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MCTP-10-56

MIT-CTP-4197

PUPT-2361

SUSY Ward identities, Superamplitudes, and Counterterms

Henriette Elvang, Daniel Z. Freedman, Michael Kiermaier

Michigan Center for Theoretical Physics, Randall Laboratory of Physics

University of Michigan, Ann Arbor, MI 48109, USA

[2mm] School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA

[2mm] Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

[2mm] Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

[2mm] Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA

[3mm] elvang@umich.edu, dzf@math.mit.edu, mkiermai@princeton.edu

Ward identities of SUSY and R-symmetry relate -point amplitudes in supersymmetric theories. We review recent work in which these Ward identities are solved in SYM and supergravity. The solution, valid at both tree and loop level, expresses any NMHV superamplitude in terms of a basis of ordinary amplitudes. Basis amplitudes are classified by semi-standard tableaux of rectangular -by- Young diagrams. The SUSY Ward identities also impose constraints on the matrix elements of candidate ultraviolet counterterms in supergravity, and they can be studied using superamplitude basis expansions. This leads to a novel and quite comprehensive matrix element approach to counterterms, which we also review.

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## 1 Introduction

Supersymmetry and R-symmetry Ward identities impose linear relations among individual amplitudes in supersymmetric theories. The first question addressed in this review is how to solve the Ward identities in SYM and supergravity and specify a basis of amplitudes that determines all others in the same NMHV class. For MHV amplitudes, the answer is simple: any one amplitude determines the entire class. However, for very little information was known until recently. We review the results of [1], where the SUSY and R-symmetry Ward identities were solved to give an expansion of the general NMHV superamplitude in terms of a minimal basis of component amplitudes that are independent under these Ward identities. In the second part of this review, we apply this expansion to the analysis of potential counterterms in supergravity [2]. Imposing the additional requirement of locality on the manifestly SUSY and R-invariant expansion of superamplitudes is at the heart of this matrix-element approach to counterterms. Just as recursion relations focus on on-shell scattering amplitudes instead of Lagrangians, the center of attention is shifted from counterterm operators to their matrix elements.

The first approach to SUSY Ward identities for on-shell amplitudes was the 1977 work of Grisaru and Pendleton [3] (see also [4, 5]). They discussed the structure of these identities and solved them for 6-point NMHV amplitudes in SUSY. Six basis amplitudes were needed to determine all 60 NMHV amplitudes.111The solution of [3] was rederived using modern spinor-helicity methods in [6].

A general solution to the and Ward identities was recently presented in [1] and will be reviewed in Sec. 4-5 below. The solution exploits the properties of superamplitudes which compactly encode all individual -point amplitudes at each NMHV level. The Ward identities can be elegantly imposed as constraints on the superamplitudes which are then expressed as sums of simple manifestly SUSY- and R-invariant Grassmann polynomials, each multiplied by an ordinary amplitude. This set of ordinary amplitudes comprise a basis for the superamplitude. Only the Ward identities of non-anomalous Poincaré SUSY and symmetry are used, so the results apply both to tree and loop amplitudes. The dual conformal and Yangian symmetries of the theory are important and have led to much new information about planar amplitudes of the theory.222 See [7] for a review in this issue. Those symmetries were not included in the analysis of [1], so that the results are valid for both planar and nonplanar amplitudes of SYM and also for supergravity.

Let us provide a preview of the structure of superamplitudes and their basis expansion with details discussed in Sec. 3-5. Superamplitudes [8, 9, 10] are generating functions for ordinary amplitudes whose bookkeeping Grassmann variables are labeled by particle number and by the -symmetry index . At level NMHV, the superamplitudes are Grassmann polymomials of order . Their coefficients are the actual scattering amplitudes.

