BDDPR (); BCDKS (); GravityThree ().

While difficulties appear to arise with extending the MHV diagram expansion to general supergravity tree amplitudes FreedmanGenerating (), they will not concern us here. Instead we rely on the Kawai-Lewellen-Tye (KLT) relations KLT (); GeneralKLT (), or their reorganization in terms of diagram-by-diagram relations TreeJacobi (), to obtain the sums over supermultiplets in supergravity cuts directly from the cuts of corresponding super-Yang-Mills theory amplitudes.

This paper is organized as follows. In section II we review on-shell superspace at tree level and introduce -symmetry index diagrams. In section III we review the modern unitarity method and present the general structure of supercuts. In section IV we explain how the supersums can be evaluated in terms of the determinant of the matrix of coefficients of a system of linear equations. This section also contains various examples of cuts of super-Yang-Mills, including those of a five-point amplitude at four loops. Section V describes supersums in terms of -symmetry index diagrams, providing pictorial means for tracking different contributions. As discussed in section VI, these diagrams allow us to relate the cuts of amplitudes with fewer supersymmetries to maximally supersymmetric ones. They also allow us construct a simple algorithm for obtaining all contributions to cuts from purely gluonic ones. Various three and four-loop examples are presented in sections V and VI. In section VII we outline the use of the KLT relations to carry over the results for the sum over states in cuts of super-Yang-Mills amplitudes to the corresponding ones of supergravity theory. Our conclusions are presented in section VIII.

Ii On-shell superspace at tree level

On-shell superspaces are useful tools for probing the properties of supersymmetric field theories, providing information on their structure without any complications due to unphysical degrees of freedom. Here we review the construction of an on-shell superspace for super-Yang-Mills amplitudes. In its original form, devised by Nair Nair (), it described maximally helicity violating (MHV) gluon amplitudes and their supersymmetric partners. While we will depart at times from Nair’s original construction, the main features will persist. This same superspace also captures general amplitudes. Indeed, there currently exists two methods for constructing general amplitudes from MHV amplitudes: the MHV vertex construction of Cachazo, Svrček and Witten CSW () and the on-shell recursion relation of Britto, Cachazo, Feng and Witten (BCFW) BCFW (). The supersymmetric extension of the former approach has been given in refs. GGK (); FreedmanGenerating (); FreedmanUnitarity (); FreedmanProof (), while that of the latter approach in refs. AHCKGravity (); RecentOnShellSuperSpace ().

To evaluate the supersum in unitarity cuts we will use an approach based on MHV vertices, along the lines taken by Bianchi, Elvang, Freedman and Kiermaier FreedmanGenerating (); FreedmanUnitarity (). We will find that supersums involving only MHV and/or tree amplitudes have a surprisingly simple structure. We will also show how the MHV vertex construction allows us to immediately carry over this simplicity, with only minor modifications, to more general cuts involving arbitrary non-MHV tree amplitudes.

The on-shell superspace of the type we will review here generalizes easily to MHV and amplitudes in supergravity. Difficulties however, appear with the MHV vertex construction of non-MHV gravity tree amplitudes because the on-shell recursions used to obtain the expansion GravityMHV () can fail to capture all contributions FreedmanGenerating (). Such amplitudes may nevertheless be found without difficulty through supersymmetric extensions AHCKGravity () of the on-shell BCFW recursion relations BCFW (); CachazoLargez (), which do carry over to supergravity. However, at present BDDPR (); GravityThree (); CompactThree () we find it advantageous to use the KLT tree-level relations KLT (); GeneralKLT () or the recently discovered diagram-by-diagram relations TreeJacobi (), to obtain supergravity unitarity cuts directly from those of super-Yang-Mills theory.

ii.1 MHV amplitudes in super-Yang-Mills

The vector multiplet of the supersymmetry algebra consists of one gluon, four gluinos and three complex scalars, all in the adjoint representation of the gauge group, which here we take to be . With all states in the adjoint representation, any complete tree-level amplitude can be decomposed as


where are tree-level color-ordered -leg partial amplitudes. The ’s are generators of the gauge group and encode the color of each external leg , with color group indices . The sum runs over all noncyclic permutations of legs, which is equivalent to all permutations keeping one leg fixed (here leg ). Helicities and polarizations are suppressed. We use the all outgoing convention for the momenta to define the amplitudes.

