A Seven-Point N{}^{2}MHV Yangian invariants

Multi-Particle Amplitudes from the Four-Point Correlator in Planar N=4 SYM

Abstract

A non-trivial consequence of the super-correlator/super-amplitude duality is that the integrand of the four-point correlation function of stress-tensor multiplets in planar super Yang-Mills contains a certain combination of -point amplitude integrands for any . This combination is the sum of products of all helicity super-amplitudes with their corresponding helicity conjugates. The four-point correlator itself is described by a single scalar function whose loop level integrands possess a hidden permutation symmetry facilitating its computation up to ten loops. We discover that assuming Yangian symmetry and an appropriate basis of planar dual conformal integrands it is possible to disentangle the contributions from the individual amplitudes from this combination. We test this up to seven points and up to two loops. This suggests that any scattering amplitude for any , with any helicity structure and at any loop order may be extractable from the four-point correlator.

Amplitudes, Correlator, Duality, Supersymmetry
1

1 Introduction

Super Yang-Mills theory (SYM) is singled out amongst all four-dimensional quantum field theories for its remarkable symmetry and mathematical structure along with its key role in the AdS/CFT correspondence. Further simplification and intriguing structures arise in the planar limit where the number of colours in the gauge group becomes large.

The fundamental objects of interest in any conformal theory are gauge-invariant operators and their correlation functions. The simplest operator in SYM is where is one of the six scalars. This is a special operator: related via superconformal symmetry to both the stress-energy tensor and the on-shell Lagrangian; protected from renormalisation, and annihilated by half of the supercharges in the theory. Two- and three-point correlators of such operators are protected from renormalisation taking their free value, therefore making the four-point correlator the first non-trivial case. This correlator has been well studied over a number of years and has been computed to three loops fully and to ten loops at the level of the integrand [1, 2, 3, 4, 5, 6, 7, 8, 9]. It was the discovery of a hidden permutation symmetry at the integrand level—vastly reducing the basis of integrands to a problem of enumerating planar graphs with certain conformal properties [5]—which made these results possible. The basis is given in terms of so-called graphs, which at loops are graphs with -vertices, containing both edges and numerator lines with resulting conformal weight four. This provides a compact representation of the correlation function for which we provide examples in subsection 2.1. Efficient graphical methods have since been developed for obtaining the coefficients of the graphs and hence fully determining the correlator.

Another set of key quantities in SYM are the integrands of scattering amplitudes, possessing remarkable mathematical structure and at the forefront of current research [10, 11, 12, 13, 14, 15, 8, 9, 16]. A feature of planar SYM relates the aforementioned objects to one another via the correlator/amplitude duality. This equates the correlator (divided by its Born contribution) in a polygonal light-like limit to the square of the scattering amplitude normalised by its MHV tree-level value [17, 18, 19]. The duality was later extended to incorporate supersymmetry [20, 21, 22].

One can consider this duality for a correlator of any number of points, but it is particularly interesting to apply it to the four-point correlator integrand. In this case, taking an -point light-like limit (involving internal integration points as well as “external” points—the permutation symmetry means there is no distinction) gives the sum of products of all -point helicity superamplitudes with their helicity conjugates. This remarkable feature makes use of the fact that the -loop 4-point correlator integrand is itself an -point -loop correlator with 4 scalar operators and Lagrangians, which are in turn related by supersymmetry to .

Concretely then, taking the -point light-like limit of the -loop, 4-point correlator, represented by (whose precise definition will be given later) we obtain the following combination of NMHV, -loop, -point superamplitudes (normalised by the MHV tree-level superamplitude), ,

(1)

where is a simple algebraic factor.

Note that this sum involves all NMHV amplitudes at loops, as well as lower-loop amplitudes. Furthermore, these are all combined together into a simple scalar function of the external momenta only, , without any complicated helicity/superspace dependence—the correlator, , is a much simpler object than the constituent amplitudes themselves.