Supercharges and defined by the simple expressions

 ˜Qa=n∑j=1|j⟩ηja,Qa=n∑j=1|j]∂∂ηja (1.1)

act directly on the superamplitudes, giving the Ward identities

 ˜QaAn=0,QaAn=0. (1.2)

It is these SUSY Ward identities combined with important constraints due to R-symmetry which are solved in [1]. The solutions derived for superamplitudes take the schematic form

 ANKMHVn = ∑IAIZI. (1.3)

The index enumerates the set of independent SUSY and R-symmetry invariant Grassmann polynomials of degree . They are constructed from two simple and familiar ingredients, which are explained in more detail below. First, each contains a factor of the well known Grassmann delta-function, , which expresses the conservation of . It is a degree polynomial which is annihilated by both and . The other ingredient is that each contains factors of the first-order polynomial

 mijk,a≡[ij]ηka+[jk]ηia+[ki]ηja, (1.4)

in which label three external lines of the -point amplitude. Every polynomial is annihilated by . The polynomial (1.4) is the essential element of the well-known 3-point anti-MHV superamplitude.

The basis amplitudes in (4.10) are matrix elements for specific particle processes within each NMHV sector. Finding the basis can be formulated as a group theoretic problem, and it has a neat solution. The number of amplitudes in the basis is the dimension of the irreducible representation of corresponding to a rectangular Young diagram with rows and columns! The independent amplitudes are precisely labeled by the semi-standard tableaux of this Young diagram.

As an example, consider the -point NMHV amplitude in SYM. There are 5 basis amplitudes which can be chosen to be the 6-point matrix elements:

 ⟨−+++−−⟩,⟨λ−λ+++−−⟩,⟨s¯s++−−⟩,⟨λ+λ−++−−⟩,⟨+−++−−⟩. (1.5)

The last 4 particles in each amplitude are ‘standardized’ by SUSY to be gluons of positive and negative helicity. In the first two positions we must allow any combination that leads to an NMHV amplitude, i.e. pairs of gluons, gluinos, and scalars. For supergravity, the analogous basis contains 9 basis amplitudes which we can again specify to contain ‘standardized’ gravitons as the last 4 particles and pairs of gravitions, gravitinos, etc. on the first two lines.

Basis amplitudes containing four gluons ‘’ on four fixed lines are particularly convenient to write down the superamplitude in closed form. Using a computer-based implementation of this superamplitude, however, one can choose any other set with the same number of linearly independent amplitudes. Linear independence, in this case, is best verified numerically. At the -point NMHV level, for example, a suitable basis of linearly independent gauge theory amplitudes is the split-helicity gluon amplitude together with of its cyclic permutations, specifically

 ⟨+++−−−⟩,    ⟨−+++−−⟩,    ⟨−−+++−⟩,    ⟨−−−+++⟩,    ⟨+−−−++⟩. (1.6)

In supergravity, the pure graviton amplitude together with permutations of its external lines represents a suitable basis. It is striking that the basis of planar SYM ( supergravity) at the -point NMHV level reduces to momentum permutations of a single all-gluon (all-graviton) amplitude.

The second major topic of this review is the application of the basis expansions of superamplitudes to candidate counterterms of the form in the loop expansion of perturbative supergravity. The matrix element method complements and extends other approaches to counterterms which work with on-shell superspace [11, 12, 13], information from string theory [14, 15, 17], and light-cone superspace [18]. The leading matrix elements of a potential counterterm must be local and gauge invariant, and this means that they are polynomials in the spinor brackets associated with the external momenta. Matrix elements of candidate counterterms at loop order are strongly constrained by the overall scale dimension and the helicities of their external particles. In many cases one can show quite simply that there are no local SUSY and R-invariant superamplitudes that satisfy these constraints. Then the corresponding operator is not supersymmetrizable and cannot appear as an independent counterterm. On the other hand, when the constraints are satisfied, the method explicitly constructs the matrix elements of a linearized supersymmetric completion of the operator.

In addition to SUSY and R-symmetry, the spontaneously broken symmetry [19] of supergravity gives additional constraints on counterterm matrix elements with external scalar particles. In particular, counterterm matrix elements must vanish in the single-soft scalar limit. These constraints were analyzed to exclude the potential -, -, and -loop counterterms , , and in the recent papers [20, 21] (see also [22, 17, 23]), which are reviewed in Sec. 6.5 below.