All states transform in antisymmetric tensor representations of the -symmetry group such that states with opposite helicities are in conjugate representations. The -symmetry and helicity quantum numbers uniquely specify all on-shell states:


where and are, respectively, the positive and negative helicity gluons and gluinos while are scalars. (The scalars are complex-valued and obey a self-duality condition which will not be relevant here.) These fields are completely antisymmetric in their displayed -symmetry indices—denoted by —which transform in the fundamental representation of , giving a total of 16 states in the on-shell multiplet.

Alternatively, we can use the dual assignment obtained by lowering the indices with a properly normalized Levi-Civita symbol , giving the fields,


We will use both representations to describe the amplitudes of super-Yang-Mills. For MHV amplitudes we will mainly use the states with upper indices in eq. (2) whereas for we will use mainly the states with lower indices in eq. (3). This is a matter of convenience, and the two sets of states may be interchanged, as we will briefly discuss later in this section.

We begin by discussing the MHV amplitudes, which we define as an amplitude with a total of eight (2 4 distinct) upper indices. (In order to respect invariance, amplitudes of the fields in eq. (2) must always come with upper indices, where is an integer. Furthermore amplitudes with four or zero indices vanish as they are related by supersymmetry to vanishing SWI () amplitudes.) Some simple examples of MHV amplitudes, which we will use in section II.3, are,


where are four distinct fundamental indices. The overall phases of these amplitudes depend on conventions. We will fix this ambiguity by demanding that the phases be consistent with the supersymmetry algebra, which is automatically enforced when using superspace. The amplitudes are written in terms of the familiar holomorphic and antiholomorphic spinor products,


where the and are commuting spinors which may be identified with the positive and negative chirality solutions and of the massless Dirac equation and the spinor indices are implicitly summed over. These products are antisymmetric, , .

Momenta are related to these spinors via


and similar formulæ hold for the expression of . We will often write simply or sometimes . The proper contractions of momenta with spinorial objects will be implicitly assumed in the remainder of the paper. Typically, we will denote external momenta by and loop momenta by .

A subtlety we must deal with is a slight inconsistency in the standard spinor helicity formalism for massless particles when a state crosses a cut. In a given cut we will always have the situation that on one side of a cut line the momentum is outgoing, but on the other side it is incoming. Thus across a cut we encounter expressions such as , which is not properly defined in our all-outgoing conventions and can lead to incorrect phases. This is because the spinor carries momentum , and thus it has an energy component of opposite sign to that carried by the spinor . This problem is due to the fact that the spinor helicity formalism does not distinguish between particle and antiparticle spinors, as has been discussed and corrected in refs. SignSubtlety () for the MHV vertex expansion, and for BCFW recursion relations with fermions. To deal with this, we use the analytic continuation rule that the change of of sign of the momentum is realized by the change in sign of the holomorphic spinor FreedmanUnitarity (),


ii.2 The MHV Superspace

The supersymmetry relations between the different MHV amplitudes may be encoded in an on-shell superspace, which conveniently packages all amplitudes into a single object—the generating function or superamplitude. Each term in the superamplitude corresponds to a regular component scattering amplitude. Depending upon the detailed formulation of the superspace, scattering amplitudes of gluons, fermions and scalars are then formally extracted either by the application of Grassmann-valued derivatives FreedmanUnitarity (), or, equivalently, by multiplying with the appropriate wave functions and integrating over all Grassmann variables Nair (); WittenTopologicalString (). Effectively, these operations amount to selecting the component amplitude with the desired external states.

The MHV generating function (or superamplitude) is defined as,


where the leg label is identified with the leg label , and are Grassmann odd variables labeled by leg and -symmetry index . As indicated by the cyclic denominator, this amplitude is color ordered (i.e., it is the kinematic coefficient of a particular color trace in eq. (1)), even though the numerator possesses full crossing symmetry having encoded all possible MHV helicity and particle assignments. We suppress the delta-function factor responsible for enforcing the overall momentum conservation.

The eightfold Grassmann delta function in (8) is a product of pairs of delta functions, each pair being associated with one of the possible values of the -symmetry index:


This expression can be further expanded,


using the usual property of Grassmann delta functions that . Each monomial in in the superamplitude corresponds to a different MHV amplitude. In this form it is clear that all terms indeed have eight upper indices, as expected for an MHV amplitude.

Similarly, one may define an on-shell superspace, whose Grassmann parameters are , in which the superamplitude takes a form analogous to (8):


The indices are now lowered, which implies that the component amplitudes are built from the external states in (3) with a total of eight lower indices.