The question we address in this paper is whether contains all the information about these constituent amplitudes, or put another way, whether one can extract all the individual superamplitudes themselves purely from the combination . We know this can be achieved at four- and five-points [5, 6, 16]. This may seem unlikely for higher points at first glance: on the left-hand side, is a purely scalar function of the momenta, whereas the superamplitudes on the right-hand side can exhibit complicated helicity structure.

Our findings are consistent with the following conjecture: assuming the tree-level MHV and anti-MHV amplitudes, parity, Yangian symmetry and a dual conformally invariant basis of planar integrands:2 all -point NMHV scattering amplitude integrands at any loop order (modulo signs3) can be obtained from the four-point correlator, which thus packages all of this information together into a simple scalar function.

Let us now make the above statement more precise and specify what information can be obtained from the correlator at each loop level. First note that the -loop correlator combination (1) involves the parity-even -loop combinations (consider and in (1))

(2)

together with lower-loop amplitudes. Thus from this combination alone, the correlator at this loop level cannot see ambiguities in the amplitude of the form

(3)

where is any combination of -loop integrands.4 Remarkably, we find that (3) is the only form of ambiguity arising from the duality at this loop level, and even more remarkably, this ambiguity is resolved by considering the correlator at one loop higher. Imposing parity reduces the ambiguity to the space of parity-odd integrands only, and imposing cyclicity further reduces this to just the space of cyclic combinations of parity-odd integrands.

More precisely then, the conjecture is that knowing all -point amplitudes fully to loops, as well as the parity-even pieces (as defined in (2)) at loops, then from the light-like limit of the -loop four-point correlator, we can extract the “parity-even” part of all -loop amplitudes along with fixing the remaining ambiguities at loops. Thus, we can recursively extract the parity-even part of the -loop amplitude from the -loop correlator, with , and the entire amplitude if we additionally use .

In this paper, we verify this statement by checking at six points and seven points up to two loops for the parity-even part.

In order to achieve this, we use a basis of planar dual conformal -loop integrands, to construct an ansatz for the superamplitudes. The integrands are functions of , where are dual momenta (external or loop co-ordinates) together with the parity-odd dual conformal covariant, most straightforwardly expressed as where the are 6d embedding dual momentum co-ordinates. At two loops we use a refinement of this basis, namely the prescriptive basis of [10, 11, 12] which we show can be written in terms of the above basis.

We also need to control the helicity structures of the superamplitudes. For this we use a basis of Yangian-invariant Grassmannian integrals, and as a technical aid, we use amplituhedron co-ordinates [23, 24, 25, 26]. We thus write an ansatz for the constituent superamplitudes of the form

(4)

which we substitute into the duality equation (1) in order to determine the coefficients .

To extract the combination (1) from this ansatz, we use a number of tools: dual-space, momentum twistors, the Grassmannian, amplituhedron co-ordinates, rules from the amplituhedron, and an understanding of the duality in question. For completeness, we review these ideas—starting with graphs and the correlator/amplitude duality. We then review momentum twistors, the Grassmannian, amplituhedron co-ordinates and proceed with the six-point extractions.

Finally, we repeat the idea for seven particles which invokes the Grassmannian to derive invariants and draw conclusions from our results.

2 Review of Key Ideas

2.1 Representing the Correlator with graphs

The four-point correlator of interest is

(5)

involving the protected operator for any scalar in the theory. We define as the the -loop integrand of (5)

(6)

where are the -loop integration variables. Note that we have a unique definition of this integrand via the -point correlator involving Lagrangian insertions in addition to the four operators in (5).

There is a powerful hidden symmetry [5] possessed by this integrand (for ) which simply states that is invariant under any permutation of its variables. This therefore places “external” variables on the same level as internal variables, . The symmetry vastly reduces the basis of potential integrands for the correlator to that of objects ,

(7)

These basis elements are equivalent to so-called graphs, which at loops are undirected graphs with -vertices composed of both solid (denominator), dashed (numerator) lines and signed degree (number of edges minus number of numerator lines leaving each vertex) equal to four. The solid edges must contribute to a simple planar graph. These graphs and their coefficients have been determined up to ten loops using combinations of various efficient graphical methods [9]. The above provides a compact representation of the correlation function with expressions up to four loops displayed below

(8)

where , , and all come with unit-magnitude coefficients up to four loops. Each graph corresponds to an algebraic expression (we are not careful to distinguish the graphs and their corresponding expressions) as follows: label the vertices with labels 1 to . Then a solid or dashed edge between vertices and is represented by or , respectively. The property of undirectedness is apparent by the property . Then sum over all possible inequivalent labellings of the graph.