The net result of the matrix element approach to counterterms, combined with the results of [13], is that there are no admissible counterterms in supergravity at loop order . The method does not exclude counterterms at loop order , but it shows that the only possible independent loop counterterm is  , whose leading matrix elements involve 4 external particles [21]; higher-point operators such as (for which we present simple explicit superamplitude expressions in Sec. 6.3) and are compatible with SUSY and R-symmetry, but have non-vanishing single-soft scalar limits and thus violate continuous symmetry [21] (see also [24]). This implies that a computation of the -point amplitude is sufficient to determine whether or not supergravity is finite at -loop order.

In Sec. 7, we discuss the structure of superamplitudes with reduced R symmetry. We focus on amplitudes that are invariant under an subgroup of ; these are relevant both for the study of single-soft scalar limits in supergravity and for closed string tree amplitudes with massless external states in 4 dimensions.

## 2 SUSY Ward identities

Particle states of the and theories transform in anti-symmetric products of the fundamental representation of the R-symmetry groups and . Thus the gluons, the 4 gluinos, and the 6 scalars of the theory can be described by annihilation operators which carry anti-symmetrized upper indices:

 B,  Ba,  Bab,  Babc,  Babcd, (2.1)

with . The tensor rank is related to the particle helicity by . The 256 particle states of supergravity are described analogously by annihilation operation operators of tensor rank . Helicity and rank are related by .

We now discuss the S-matrix elements and Ward identities for the simpler theory. The extension to is straightforward. One can suppress indices and simply use for any annihilation operator from the set in (2.1). A generic -point amplitude may then be denoted by

 An(1,2,…,n) = ⟨B⋯1B⋯2⋯B⋯n⟩. (2.2)

invariance requires that the total number of (suppressed) indices is a multiple of 4, i.e. . Furthermore each index value must appear times among the operators . The integer determines the NMHV class by

In general one considers complex null momenta , described by a bi-spinor . For real momenta, when angle and square spinors are related by complex conjugation, each describes a physical amplitude in which particles in the final state have positive null and particles in the initial state have negative null . In scattering theory the -matrix describes particle states in the limit of infinite past and future in which wave packets separate and interactions can be neglected. Therefore the SUSY charges that act on asymptotic states are determined by the free field limit of the transformation rules of the field theory.

In this section it is convenient to define chiral supercharges and , which include contraction with the anti-commuting parameters of SUSY transformations. The commutators of the operators and with the various annihilators are given by:

 [˜Qa,B]=0,[˜Qa,Bb]=⟨ϵp⟩δbaB,[˜Qa,Bbc]=⟨ϵp⟩2!δ[baB\raisebox1.707165pt$c]$,[˜Qa,Bbcd]=⟨ϵp⟩3!δ[baB\raisebox1.707165pt$cd]$,[˜Qa,Bbcde]=⟨ϵp⟩4!δ[baB\raisebox1.707165pt$cde]$,[Qa,B]=[pϵ]Ba,[Qa,Bb]=[pϵ]Bab,[Qa,Bbc]=[pϵ]Babc,[Qa,Bbcd]=[pϵ]Babcd,[Qa,Bbcde]=0, (2.3)

Note that raises the helicity of all operators and involves the spinor angle bracket . Similarly, lowers the helicity and spinor square brackets appear.

The Ward identities that relate S-matrix elements are obtained from

 0 = ⟨[˜Qa,B⋯1B⋯2⋯B⋯n]⟩, (2.4) 0 = ⟨[Qa,B⋯1B⋯2⋯B⋯n]⟩. (2.5)

The overall expressions vanish because supercharges annihilate the vacuum state. One then obtains concrete relations among amplitudes by moving the supercharges to the right, using the appropriate entry from (2.3) to evaluate or . To obtain a non-trivial relation, the product of operators must contain an odd number of fermions. There are further constraints from invariance. In (2.4). the distinguished index value must appear times among the , while other index values appear times each. Similarly, in (2.5), the index value must appear times and other index values times each. One thus sees that the Ward identities relate amplitudes within an NMHV class.