We note that the arguments of the MHV delta functions are the super-momenta , and for are similarly the conjugate super-momenta ,


where the index runs over all the external legs of the amplitude. Thus the purpose of the delta functions is to enforce super-momentum conservation constraint in the respective superspaces. For later purposes we define the individual super-momenta of the external legs,


The two superspaces can be related. Following ref. FreedmanUnitarity () we can rewrite the superamplitudes in the MHV superspace (or -superspace) via a Grassmann Fourier transform. For this purpose we define FreedmanUnitarity () the operator,


which realizes this Fourier transform. Then, following FreedmanUnitarity (), the superamplitude in the -superspace can be written as


From this perspective, the Grassmann Fourier transform is then easily expressed as the rule,


Here the spinors and are understood as being contracted after they are brought next to each other by anticommuting them past the various factors. While the spinors are generally taken as Grassmann-even, for the purposes of this rule it is convenient to treat them as Grassmann-odd.

However, in the above Fourier transformed amplitude the notion of the numerator as a supermomentum conservation constraint has been obscured. This can be somewhat cured using a second alternative presentation of the superamplitude in which we consider an integral representation of the ,


where are Grassmann odd integration parameters, . The action of the Grassmann Fourier transform (14) yields immediately KorchemskyOneLoop () a product over one-dimensional Grassmann delta functions, one for each external leg:


While somewhat obsfucated, for later purposes it is important to note the right-hand side of this equation is proportional to the -space supermomentum conservation constraint for . This relation may be exposed by taking an appropriate linear combination KorchemskyOneLoop () of the arguments of the delta functions:


upon using the momentum conservation constraint . (For the Fourier transformed amplitude is not proportional to . Even so, this amplitude still conserves supermomentum and is invariant under -supersymmetry KorchemskyOneLoop ().) While these manipulations may be explicitly carried out at the expense of introducing a Jacobian factor, it is frequently more convenient not to do so. Indeed, we will more often work directly with equation (18).

ii.3 Diagrammatic representation of MHV superamplitude

As mentioned, we are interested in simplifying the evaluation of sums over the members of the multiplet and uncovering their structure. For this purpose we introduce a diagrammatic approach for capturing the superspace properties of MHV amplitudes. These diagrams will be in one-to-one correspondence with the contributions to any given cut amplitude, allowing us to map out the structure of its supersum. We will give rules for translating the diagrams into algebraic results, including those for the Grassmann parameters needed to obtain the correct relative signs. While constructed for the maximally supersymmetric Yang-Mills theory in four dimensions, the ideas behind this method extend to theories with reduced supersymmetry (see section VI.1), being particularly well-suited for studying deformations of super-Yang-Mills theory.

Inspecting the eightfold Grassmann delta function, as given in eq. (10), we recognize that the basic building block of the MHV amplitude numerators is the spinor product of supermomenta,


For each index, the delta function in eq. (10) is simply the sum over all such products. We represent the supermomentum product graphically by a shaded (blue) line connecting point and , as in fig. 1(a). We will call this object “index line”.

Figure 1: For an MHV amplitude the shaded (blue) “index line” (a) connecting leg to leg represents . The two endpoints (and line) carry the same index. A solid (black) line (b) without endpoint dots represents a spinor product in the denominator.

In addition to the Grassmann delta function, color-ordered MHV amplitudes also have another important structure, the cyclic spinor string in the denominator,


This object has the same order as the trace of color-group generators, and can be thought of as being in one-to-one correspondence with this color structure. The spinor products in the denominator of MHV amplitudes will be represented by solid (black) lines without endpoint dots shown in fig. 1(b). The cyclicity of the MHV denominator implies that these lines form closed loops, except for the small gaps that we take to represent external states. It is convenient to draw the diagrams in a form reminiscent of string theory world-sheets, as displayed in fig. 2. The main role of the solid (black) lines will be to span the background, or canvas, on which the shaded (blue) index lines are drawn. The presentation of amplitudes in this world-sheet-like fashion provides the necessary room to draw the index lines without cluttering the figures. These diagrams—which we will call “index diagrams”—capture the spinor structures of MHV tree amplitudes along with the relative signs encoded by the superspace.

Given an MHV tree -point amplitude with specified external states, the rules for drawing the index diagram are simple: First draw the solid (black) lines representing the cyclic spinor string of the MHV amplitude denominator. Leave gaps between these lines to represent the external states, or legs. Label these legs with the appropriate momentum, helicity and indices. If the same index appears on external legs they should be connected by a shaded (blue) line with endpoint dots. This completes the diagram.

Figure 2: Examples of index diagrams for specifying numerator factors of MHV tree amplitudes. The diagrams (a), (b) and (c) correspond to the amplitudes in eq. (22). The shaded (blue) line connecting leg to leg represents a factor of respectively, and solid (black) lines represents . The white “” label on black background indicates that the amplitude is holomorphic, or MHV.