For example, the three-loop correlator is explicitly given as

The above example for has a single dashed line coming from the in the numerator which is only partially cancelled by the denominator.

Summing over inequivalent labellings is the same as summing over all permutations and dividing by the size of the automorphism group (symmetry factor) of the graph (20 in the above example). This representation keeps graph labels implicit, so that a given unlabelled graph is defined by a labelled expression summed over all symmetric permutations.

2.2 The Correlator/Amplitude Duality

The fully supersymmetric correlator/amplitude duality is conjectured to relate integrands for the -point super-correlator to the square of -point super-amplitude integrands in an -gon light-like limit. However, the maximally nilpotent piece of the -point super-correlator is equivalent to the 4-point correlator. We thus consider the four-point correlator, under various -gon light-like limits for . Remarkably, this correlator contains information about all higher point amplitudes in the -gon limits [20, 22, 16].

It is natural to study amplitude integrands as functions of external and loop momenta. It was however observed in [27] that it is often useful to reparametrise to “dual momenta” via where are understood mod . These dual momenta of the amplitude are identified with the Minkowski co-ordinates of the correlator via the duality. The conformal invariance of correlation functions then implies a hidden dual conformal invariance known to be a property of planar amplitudes in SYM [28, 29, 30].

The simplest case of the duality then states that the -point light-like limit (where four consecutive points become light-like separated, ) outputs four-particle amplitudes squared. At the integrand level at fixed loop level, we get

(9)

for . Note that both sides are interpreted as integrands: on the right-hand side, depends on integration variables and depends on the remaining integration variables. The result is then symmetrised over all integration variables. Furthermore, all amplitudes (throughout the paper) are understood to be divided by the corresponding tree-level MHV superamplitude.

At higher points, it is convenient to consider superamplitudes written in the chiral superspace formalism of dual Minkowski superspace [29]:

(10)

for index , where are Nair Grassmann-odd super-momentum variables and spinor helicity variables. In complete analogy to region momenta, we define “dual/region super momenta”, so that super-momentum conservation is trivialised.

The higher-point generalisation of (9) then involves the -point light-like limit ( ) giving

(11)

Here is the superamplitude divided by the tree-level MHV superamplitude so that , and generalises as follows

(12)

Similarly to the four-point case, on the right-hand side of (11), is understood to depend on external variables together with loop variables and to depend on the same external variables, but the other loop variables—these are then symmetrised over. Note that the numerator on the right-hand side is a maximally nilpotent superconformal invariant. Since there is a unique maximally nilpotent invariant, this is proportional to the maximally nilpotent invariant amplitude and therefore the ratio in (11) makes sense and removes all dependence. We will see explicit examples of the use of this equation shortly and find amplituhedron variables to be the most useful way of dealing with the Grassmann-odd structure.

2.3 (Super) Momentum Twistors and Amplituhedron Coordinates

It is convenient to recast in terms of “momentum twistors” [23, 31]. Complex Minkowski space is equivalent to the Grassmannian of 2-planes in , where , and

(13)

for any matrix . Here the rows and are basis elements of the 2-plane and the equivalence relation corresponds to a change of basis. One choice of basis fixes the first 2x2 block to the identity and the next 2x2 block corresponds to standard co-ordinates for Minkowski space (in spinor notation)

(14)

Two Minkowski co-ordinates that are light-like separated correspond to two planes which intersect. In the case of the light-like limit of the correlator where we have consecutively light-like separated co-ordinates, it makes sense to choose the basis for the corresponding 2-planes to be the lines of intersection. Thus we have

(15)

where is the particle number. In this case, the are known as momentum twistors.

Figure 1: The transformation between dual-momentum space and momentum-twistor space.