At the MHV level, the SUSY Ward identities give very simple and transparent relations. For example, consider

 0 = ⟨[˜Q1,−−B1+⋯+]⟩ = ⟨ϵ1⟩⟨B234−B1+⋯+⟩+⟨ϵ2⟩⟨−B234B1+⋯+⟩+⟨ϵ3⟩⟨−−++⋯+⟩, (2.6)

where we used that negative helicity gluons ‘’ transform as , while positive helicity gluons ‘’ are annihilated by the supercharge,  . There are three contributions on the right hand side of (2.6): the first two are gluon pair amplitudes and the last one is the -gluon MHV amplitude. However, there are two linearly independent choices of the SUSY spinor . If we choose , then (2.6) yields the relation

 ⟨B234−B1+⋯+⟩ = ⟨23⟩⟨12⟩⟨−−++⋯+⟩. (2.7)

between a gluino pair amplitude and the -gluon amplitude. If we choose , then we find a similar relation for the other gluon pair amplitude. For every set of operators in in which the index appears three times and the indices  twice each, the Ward identity contains three terms. By choice of one obtains two independent relations similar to (2.7). By combining the various relations, one can show that any MHV -point amplitude can be expressed as a rational function of angle brackets times the -gluon amplitude . Another fact about MHV amplitudes is that the Ward identities are automatically satisfied when the relations from the WI’s are incorporated. These key properties of the MHV sector are best seen from the MHV generating function discussed in the next section.

The situation in the NMHV sector is very different, as we can see by examining the Ward identity

 0 = = ⟨ϵ1⟩⟨B234−−B1++⟩+⟨ϵ2⟩⟨−B234−B1++⟩+⟨ϵ3⟩⟨−−B234B1++⟩+⟨ϵ4⟩⟨−−−+++⟩.

Now there are four terms, while can take two independent values. Thus, one obtains two independent equations for four amplitudes, which is a linear system of “defect two”. Every NMHV compatible choice of the operators in produces a similar pair of linear equations. Thus one obtains a large coupled set of such relations, and the overall rank of the system is difficult to ascertain. This problem is indeed best addressed in the language of superamplitudes, which we introduce in the next section. Please read on.

## 3 Superamplitudes and their Symmetries

### 3.1 Superamplitudes and supersymmetry constraints

The annihilation operators of the 16 massless states — gluons, gluinos, scalars — of the supermultiplet can be encoded in the ‘on-shell superfield’

 Φ=B+ηaBa−12!ηaηbBab−13!ηaηbηcBabc+η1η2η3η4B1234, (3.1)

in which the bookkeeping Grassmann variables are labeled by indices . The supercharges and act on by multiplication or differentiation. They ‘move’ operators to the right or left in (3.1) to reproduce the commutation relations (2.3). The anticommutator of the two supercharges is and thus realizes Poincaré SUSY.

The amplitudes for all -point processes within a given NMHV class are collected into superamplitudes , which are polynomials in the . The superamplitudes we discuss here must be invariant. In particular, an NMHV superamplitude is a degree polynomial in the in which each index value appears times in every monomial term. Any desired amplitude can be projected out from by acting with the differential operators [6] that select the desired external state from each . The total derivative order is .

The construction for supergravity is completely analogous: the 256 massless states are encoded into superfields using Grassmann variables labelled by the global R-symmetry group . The NMHV superamplitudes are degree polynomials in the ’s. In the rest of this section we study the maximally supersymmetric gauge and gravity theories ( and ) jointly.

In [1] it is shown that the SUSY Ward identities (1.2) can be satisfied if superamplitudes are constructed from two basic ingredients. The first ingredient is the well-known Grassmann -function

 δ(2N)(˜Q) ≡ δ(2N)(n∑i=1|i⟩ηia) = 12NN∏a=1n∑i,j⟨ij⟩ηiaηja. (3.2)

is fully supersymmetric. Indeed, it is clear that , while momentum conservation ensures that . We will show below that is also invariant.

The -function is the only element needed to construct MHV superamplitudes. Note that it has the correct polynomial order, namely . The -point MHV superamplitude is simply given by

 AMHVn=δ(2N)(˜Q) ⟨++⋯+−−⟩⟨n−1,n⟩N. (3.3)

It has one ‘basis amplitude,’ namely the pure gluon/graviton MHV amplitude . When the order- differential operator, which selects a given process, is applied, angle brackets are produced from the -function, and the chosen amplitude is then times the basis amplitude.