Consider, for example, the tree amplitudes in eq. (4), whose corresponding diagrams are shown in fig. 2. The “” and “” labels on the external states indicate the helicities, while the black-and-white-inverted “” and “” labels internal to the diagram indicates whether it is an MHV or amplitude, respectively. We will refer to this property of being either MHV or as an amplitude’s holomorphicity, as MHV amplitudes are built from holomorphic spinors and amplitudes are constructed from anti-holomorphic spinors. From the above construction it follows that the index lines in the diagrams of fig. 2 are in one-to-one correspondence to components in the MHV superamplitude, including the Grassmann parameters. Translating from the figures to analytic expressions using the rules of fig. 1, we can easily write down these component amplitudes,


where we have labeled the color ordered amplitudes (including Grassmann parameters) using a “correlator” notation on the left hand side. Repeated indices are not summed over their values; rather, their values are fixed and correspond to the particular choice of labels identifying the external states. For the amplitude to be nonvanishing, the labels must be distinct.

Figure 3: The same amplitudes as in fig. 2, now in the representation. The shaded (blue) line connecting leg to leg represents a factor of respectively, and solid (black) lines represents . The white “” label on black background indicates that the amplitude is anti-holomorphic, or .

Diagrams tracking the indices for amplitudes are similar. As a simple example, consider the same amplitudes as above, but reinterpreted as amplitudes—for four-point amplitudes (but no others) this is always possible. In the form the amplitudes are,


where are the conjugate supermomenta defined in eq. (13). The index diagrams corresponding to these expressions are displayed in fig. 3. Now the lines are interpreted in terms of conjugate or anti-holomorphic spinors and Grassmann parameters. As mentioned above, this is indicated by the black-and-white-inverted “” label on each diagram.

If we wish to work entirely in the -superspace for both MHV and amplitudes, we must map the parameters to ’s using the Grassmann Fourier transform in eq. (14). This transformation is conveniently captured by the rule in eq. (16), giving,


While perhaps less obvious for the time being, the utility of the index diagrams will become apparent in section V, where they will allow a transparent bookkeeping of the helicity states in unitarity cuts of multi-loop (super)amplitudes.

ii.4 MHV superrules for non-MHV superamplitudes

Figure 4: The MHV vertex construction builds non-MHV superamplitudes from MHV superamplitudes. The blobs are MHV superamplitudes, and the dots signify an arbitrary number of external legs, of which a few are drawn explicitly.

The MHV vertex construction generates non-MHV amplitudes from the MHV ones via a set of simple diagrammatic rules. Their validity has been proven in various ways, including the use of on-shell recursion Risager () and by realizing the MHV vertex rules as the Feynman rules of a Lagrangian CSW_Lagrangian (); CSW_QCD (). The former approach was recently shown to hold, with certain modifications, for all amplitudes of super-Yang-Mills theory FreedmanProof (), proving the validity of the MHV vertex construction for the complete theory. The latter approach was also extended CSW_LagrangianSusy () to the complete Lagrangian by carrying out an supersymmetrization of the MHV Lagrangian of refs. CSW_Lagrangian ().

The -point NMHV gauge theory superamplitude (where the “N” stands for “Next-to-”) contains gluon amplitudes with negative helicity gluons. One begins its construction by drawing all tree graphs with vertices, on which the external legs are distributed in all possible inequivalent ways while maintaining the color order. Examples of these graph topologies are shown in fig. 4.

To each vertex one associates an MHV superamplitude (8). As in the bosonic MHV rules, the holomorphic spinor associated to an internal leg is constructed from the corresponding off-shell momentum using an arbitrary (but the same for all graphs) null reference antiholomorphic spinor ,


Alternatively, the holomorphic spinor can be defined in terms of a “null projection” of , given by K_proj (); BBK (),


where is a null reference vector. In this form it is clear that the momenta of every vertex are on shell, thus, at this stage, the expression corresponding to each graph is a simple product of well-defined on-shell tree superamplitudes. (The analogous construction for gravity amplitudes is more complicated due to the fact that MHV supergravity amplitudes are not holomorphic GravityMHV ().)

