The All-Loop Integrand For Scattering
[1pt] Amplitudes in Planar SYM
N. Arkani-Hamed, J. Bourjaily, F. Cachazo, S. Caron-Huot, J. Trnka
School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
Department of Physics, Princeton University, Princeton, NJ 08544, USA
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J W29, CA
We give an explicit recursive formula for the all -loop integrand for scattering amplitudes in SYM in the planar limit, manifesting the full Yangian symmetry of the theory. This generalizes the BCFW recursion relation for tree amplitudes to all loop orders, and extends the Grassmannian duality for leading singularities to the full amplitude. It also provides a new physical picture for the meaning of loops, associated with canonical operations for removing particles in a Yangian-invariant way. Loop amplitudes arise from the “entangled” removal of pairs of particles, and are naturally presented as an integral over lines in momentum-twistor space. As expected from manifest Yangian invariance, the integrand is given as a sum over non-local terms, rather than the familiar decomposition in terms of local scalar integrals with rational coefficients. Knowing the integrands explicitly, it is straightforward to express them in local forms if desired; this turns out to be done most naturally using a novel basis of chiral, tensor integrals written in momentum-twistor space, each of which has unit leading singularities. As simple illustrative examples, we present a number of new multi-loop results written in local form, including the 6- and 7-point 2-loop NMHV amplitudes. Very concise expressions are presented for all 2-loop MHV amplitudes, as well as the 5-point 3-loop MHV amplitude. The structure of the loop integrand strongly suggests that the integrals yielding the physical amplitudes are “simple”, and determined by IR-anomalies. We briefly comment on extending these ideas to more general planar theories.
- 1 The Loop Integrand for SYM Amplitudes
- 2 Canonical Operations on Yangian Invariants
- 3 Loops From Hidden Entanglement
- 4 Recursion Relations For Arbitrary Loop Amplitudes
- 5 The Loop Integrand in Local Form
- 6 Multi-Loop Examples
- 7 Outlook
- A The BCFW-Form of the 1-Loop 6-Particle NMHV Amplitude
- B The Local 2-Loop 7-Particle NMHV Amplitude
1 The Loop Integrand for SYM Amplitudes
Scattering amplitudes in gauge theories have extraordinary properties that are completely invisible in the textbook formulation of local quantum field theory. The earliest hint of this hidden structure was the remarkable simplicity of the Parke-Taylor formula for tree-level MHV amplitudes [1, 2]. Witten’s 2003 proposal of twistor string theory  gave a strong impetus to rapid developments in the field, inspiring the development of powerful new tools for computing tree amplitudes, including CSW diagrams  and BCFW recursion relations [5, 6, 7, 8]. At one-loop, very efficient on-shell methods now exist [9, 10] and at higher-loop level generalizations of the unitarity-based method [11, 12, 13, 14] have made a five-loop computation possible , which should soon determine the five-loop cusp anomalous dimension .
The BCFW recursion relations in particular presented extremely compact expressions for tree amplitudes using building blocks with both local and non-local poles. In a parallel development, an amazing hidden symmetry of planar SYM—dual conformal invariance—was noticed first in multi-loop perturbative calculations  and then at strong coupling , leading to a remarkable connection between null-polygonal Wilson loops and scattering amplitudes [18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. It was quickly realized that the BCFW form of the tree amplitudes manifested both full superconformal and dual superconformal invariance, which together close into an infinite-dimensional Yangian symmetry algebra . Understanding the role of this remarkable integrable structure in the full quantum theory, however, was clouded by the IR-divergences that appear to almost completely destroy the symmetry at loop-level, leaving only the anomalous action of the (Bosonic) dual conformal invariance [29, 30, 31, 32].
1.1 Grassmannian Duality for Leading Singularities
In , a strategy for making progress on these questions was suggested. The idea was to find objects closely associated with scattering amplitudes which are completely free of IR-divergences; the action of the symmetries would be expected to be manifest on such objects, and they would provide “data” that might be the output of a putative dual theory of the S-Matrix.