The second basic ingredient that is needed to construct NMHV superamplitudes is the simple polynomial of (1.4). The Schouten identity ensures , and this holds for any choice of three lines , adjacent or non-adjacent, independent of momentum conservation.

We write the NMHV superamplitude

 ANKMHVn = δ(2N)(˜Q) PN×K, (3.4)

where is a polynomial of degree in the variables. The delta-function (3.2) ensures that . Since commutes with the delta-function, the only remaining SUSY constraint is . This is a non-trivial condition, but we show that its general solution can be expressed in terms of products of the polynomials . The solution depends on the R-symmetry Ward identities, which we discuss next.

### 3.2 R-symmetry

To establish invariance of a function of the -variables it is sufficient to impose invariance under transformations acting on all choices of a pair of the indices . To be specific, consider infinitesimal transformations in the -plane:

 σ1:{δRηia=θηibδRηib=θηia,σ2:{δRηia=−iθηibδRηib=iθηia,σ3:{δRηia=θηiaδRηib=−θηib. (3.5)

Here is the infinitesimal transformation parameter.

As a warm-up to further applications, we show that the -function (3.2) is invariant; this implies that MHV superamplitudes necessarily preserve the full -symmetry. Since any monomial of the form is invariant under a -transformation, so is the -function. A -transformation in the 12-plane gives

 (3.6)

Anticommutation of the (highlighted) Grassmann variables antisymmetrizes the sum over and then vanishes by Schouten identity. The “” stands for independent terms from acting on and . These terms can be treated the same way. Invariance under -transformations follows directly from invariance and needs no further proof.

The R-symmetry constraints play an important role in the analysis of the SUSY Ward identities beyond the MHV level.

The analysis of the R-symmetry Ward identities also leads to a set of new cyclic identities for amplitudes. The identities encode relationships among amplitudes with the same types of external states, but with their R-symmetry indices distributed in different ways. An example is the following 4-term relation among SYM NMHV amplitudes with gluinos and scalars :

 0 = A6(λ123λ3λ123λ4s14s24) + A6(λ123λ4λ123λ3s14s24) (3.7) + A6(λ123λ4λ123λ4s13s24) + A6(λ123λ4λ123λ4s14s23).

We call this a cyclic identity because the four boldfaced indices are cyclically permuted.

## 4 Basis expansion of superamplitudes in N=4 Sym

We outline the strategy used to solve the SUSY- and R-symmetry Ward identities and construct a particular basis for the amplitudes at NMHV level for SYM. We then present the representations of superamplitudes using this basis. We emphasize results below and leave full details to App. A.

### 4.1 Strategy for solving the SUSY Ward identities

The initial form of the NMHV superamplitude is

 ANMHVn = δ(8)(˜Q) P4. (4.1)

Our task is to construct a minimal basis for all 4th order Grassmann polynomials that are invariant and satisfy . Let’s get to work.

1. First consider the constraints of R-symmetry invariance discussed in Sec. 3.2. The -transformations require to be a linear combination of monomials, so we write

 P4 =n∑i,j,k,l=1qijklηi1ηj2ηk3ηl4. (4.2)

The action of the -rotation in the -plane gives

 δR(qijklηi1ηj2ηk3ηl4)=θqijkl(ηi2ηj2+ηi1ηj1)ηk3ηl4. (4.3)

This quantity must vanish; hence the coefficients must be symmetric in indices and , . A similar argument for any generator of implies that is a totally symmetric tensor.

2. The superamplitude (4.1) includes the -function as a factor, so the 8 conditions it imposes can be used to eliminate a total of distinct , namely any choice of two for each . A convenient choice (which we make) is to eliminate the 4+4 Grassmann variables associated with lines and . Then  will then not depend on and , and we write

 P4=1⟨n−1,n⟩4n−2∑i,j,k,l=1cijklηi1ηj2ηk3ηl4. (4.4)

The ’s are linear combinations of the ’s; we will not need their detailed relationship. The coefficient in (A.3) could be absorbed by a redefinition of the , but we keep it for later convenience. As in step 1, R-symmetry requires the ’s to be fully symmetric, so the number of needed inputs at this stage is  .