To each internal line connecting two vertices one associates a super-propagator which consists of the product between a standard scalar Feynman propagator and a factor which equates the fermionic coordinates of the internal line in the two vertices connected by it. The structure of the propagator depends on the precise definition of the superspace, but such details are not important for the following. Upon application of the precise rules for assembling the MHV vertex diagrams, the expression for the NMHV superamplitude is given by


where the integral is over the internal Grassmann parameters () associated with the internal legs, and each is the (off-shell) momentum of the ’th internal leg of the graph. The MHV superamplitudes appearing in the product correspond to the vertices of the graph. The momentum and dependence of the MHV superamplitudes is suppressed here. We note, however, that the null projection of each internal momentum and the Grassmann variable appear twice, in the form,


Each integration in eq. (27) selects the configurations with exactly four distinct -variables on each of the internal lines. Since a particular can originate from either of two MHV amplitudes, as per eq. (28), there are possibilities that may give non-vanishing contributions. These contributions correspond to the 16 states in the multiplet, making it clear that the application of indeed yields the supersum. However, for a given choice of external states, each term corresponding to a distinct graph in (27) receives nonzero contributions from exactly one state for each internal leg.

Note that as far as sewing of amplitudes is concerned, it makes no difference whether an intermediate state is put on-shell due to a cut or due to the MHV vertex expansion. This observation, implying that sewing of general amplitudes proceeds by integrating over common variables, will play an important role in our discussion of cuts of loop amplitudes.

Figure 5: The index lines for two out of the nine diagrams of the MHV vertex expansion for the amplitude . The dashed vertical (red) line signifies that the intermediate state is on-shell. The integration will force exactly four index lines to end (or start) on the intermediate on-shell state.

We now illustrate the index diagrams, introduced in the previous section, for the MHV-vertex expansion of an NMHV example. Since the index diagrams represent component amplitudes these diagrams clarify the details of the state sum. First we note that according to eq. (27) an NMHV amplitude is a polynomial in of degree , since there are MHV amplitudes—which by definition contain eight ’s with upper indices—and the Grassmann integration removes of them. Thus, an NMHV amplitude must have 12 ( distinct) upper indices.

Let us consider the seven-point amplitude which is of this form. There are a total of nine non-vanishing diagrams, of which two are displayed as index diagrams in fig. 5, illustrating the sewing of gluonic and fermionic states, respectively. Summing over the diagrams gives us the amplitude




and we suppress all but the contributions of the two diagrams in fig. 5. In the last equality we carried out the Grassmann integration, which here only serves to convert the internal four powers of to factors of . When using the MHV diagrams expansion in unitarity cuts of loop amplitudes, as we will see in section IV, it is generally convenient to delay carrying out the Grassmann integrations until the complete cut is assembled.

We note that it is convenient to collect the various NMHV tree superamplitudes into a single generating function,


where is the number of external legs, and the sum terminates with the amplitude, here written as an NMHV amplitude in superspace. The number of terms in this sum is for . The three-point case should be treated separately since it contains two terms, MHV and , which cannot be supported on the same kinematics.

Iii Evaluation of Loop Amplitudes using the Unitarity Method

The direct evaluation of generalized unitarity cuts of super-Yang-Mills scattering amplitudes requires summing over all possible intermediate on-shell states of the theory. Various strategies for carrying out such sums over states have recently been discussed in refs. KorchemskyOneLoop (); AHCKGravity (); FreedmanUnitarity (). Here we review our current approach, which is closely related to the generating function ideas of ref. FreedmanGenerating (); FreedmanUnitarity (). Additionally, we present an analysis of the structure of the resulting factors and expose various universal features.

iii.1 Modern unitarity method

The modern unitarity method gives us a means for systematically constructing multi-loop amplitudes for massless theories. This method and its various refinements have been described in some detail in references UnitarityMethod (); Fusing (); BRY (); GeneralizedUnitarity (); TwoLoopSplit (); BDDPR (); BCFGeneralized (); FiveLoop (); FreddyMaximal (), so here we will mainly review points salient to the sums over all intermediate states appearing in maximally supersymmetric theories.

Figure 6: Some examples of generalized cuts at four loops. Every exposed line is cut and satisfies on-shell conditions. Diagrams (b) and(c) are near maximal cuts. In four dimensions only MHV or tree amplitudes appear in cuts (a)-(c) while in cut (d) non-MHV tree amplitudes appear.

The construction starts with an ansatz for the amplitude in terms of loop momentum integrals. We require that the numerator of each integral is a polynomial in the loop and external momenta subject to certain constraints, such as the maximum number of factors of loop momenta that can appear. The construction of such an ansatz is simplest for the super-Yang-Mills four-point amplitudes where it turns out that the ratio between the loop integrand and the tree amplitudes is a rational function solely of Lorentz invariant scalar products BRY (); BCDKS (); FiveLoop (). For higher-point amplitudes similar ratios necessarily contain either spinor products or Levi-Civita tensors, as is visible even at one loop UnitarityMethod ().