One often considers projective twistor space and so points in dual momentum space are associated to projective lines in which intersect if the corresponding space-time points are light-like separated. Thus for the -gon light-like limit we get the picture illustrated in Figure 1. Loop variables in space also correspond to lines in momentum-twistor space which will not intersect with other lines. These can be specified by two twistors each in the same way.

Dual conformal symmetry acts linearly on the index of the momentum twistors and so a natural dual conformal covariant is provided by the (momentum) twistor four-bracket defined as the determinant of the square matrix formed by four twistors

(16)

Another way of thinking of Minkowski space is to take an invariant combination of the co-ordinates

(17)

Since is anti-symmetric in its indices, it is equivalent to a 6-vector and these are precisely the 6d embedding co-ordinates for Minkowski space. Notice that the above relations imply

(18)

In a similar way, chiral superspace can be thought of as the Grassmannian of 2-planes in

(19)

and the entire discussion above gets similarly uplifted into . We obtain super-momentum twistors, , living in ,

(20)

Beyond the MHV sector, dual superconformal symmetry implies that the superamplitudes can be written in terms of dual superconformal invariants [29]. For example, at the NMHV level, these are known as invariants and defined by a (dual) conformal ratio of four brackets and a Grassmann-odd delta function

(21)

It is also convenient to further change variables to “bosonised dual momentum superspace co-ordinates” or “amplituhedron co-ordinates” following [31, 25]. For an NMHV superamplitude, these variables live in a -dimensional purely bosonic space and are defined as

(22)

where we have introduced the global Grassmann-odd variables , which converts the co-ordinate into additional augmented momentum-twistor co-ordinates.

For example, for , we convert supermomenta to 5-dimensional co-ordinates, then the superconformal invariant (21) becomes the ratio of five-brackets and four-brackets:

(23)

Here the 5-brackets are the obvious 5d generalisations of the 4-brackets:

(24)

This rewriting will be useful later as it will trivialise the multiplication of invariants which we will need when considering products of amplitudes. The expression (23) becomes equal to (21) after integrating out the additional co-ordinate.

Note that the anti-MHV -point tree-level superamplitude has the following simple form in amplituhedron co-ordinates

(25)

2.4 Yangian Invariants from the Grassmannian

We will need to expand higher amplitudes in terms of higher analogues of the invariants (21), (23). For any , these superconformal (indeed Yangian) invariants can be understood as residues of a Grassmannian integral in planar SYM [24, 32, 33, 34, 35]. The main goal here is to introduce the tools needed to take the residues of the Grassmannian, directly in amplituhedron space and thus derive covariant forms for higher analogues of the invariants (23). Let us therefore introduce the Grassmannian representation for -particle NMHV Yangian invariants [35]:

(26)

is the matrix defining a Grassmannian of -planes in dimensions, and are super twistor co-ordinates. The -redundancy reflects a change of basis for planes. The denominator is simply given as -minors constructed from columns of

(27)

We need a contour of integration. Note that the integral is dimensional (after division by vol), and there are bosonic delta functions, leaving non-trivial integrals. The non-trivial contributions to these integrals arise from -dimensional poles of the integrand. A spanning set of all possible integrals of this form is thus provided by the residues of these poles, which define a codimension integration region. This then corresponds to a dimensional “cell” of . These are in turn classified by permutations (see [35], in particular section 12). From this formalism, one can obtain (positive)5 canonical co-ordinates for this cell inside such that the measure in (26), reduces to the simple dlog form

(28)

Now we wish to write these Yangian invariants in amplituhedron co-ordinates (which in particular makes multiplying invariants together much simpler). In amplituhedron co-ordinates, the Grassmannian integral (26), translates simply to

(29)

Here we have defined

(30)

where is defined in (22) and we have split the index into an ordinary twistor index and additional indices . Note that , and is the natural Grassmannian invariant -function whose precise definition can be found in [25].

The natural brackets in amplituhedron space, , are -brackets, but using we can form -brackets with four ’s and , for example

(31)

We could equally replace in (31) with s to form -independent -brackets.