It is a consequence of our choice to eliminate and that all basis amplitudes have negative helicity gluons on lines and .

3. The polynomial in (A.3) satisfies the Ward identity if and only if the linear relations hold for any triple . Since is a 2-component spinor, there are two independent constraints which allow us to eliminate a choice of two lines and completely from the indices of the ’s in . This is analogous to the use of the Ward identities to eliminate two sets of -variables in step 2, and a consequence is that lines and are positive helicity gluons in all basis amplitudes. In the following we choose and .

We rewrite in terms of ’s with and find that this naturally leads to the appearance of the polynomials , defined in (1.4). The result (see appendix for details) is the following form of the NMHV superamplitude:

 ANMHVn =∑1≤i≤j≤k≤l≤n−4cijklX(ijkl) with X(ijkl)≡∑P(i,j,k,l)Xijkl, (4.5)

where the are -polynomials of degree 12 that are annihilated by both and :

 Xijkl ≡ δ(8)(˜Qa) mi,n-3,n-2;1mj,n-3,n-2;2mk,n-3,n%−2;3ml,n-3,n-2;4[n−3,n−2]4⟨n−1,n⟩4. (4.6)

The sum over permutations in the definition of is over all distinct arrangements of fixed indices . For instance, we have . Likewise, contains the 6 distinct permutation of its indices, and has 12 terms.333The number of distinct permutations of a set with repeated entries is a multinomial coefficient [25].

4. The coefficients with parameterize the most general SUSY and -symmetry invariant NMHV superamplitude. The last step is to relate to actual amplitudes which then become the basis amplitudes. By direct application of the appropriate Grassmann derivatives, we find that each is identified as a single amplitude

 cijkl=An({i,j,k,l}++−−)≡⟨⋯B1i⋯B2j⋯B3k⋯B4l⋯+n-3+n-2−n-1−n⟩, (4.7)

with . Let us clarify the notation: means that line carries index 1, line carries index 2 etc. If , this means that the line carries both indices 1 and 2, and the notation should then be understood as . Furthermore the dots indicate positive-helicity gluons in the unspecified positions, specifically any state is a positive helicity gluon. For example, . For clarity, we have used a ‘’ to separate the first states from the last four gluon states, which are the same for all basis amplitudes.

Our final result for the manifestly SUSY and R-symmetric SYM NMHV superamplitude is

 ANMHVn =∑1≤i≤j≤k≤l≤n−4An({i,j,k,l}++−−) X(ijkl). (4.8)

One might say that we have used the SUSY generators and to ‘rotate’ two states, and , to be positive helicity gluons and two other states and , to be negative helicity gluons. Any NMHV amplitude can be obtained from (4.8) by applying the 12th-order Grassmann derivative that corresponds to its external states. The amplitude will then be expressed as a linear combination of the independent basis amplitudes . The collection of these amplitudes is what we define as the algebraic basis.

Let us consider examples of superamplitudes in the basis (4.8). For we have to distribute the four -indices on lines: there is only one choice, namely to put them all on line 1, which then must be a negative helicity gluon. Thus the 5-point NMHV superamplitude is described in terms of a single basis element ; this is of course not surprising, since the 5-point NMHV sector is equivalently described as anti-MHV. The superamplitude takes the form .

Next, let us write the 6-point superamplitude in the basis (4.8). The four indices should now be distributed in all inequivalent ways on lines and . There are five ways to do this — , , , and — giving five basis amplitudes. The 6-point NMHV superamplitude can thus be written

 ANMHV6 = (4.9) + ⟨λ1λ234++−−⟩X(1222) + ⟨+−++−−⟩X2222.

Here, we use a notation where denotes a gluino ( or ) with the indicated indices, and denotes the scalar .