The arbitrary coefficients appearing in the ansatz are systematically constrained by comparing generalized cuts of the ansatz to cuts of the loop amplitude. Particularly useful are cuts composed of tree amplitudes of form,


evaluated using kinematic configurations that place all cut momenta on shell, . Cuts which break up loop amplitudes into products of tree amplitudes are generally the simplest to work with to determine an amplitude, although one can also use lower-loop amplitudes in the cuts as well. In special cases, such as when there is a four-point subamplitude, this can be advantageous FourLoopNonPlanar (). In fig. 6, we display a few unitarity cuts relevant to four loops. If cuts of the ansatz cannot be made consistent with the cuts of the amplitude, then it is, of course, necessary to enlarge the ansatz.

The reconstruction of an amplitude from a single cut configuration is typically ambiguous as the numerator may be freely modified by adding terms which vanish on the cut in question. Consider, for example, a particular two-particle cut with cut momenta labeled and . No expressions proportional to and are constrained by this particular cut. Such terms are instead constrained by other cuts. After information from all cuts is included, the only remaining ambiguities are terms which are free of cuts in every channel. In the full amplitude these ambiguities add up to zero, representing the freedom to re-express the amplitude into different algebraically equivalent forms. Using this freedom one can find representations with different desirable properties, such as manifest symmetries or explicit power counting GravityThree (); CompactThree ().

For multi-loop calculations, generally it is best to organize the evaluation of the cuts following to the method of maximal cuts FiveLoop (). In this procedure we start from generalized cuts GeneralizedUnitarity (); TwoLoopSplit (); BCFGeneralized () with the maximum number of cut propagators and then systematically reduce the number of cut propagators FiveLoop (). This allows us to isolate the missing pieces of the amplitude, as well as reduce the computational complexity of each cut. A related procedure is the “leading-singularity” technique, valid for maximally supersymmetric amplitudes FreddyMaximal (); LeadingApplications (). These leading singularities, which include additional hidden singularities, have been suggested to determine any maximally supersymmetric amplitude AHCKGravity ().

At one loop, all singular and finite terms in amplitudes of massless supersymmetric theories are determined completely by their four-dimensional cuts Fusing (). Unfortunately, no such property has been demonstrated at higher loops, although there is evidence that it holds for four-point amplitudes in this theory through five loops BCDKS (); GravityThree (); FiveLoop (). We do not expect that it will continue for higher-point amplitudes. Indeed, we know that for two-loop six-point amplitudes terms which vanish in do appear TwoLoopSixPt (). Even at four points, Gram determinants which vanish in four dimensions, but not in -dimensions, could appear at higher-loop orders.

At present, -dimensional evaluation of cuts is required to guarantee that integrand contributions which vanish in four dimensions are not dropped. -dimensional cuts DDimUnitarity () make calculations significantly more difficult, because powerful four-dimensional spinor methods SpinorHelicity () can no longer be used. (Recently, however, a helicity-like formalism in six dimensions has been given D6Helicity ().) Some of this additional complexity is avoided by performing internal-state sums using the (simpler) gauge supermultiplet of super-Yang-Mills theory instead of the multiplet. In any case, it is usually much simpler to verify an ansatz constructed using the simpler four-dimensional analysis, than to construct the amplitude directly from its -dimensional cuts.

For simple four-dimensional cuts, the sum over states in eq. (32), can easily be evaluated in components, making use of supersymmetry Ward identities SWI (), as discussed in ref. BDDPR (). In some cases, when maximal or nearly maximal number of propagators are cut, it is possible to choose “singlet” kinematics which force all or nearly all particles propagating in the loops to be gluons in the super-Yang-Mills theory FiveLoop (). However, for more general situations, we desire a systematic means for evaluating supersymmetric cuts, such as the generating function approach of ref. FreedmanGenerating (); FreedmanUnitarity ().

iii.2 General structure of a supercut

Using superamplitudes, integration over the parameters of the cut legs represents the sum over states crossing the cuts in eq. (32). The generalized supercut is given by,


where are generating functions (31) connected by on-shell cut legs. The supercut incorporates all internal and external helicities and particles of the multiplet. In most cases it is convenient to restrict this cut by choosing external configurations, e.g. external MHV or sectors (or even external helicities), etc. In many cases it is also convenient to expand out each into its NMHV components, and consider each term–consisting of a product of such amplitudes—as a separate contribution. We will focus our analysis on such single terms, since as we will see they form naturally distinct contributions, each being an invariant KorchemskyOneLoop () expression. As these contributions correspond to internal quantities they must be summed over. We note that although in this discussion we restrict to cuts containing only trees, it can sometimes be advantageous to consider cuts containing also four and five-point loop amplitudes, since they satisfy the same supersymmetry relations as the tree-level amplitudes.