There is an efficient way to arrive at a fully covariant form for a Yangian invariant corresponding to a particular residue via the canonical co-ordinates for this residue. To do this, we think of the reduced measure as a differential form on (simply a change of co-ordinates). Therefore

(32)

where is a function of weight in , rendering -weightless. Here is the natural Grassmannian invariant measure, using (31) but with the anti-symmetric differential form in the last 4 slots of the -bracket.

If we can write in this way, the Yangian invariant (29) is simply

(33)

noting that the brackets involving then reduce to 4-brackets .

In fact, we will be able to jump directly from the canonical co-ordinates and corresponding dlog form (28) to the Yangian invariant by a covariantisation procedure. We illustrate this with the example of seven-point Yangian invariants in section 4.

Note that the amplituhedron (bosonised) form for super-invariants have a number of advantages over the standard form. In particular, non-trivial identities which are very hard to see in the superspace formalism arise naturally as Schouten-like identities of the bosonised quantities. One potential question is how to extract components from this form. There is a straightforward way to think of this without first converting back to the standard form for the super-invariant in terms of ’s. This is particularly straightforward if we seek a component of the form . Such components are extractable in a canonical way by placing the points adjacent to one another in the six-bracket representation and simply removing them, thus projecting to four brackets, e.g.

3 Six-Point Integrands

Let us now consider the hexagonal light-like limit of the four-point correlator, taking six points of the correlator to be consecutively light-like separated: . The duality, (11) becomes

(34)

We will restrict this statement to various orders of perturbation, using the known correlator to predict amplitude integrands on the right-hand side. This leads to a simple linear algebra problem for matching coefficients from a sensible ansatz for the amplitude to the known correlator.

3.1 Tree Level

At tree level, , the duality (34) becomes

(35)

recalling that all amplitudes are understood to be divided by the tree-level MHV amplitude and thus . Evaluating the left-hand side of (35) amounts to symmetrising over , multiplying by and applying the 6-gon limit. Using (8), one straightforwardly obtains

(36)

Equating (35) and (36) then gives a prediction for in terms of finite cross ratios. We now wish to derive the NMHV tree-level amplitude itself, from this combination. We start with an ansatz for in terms of invariants (the six-point Yangian invariants). At six points, an invariant in the square-bracket notation is uniquely specified by the index it is missing

(37)

we will use this notation for the rest of this section. These six invariants are not independent since

(38)

so we use only five of these in our basis. Thus we have the following ansatz

(39)

with arbitrary coefficients . Since , the square is:

(40)

To proceed, we need a rule for multiplying two NMHV invariants to produce the numerator of the unique six-point NMHV invariant. In the numerator of the above, we have combinations such as

(41)

For six external points, there is a unique non-trivial six bracket. As the above is NMHV, the right-hand side must contain . Dual conformal invariance then uniquely fixes the remaining 4-brackets. This rule gives all products in terms of . Equation (35) requires (40) to be divided by the NMHV tree-level amplitude, . This is the anti-MHV amplitude (25) which at six points is6

(42)

As an example, consider the product

where in the second line, the amplituhedron rule (41) was used.

Proceeding in a similar way for all other products in (40), we obtain simple rules for all products of invariants divided by the anti-MHV tree-level amplitude (42) in terms of ordinary bosonic twistor brackets:

(43)

together with cyclic permutations of these.

Plugging these products into the ansatz for the square of the NMHV amplitude (40) and then into the duality equation (35), we equate the resulting expression7 to the known correlator (36) (with the replacement , see (18)).

The resulting system of equations has the following solution:

(44)

so that

(45)

Thus we have derived the NMHV six-point tree-level amplitude from the 4-point correlator up to an overall sign. Both signs yield the desired result for the correlator

The known result is indeed given by (45) with the positive sign choice [29]. This sign can clearly never be predicted purely by the correlator since the procedure predicts the square of the amplitude. If on the other hand we choose the wrong sign at tree level, this error will persist at higher loops and we will obtain the entire NMHV amplitude to all loops but with the wrong sign.