The amplitudes of the algebraic basis used in (4.8) are of the schematic form . The states can be any particles of the theory, subject to the NMHV level constraint that each index appears once among the . As in any vector space, there many other ways to specify a basis. One can choose any other set with the same number of amplitudes, provided that they are linearly independent under the SUSY and R-symmetry Ward identities. To verify linear independence of a putative set of basis amplitudes one can project them from the superamplitude (4.8) using the appropriate differential operators and then check that the matrix which relates the new set to the original basis has maximal rank. Due to algebraic complexity, this check is best done numerically using a computer-based implementation of the superamplitude.

At the -point NMHV level, for example, any choice of linearly independent SYM amplitudes form a valid basis that completely determines the superamplitude. We have verified that the split-helicity amplitude together with of its cyclic permutations is a suitable basis of -point NMHV amplitudes. Similarly, there are pure-gluonic algebraic basis for NMHV amplitudes with and external legs. At , however, the 84 distinct gluonic amplitudes span a 69-dimensional subspace of the 70-dimensional algebraic basis. For the dimension of the algebraic basis even exceeds the number of pure-gluon amplitudes, which immediately rules out the possibility of a purely gluonic basis.

### 4.2 Functional bases and single-trace amplitudes

The representation (4.8) contains a sum over basis amplitudes which are algebraically independent under the symmetries we have imposed. However, we have not yet included possible functional relations among amplitudes, that is relations which involve reordering of particle momenta. The cyclic and reflection symmetries of single trace color ordered amplitudes are examples of such relations.

For amplitudes in the single-trace sector, the cyclic permutations are functionally dependent; they can be computed from cyclic momentum relabelings. Thus the all-gluon algebraic basis of single-trace -point NMHV amplitudes discussed above reduces to a functional basis containing the single amplitude . (Note that functional relations among amplitudes do not invalidate their use in an algebraic basis.)

For , the functional basis in the single-trace sector cannot consist of a single amplitude. Indeed, dihedral symmetry relates amplitudes, which, for , is smaller than the number of algebraic basis amplitudes. For example, for dihedral symmetry generates a set of at most 14 amplitudes from any one given amplitude, but amplitudes are needed needed to form an algebraic basis. It is an open problem to find a simple expression of the superamplitude in terms of the minimal functional basis. However, it is possible to write down superamplitudes whose algebraic basis amplitudes are pairwise functionally related by dihedral symmetry. We refer the reader to [1] for details of this construction.

### 4.3 Beyond NMHV: superamplitudes and Young tableaux

The NMHV basis amplitudes of (4.8) are labeled by four integers in the range . These numbers are conveniently arranged in the semi-standard tableaux jjjj of the Young diagram with one row and four columns. It was shown in [1] that semi-standard Young tableaux provide the general organizing principle for NMHV superamplitudes. These superamplitudes can be written in the schematic form

 ANKMHVn = ∑IAIZI, (4.10)

in which index enumerates the semi-standard tableaux of the rectangular Young diagram with rows and columns. The number of such semi-standard tableaux is the dimension of the irrep corresponding to the Young diagram . For each tableau there is a basis amplitude and a manifestly SUSY and -invariant -polynomial . To illustrate this structure, we discuss the NMHV superamplitudes of SYM.

The basis amplitudes of the -point NMHV superamplitude are labeled by semi-standard Young tableaux with two rows and four columns,

 (4.11)

Each row is non-decreasing () and each column is strictly increasing ( , etc.). Each tableau corresponds to a basis amplitude with the specified gluons on the last four lines and with index on lines and , index on lines and , etc. For example,

 \framebox[9.958465pt][c]1\framebox[9.958465pt][c]1\framebox[9.958465pt][c]1\framebox[9.958465pt][c]3\framebox[9.958465pt][c]2\framebox[9.958465pt][c]2\framebox[9.958465pt][c]2\framebox[9.958465pt][c]5.    ⟷    A9({11132225}++−−) = A9(λ123λ123λ4+λ4++−−). (4.12)