If all tree amplitudes in the supercut have fewer than six legs then each supercut contribution is of the form,


where uses the Grassmann Fourier transform in eq. (14). For cuts where there are tree amplitudes with more than five legs present, some cut contributions include non-MHV tree amplitudes. For these we apply the MHV vertex expansion (27), which reduces these more complicated cases down to a sum of similar expressions as eq. (34) with only MHV and amplitudes (and additional propagators).

Certain properties of the super-Yang-Mills cuts can be inferred from the structure of generalized cuts and the manifest -symmetry and supersymmetry of tree-level superamplitudes. First we note that a cut contribution that corresponds to a product of only MHV tree amplitudes consists of a single term of the following numerator structure,


where we have made it explicit that the product over the indices can be commuted past both the product over internal cut legs and the product over tree amplitudes labeled by . Here is the total supermomentum of superamplitude , where runs over all legs of . For convenience we have also suppressed the spinor index. From the right-hand-side of eq. (35), we conclude that the numerator factor arising from the supersum of a cut contribution composed of only MHV amplitudes is simply the fourth power of the numerator factor arising from treating the index in as taking on only a single value.

A cut contribution constructed from only MHV and tree amplitudes has similar structure, though the details are slightly different. Using the fermionic Fourier transform operator (14) any -point tree amplitude can manifestly be written as a product over four identical factors, each depending on only one value of the -symmetry index,


Consequently, just as for cut contributions constructed solely from MHV tree amplitudes, for the cases where only MHV and tree amplitudes appear in a cut, the end result is that the fourth power of some combination of spinor products appears in the numerator. This feature will play an important role in section V, simplifying the index diagrams that track the -symmetry indices.

The super-MHV vertex expansion generalizes this structure to generic cuts of loop amplitudes. As already mentioned, any non-MHV tree superamplitude can be expanded as a sum of products of MHV superamplitudes. If we insert this expansion into a generalized cut, we obtain a sum of terms where the structure of each term is the same as a cut contribution composed purely of MHV amplitudes. All that changes is that the momenta carried by some spinors are shifted according to eq. (26), and some internal propagators are made explicit. We immediately deduce that the numerator of each term is given by a fourth power of the numerator factor arising when treating the index of as having a single value. This general observation is consistent with results found in ref. FreedmanUnitarity ().

The structure of the constraints due to supersymmetry may be further disentangled. It is not difficult to see that the cut of any super-Yang-Mills multi-loop amplitude is proportional to the overall super-momentum conservation constraint on the external supermomenta. Similar observations have been used in a related context in ref. RSV_3 (); KorchemskyOneLoop (); FreedmanProof (). This property is a consequence of supersymmetry being preserved by the sewing, which is indeed manifest on the cut, as we now show. Consider an arbitrary generalized cut constructed entirely from tree-level amplitudes; using the MHV-vertex super-rules, this cut may be further decomposed into a sum of products of MHV tree amplitudes. Each term in this sum contains a product of factors of the type (9), one for each MHV amplitude in the product. Using the identity each such product of delta functions may be reorganized by adding to the argument of one of them the arguments of all the other delta functions:


where is the number of MHV trees amplitudes—including those from a single graph of each MHV-vertex expansion. In the conventions (7) in which a change of the sign of the four-momentum translates to a change of sign of the holomorphic spinor , and therefore also in , we immediately see that in the first delta function all corresponding to internal lines occurs pairwise with opposite sign, and thus cancel, leaving only external variables,


where denotes the set of external legs of the loop amplitude whose cut one is computing. Thus, this delta function depends only on the external momentum configuration and is therefore common to all terms appearing in this cut. The generalized cuts involving only tree amplitudes are sufficient for reconstructing the complete loop amplitude TwoLoopSplit (), therefore it is clear that the superamplitude and all of its cuts are proportional to , assuming four-dimensional kinematics.

As can be seen from eqs. (18) and (19), the discussion above, showing supermomentum conservation, goes through unchanged for cuts containing -point tree-level amplitudes with . This includes all cuts with real momenta. For , from ref. KorchemskyOneLoop (), we see that the supermomentum conservation constraint of three-point amplitudes may be obtained from their fermionic constraint upon multiplication by a spinor corresponding to one of the external lines. Using this observation, it is then straightforward to show that for eqs. (37) and (38) continue to hold.