3.2 One Loop

At one loop, the duality (34) reads:

(46)

The first two terms form the MHV amplitude plus its parity conjugate whilst the last term is a product of NMHV tree- and one-loop amplitudes.8

As mentioned in the introduction, in order to go beyond tree level we require a basis of integrands. At one loop we have the following basis of 23 independent planar boxes and parity-odd pentagons:

one mass (6)
two-mass hard (6)
two-mass easy (3)
two-mass easy (3)
parity-odd pentagon (5) (47)

where the list is understood to include all those related by cycling the six external variables (the numbers of independent integrands in each class is given in parentheses after each). Note that there are only 5 independent parity-odd pentagons rather than 6 that one would expect from cyclicity. This is because there is an identity of the form

(48)

which we use to solve for in terms of the others. This identity is easily understood in the 6d embedding formalism where it can be written as

(49)

Here the square bracket indicates anti-symmetrisation over 7 variables which yields zero in 6 dimensions.

Our one-loop ansätze (see (4)) for the amplitudes thus reads

(50)

The problem now involves solving a system of equations for the coefficients obtained by plugging these ansätze together with the previously found tree-level result (45) into (46). We will require the products of invariants (43). Moreover, we can use parity and cyclicity to immediately reduce the number of free coefficients.

Equation (46) can be evaluated at generic kinematic configurations. The Mathematica package in [11] generates convenient configurations of small magnitude in random rational numbers. This process is repeated many times yielding a quadratic system over the rational numbers.

Solving the system of equations with 161 coefficients arising from (46) we obtain a solution with free coefficients. Remarkably, the NMHV sector is entirely (and correctly) solved consistent with the comment in footnote 3. The anti-MHV sector is then fixed in terms of the MHV sector which is itself completely unfixed (hence 23 free coefficients—one for each integrand) and consistent with the ambiguity (3). Imposing parity invariance, which takes then reduces the number of free coefficients down to 5—the number of parity-odd integrands. Further imposing cyclicity reduces this down to just 1 free coefficient.

The resulting solution can be written as

(51)

In the (anti-)MHV sector, we recognise the well known 1-loop result of a sum over one-mass and two-mass easy boxes together with an as yet undetermined parity-odd sector. The NMHV amplitude on the other hand is completely determined in terms of one-mass, two-mass hard boxes and parity-odd pentagons.

This prediction (51) agrees precisely with the known answer for . We will return to this as yet undetermined parameter in the next subsection.

3.3 Two Loops

We now proceed to two loops, using as input the one-loop solution obtained above (51). We first need a basis of two-loop integrals. A natural basis purely in position space is provided by dual conformal parity-even planar double boxes, pentaboxes, and pentapentagons, with all possible numerators, together with parity-odd pentaboxes and pentapentagons involving the 6d -tensor.

However, a convenient alternative dual conformal basis has been provided (together with an associated Mathematica package) in [11, 12] called the prescriptive basis. Although originally given in twistor space, all elements of this two-loop prescriptive basis can be rewritten in dual momentum space in terms of the planar basis described in the previous paragraph. We attach this -space translation as a file to the work’s arXiv submission.

The prescriptive basis at two loops consists of elements. These integrands we simply label as with .

We now insert the ansätze for the two-loop six-point amplitudes

(52)

comprising of free coefficients, into the duality formula (34) which at this loop level reads

(53)

Like the one-loop case, the whole NMHV sector at two loops is completely fixed by this equation. There are free undetermined coefficients in total, the anti-MHV sector being completely fixed in terms of the MHV sector, but the MHV sector itself being completely unfixed. This is precisely as expected in (3) and the accompanying footnote. Imposing parity then reduces the number of free coefficients to 36—the number of parity-odd two-loop planar dual conformal integrands. Further imposing cyclicity reduces this down to 6—the number of cyclic classes of parity-odd integrands. We expect these to be determined at the next loop order and cannot see any obstructions going to higher order.

The equations also (almost) determine the value of in (51): the ambiguity at one loop. The equations are clearly quadratic in one-loop parameters and in fact, this gives rise to two possible solutions. This is evident as the correlator determines only the parity-symmetric product

(54)

for and where , are the MHV, anti-MHV amplitudes normalised by their respective tree-level amplitudes (so ,