The leading singularities of scattering amplitudes are precisely objects of this sort. Thinking of loop amplitudes as multi-dimensional complex integrals, leading singularities arise from performing the integration not on the usual non-compact ‘contours’ over all real loop-momenta, but on compact contours ‘encircling’ isolated (and generally complex) poles in momentum space. As such, they are free of IR-divergences and well-defined at any loop order, yielding algebraic functions of the external momenta. Leading singularities were known to have strange inter-relationships and satisfy mysterious identities not evident in their field-theoretic definition. Morally they are also expected to be Yangian-invariant, although even this is not completely manifest111Indeed we will give a proof of this basic fact in the next section; a different argument for the same result is given in .. Thus leading singularities seem to be prime candidates for objects to be understood and computed by a dual theory.
Such a duality was proposed in , connecting leading singularities of color-stripped, -particle NMHV scattering amplitudes in SYM to a simple contour integral over the Grassmannian :
Here labels the external particles, and are variables in . The original formulation of this object worked with twistor variables , and was given as . This was quickly realized  to be completely equivalent to a second form in momentum twistor space , with . Here the variables are the “momentum-twistors” introduced by Hodges , which are the most natural variables with which to discuss dual superconformal invariance. Furthermore, these momentum twistors are simple algebraic functions of the external momenta, upon which scattering amplitudes conventionally depend222 To quickly establish notation and conventions, the momentum of particle is given by , and the point in the dual co-ordinate space is associated with the line in the corresponding momentum-twistor space. This designation ensures that the lines and intersect, so that correspondingly, is null. (Bosonic) dual-conformal invariants are made with 4-brackets . An important special case is ; 2-brackets are computed using the upper-two components of and cancel out in dual-conformal expressions. For more detail see [37, 35, 36]..
Since the Grassmannian integral is invariant under both ordinary and dual superconformal transformations, it enjoys the full Yangian symmetry of the theory, as has been proven more directly in . In fact, it has been argued that these contour integrals in generates all Yangian invariants, 333The residues of are Yangian-invariant for generic momenta away from collinear limits. See [39, 40] for important discussions of the fate of Yangian invariance in the presence of collinear singularities.[41, 42].
Leading singularities are associated with residues of the Grassmannian integral. Residue theorems  imply many non-trivial and otherwise mysterious linear relations between leading singularities. These relations are associated with important physical properties such as locality and unitarity .
Further investigations  identified a new principle, the Grassmannian “particle interpretation”, which determines the correct contour of integration yielding the BCFW form of tree amplitudes . Quite remarkably, a deformation of the integrand connects this formulation to twistor string theory [46, 44, 47]. Furthermore, another contour deformation produces the CSW expansion of tree amplitudes , making the emergence of local space-time a derived consequence from the more primitive Grassmannian starting point.
The Grassmannian integral and Yangian-invariance go hand-in-hand and are essentially synonymous; indeed, the Grassmannian integral is the most concrete way of thinking about Yangian invariants, since not only the symmetries but also the non-trivial relationship between different invariants are made manifest; even connections to non-manifestly Yangian-invariant but important physical objects (such as CSW terms) are made transparent.
Given these developments, we are encouraged to ask again: is there an analogous structure underlying not just the leading singularities but the full loop amplitudes? Does Yangian-invariance play a role? And if so, how can we see this through the thicket of IR-divergences that appear to remove almost all traces of these remarkable symmetries in the final amplitudes?
1.2 The Planar Integrand
Clearly, we need to focus again on finding well-defined objects associated with loop amplitudes. Fortunately, in planar theories, there is an extremely natural candidate: the loop integrand itself!
Now, in a general theory, the loop integrand is not obviously a well-defined object. Consider the case of 1-loop diagrams. Most trivially, in summing over Feynman diagrams, there is no canonical way of combining different 1-loop diagrams under the same integral sign, since there is no natural origin for the loop-momentum space. The situation is different in planar theories, however, and this ambiguity is absent. This is easy to see in the dual -space co-ordinates . The ambiguity in shifting the origin of loop momenta is nothing other than translations in -space; but fixing the of the external particles allows us to canonically combine all the diagrams. Alternatively, in a planar theory it is possible to unambiguously define the loop momentum common to all diagrams to be the one which flows from particle “1” to particle “2”.