The explicit presence of the overall supermomentum conservation constraint eq. (38) is sufficient to exhibit the finiteness Mandelstam () of super-Yang-Mills theory. Since, as we argued, the same overall delta function appears in all cuts, it follows that the complete amplitude also has it as an overall factor. In fact, there is a strong similarity between the superficial power counting that results from this and the super-Feynman diagrams of an off-shell superspace. Indeed, the count corresponds to what we would obtain from the Feynman rules of a superspace form of the MHV Lagrangian CSW_LagrangianSusy () which manifestly preserves half of the supersymmetries.

More concretely, for any renormalizable gauge theory with no more than one power of loop momentum at each vertex, the superficial degree of divergence is,


where is the number of loops, the dimension, the number of external legs and the number of powers of momentum that can be algebraically extracted from the integrals as external momenta. For each power of numerator loop momentum that can be converted to an external momentum, the superficial degree is reduced by one unit. Taking and , corresponding to the four powers of external momentum implicit in the overall delta function (38), we see that for all loops and legs. This also implies that super-Yang-Mills amplitudes cannot contain any subdivergences as all previous loop orders are finite. It then follows inductively that the negative superficial degree of divergence, for all loop amplitudes, is sufficient to demonstrate the cancellations needed for all order finiteness. We note that although this displays the finiteness of super-Yang-Mills theory, not all cancellations are manifest, and there are additional ones reducing the degree of divergence beyond those needed for finiteness BDDPR (); FiveLoop (); HoweStelleNew ().

A similar analysis can be carried out for supergravity; in this case the two-derivative coupling leads to a superficial degree of divergence which monotonically increases with the loop order. Without additional mechanisms for taming its ultraviolet behavior, this would lead to the conclusion of that the theory is ultraviolet divergent. As discussed in refs. Finite (); GravityThree (); CompactThree () direct evidence to all loop orders indeed points to the existence of much stronger ultraviolet cancellations.

Iv The supersum as a system of linear equations

We now address the question of how to best carry out the evaluation of multiple fermionic integrals, which can become tedious for complicated multi-loop cuts. An approach to organizing this calculation, discussed in the following sections, is to devise effective diagrammatic rules for carrying out these integrals. Another complementary approach, discussed in this section, relies on the observation that the fermionic delta functions may be interpreted as a system of linear equations determining the integration variables (i.e. the variables corresponding to the cut lines) in terms of the variables associated with the external lines of the amplitude. From this standpoint, the integral over the internal ’s may be carried out by directly solving an appropriately chosen system of equations and evaluating the remaining supersymmetry constraints on the solutions of this system. While the relation between the fermionic integrals and the sum over intermediate states in the cuts is quite transparent, as we will see in later sections, it is rather obscure to identify the contribution of one particular particle configuration crossing the cut in the solution of the linear system.

iv.1 Cuts involving MHV and MHV vertex expanded trees

Simple counting shows that after the overall supermomentum conservation constraint is extracted, the number of equations appearing in cuts of MHV amplitudes equals the number of integration variables. For such cuts the result of the Grassmann integration is then just the determinant of the matrix of coefficients of that linear system. The same counting shows that the number of fermionic constraints appearing in cuts of NMHV amplitudes is larger than the number of integration variables. One way to evaluate the integral is to determine the integration variables by solving some judiciously chosen subset of the supermomentum constraints and substitute the result into the remaining fermionic delta functions. Care must be taken in selecting the constraints being solved, as an arbitrary choice may obscure the symmetries of the amplitude. One approach is to take the average over all possible subsets of constraints determining all internal fermionic variables. Another general strategy is to select the fermionic constraints with as few external momenta as possible. Since the integration variables are determined as ratio of determinants, all identities based on over-antisymmetrization of Lorentz indices, such as Schouten’s identity, are accounted for automatically, generally yielding simple expressions.

Figure 7: Supercut of a one-loop -point MHV amplitude. The white “” labels on the black background signify that these blobs are holomorphic vertices, or MHV superamplitudes. The dashed (red) line marks the cut, enforcing that momenta crossing the cut are on shell.

To illustrate this approach let us consider the example, shown in fig. 7, of the supercut of the one-loop -point MHV superamplitude


The only contribution to this cut is where both tree superamplitudes are MHV; together they contain the two delta functions,


Adding the argument of the first delta function to the second one, as discussed in (37), exposes the overall supermomentum conservation


then, the value of the fermionic integral in eq. (40) is the determinant of the matrix of coefficients of the following system of linear equations,


interpreted as a system of equations for and ; its determinant is


Thus, the resulting cut superamplitude is just