From the hook rule [26] it follows that the

 \#(N2MHV n-pt basis amplitudes) = dimSU(n−4)\framebox[9.958465pt][c]1\framebox[9.958465pt][c]1\framebox[9.958465pt][c]1\framebox[9.958465pt][c]1\framebox[9.958465pt][c]1\framebox[9.958465pt][c]1\framebox[9.958465pt][c]1\framebox[9.958465pt][c]1 = (n−5)(n−4)2(n−3)2(n−2)2(n−1)4!5!. (4.13)

The NMHV superamplitude can be written in terms of basis amplitudes as

 AN2MHVn = 116∑semi-% standard~{}~{}tableaux Y(−)YAn({i1j1k1l1i2j2k2l2}++−−) Zi1j1k1l1i2j2k2l2, (4.14)

where the ’s are manifestly SUSY- and R-symmetry invariant -polynomials similar to the ’s in (4.6), but contain eight instead of four powers of . The -polynomials and the sign factor are defined in [1].

Let us comment on the detailed information contained in the semi-standard tableaux labels of the basis amplitudes. In the example (4.12), line labels appeared times, respectively. This is a particular (ordered) partition of the boxes of the Young diagram; each semi-standard tableau corresponding to a 3+3+1+0+1 partition of 8 corresponds to a process with the same particles types for the external states: states 1 and 2 are negative helicity gluinos, states 3 and 5 are positive helicity gluinos, and state 4 is a positive helicity gluon. How many independent basis amplitudes are there corresponding to this partition? — In other words, how many -inequivalent ways are there to arrange the two sets of -indices on the two negative helicity gluinos and the two positive helicity gluinos? The answer to this question is the combinatorial quantity called the Kostka number.444The Kostka number depends on a Young tableaux with boxes and a partition of . The partition is a weight that dictates the number of times each number is used in the construction of the semi-standard tableaux of . The Kostka number counts the number of semi-standard tableaux of with weight . For the partition of the 2-by-4 rectangular Young diagram, the Kostka number is 2: in addition to (4.12) there is a second basis amplitude with the same particle types on each external line, namely

 \framebox[9.958465pt][c]1\framebox[9.958465pt][c]1\framebox[9.958465pt][c]1\framebox[9.958465pt][c]2\framebox[9.958465pt][c]2\framebox[9.958465pt][c]2\framebox[9.958465pt][c]3\framebox[9.958465pt][c]5    ⟷    A9({11122235}++−−) = A9(λ123λ124λ3+λ4++−−). (4.15)

Note that a different ordering of the partition is also possible, namely . The Kostka number is independent of the ordering, so there are also two semi-standard tableaux associated with this second ordering; they are just obtained from those in (4.12) and (4.15) by exchanging and . The corresponding basis amplitudes have a positive helicity gluino on line 4 and a positive helicity gluon on line 5. The structure outlined in this example generalizes to characterize all basis amplitudes of NMHV superamplitudes. Further details can be found in [1].

## 5 Basis expansion of superamplitudes in N=8 supergravity

The generalization of the above results to supergravity is straightforward. The MHV sector is particularly simple because the superamplitude contains only one basis amplitude which we take to be the -graviton amplitude . The superamplitude is the 16th order Grassmann polynomial

 CMHVn=δ(16)(∑i|i⟩ηai)Mn(−−+⋯+)⟨12⟩8. (5.1)

The amplitude must be bose symmetric under exchange of helicity spinors for the two negative helicity particles and for any pair of positive helicity particles. However the superamplitude must have full permutation symmetry, and so must the ratio  .

At the NMHV level, the amplitudes of the algebraic basis are now characterized by the semi-standard tableaux of a rectangular -by- Young diagram. The SUSY- and R-invariant Grassmann polynomials multiplying each basis amplitude are order ; they are constructed as in , but with twice as many ’s. NMHV -point superamplitudes must also have permutation symmetry We now discuss the NMHV sector in more detail.

NMHV amplitudes in supergravity
The identification of an algebraic basis in supergravity proceeds as in gauge theory and leads to a representation of NMHV superamplitudes analogous to (4.8) namely

 MNMHVn=∑1≤i≤j≤⋯≤v≤n-4cijklpquv X(ijklpquv), (5.2)

with symmetrized versions of the - and -invariant polynomial

 Xijklpquv = δ(16)(˜Qa) mi,n-3,n