At two-loops and above, we have a number of loop integration variables in the dual space , and the well-defined loop integrand is completely symmetrized in these variables.
So the loop integrand is well-defined in the planar limit, and any dual theory should be able to compute it. All the symmetries of the theory should be manifest at the level of the integrand, only broken by IR-divergences in actually carrying out the integration—the symmetries of the theory are broken only by the choice of integration contour.
1.3 Recursion Relations for All Loop Amplitudes
Given that the integrand is a well-defined, rational function of the loop variables and the external momenta, we should be able to determine it using BCFW recursion relations in the familiar way 444We note that  have conjectured that the loop amplitudes can be determined by CSW rules, manifesting the superconformal invariance of the theory.. At loop-level the poles have residues with different physical meaning. The first kind is the analog of tree-level poles and correspond to factorization channels. The second kind has no tree-level analog; these are single cuts whose residues are forward limits of lower-loop amplitudes. Forward limits are naïvely ill-defined operations but quite nicely they exist in any supersymmetric gauge theory, as was shown to one-loop level in . There it was also argued that forward limits are well-defined to higher orders in perturbation theory in SYM. In principle, this is all we need for computing the loop integrand in SYM to all orders in perturbation theory. However, our goal requires more than that. We would like to show that the integrand of the theory can be written in a form which makes all symmetries—the full Yangian—manifest. The Yangian-invariance of BCFW terms at tree-level becomes obvious once they are identified with residues of the Grassmannian integral, we would like to achieve the same at loop-level.
This is exactly what we will do in this paper. We will give an explicit recursive construction of the all-loop integrand, in exact analogy to the BCFW recursion relations for tree amplitudes, making the full Yangian symmetry of the theory manifest.
The formulation also provides a new physical understanding of the meaning of loops, associated with simple operations for “removing” particles in a Yangian-invariant way. Loop amplitudes are associated with removing pairs of particles in an “entangled” way. We describe all these operations in momentum-twistor space, since this directly corresponds to familiar momentum-space loop integrals; presumably an ordinary twistor space description should also be possible.
As is familiar from the BCFW recursion relations at tree-level, the integrand is expressed as a sum over non-local terms, in a form very different than the familiar “rational function scalar integral” presentation that is common in the literature. Nonetheless, the Yangian-invariance guarantees that every term in the loop amplitude has Grassmannian residues as its leading singularities.
The integrands can of course be expressed in a manifestly-local form if desired, and are most naturally written in momentum-twistor space [51, 52]. As we will see, the most natural basis of local integrands in which to express the answer is not composed of the familiar scalar loop-integrals, but is instead made up of chiral tensor integrals with unit leading-singularities, which makes the physics and underlying structure much more transparent.
Of course the integrand is a well-defined rational function which is computed in four-dimensions without any regulators. The regularization needed to carry out the integrations is a very physical one, given by moving out on the Coulomb branch  of the theory. This can be beautifully implemented, both conceptually and in practice, with the momentum-twistor space representation of the integrand [51, 52].
Quite apart from the conceptual advantages of this way of thinking about loops, our new formulation is also completely systematic and practical, taking the “art” out of the computation of multi-loop amplitudes in SYM. As simple applications of the general recursive formula, we present a number of new multi-loop results, including the two-loop NMHV 6- and 7-particle integrands. We also include very concise, local expressions for all 2-loop MHV integrands and for the 5-particle MHV integrand at 3-loops. All multiplicity results for the so-called “parity even” part of two-loop amplitudes in the MHV sector were obtained by Vergu in , extending previous work done for 5-particles  and 6-particles [25, 55] in dimensional regularization. The “parity even” part of the 6-particle amplitude in dimensional regularization has been computed in work in progress by Kosower, Roiban, and Vergu . Complete integrands have been computed at two-loop order for 5-particles in  using -dimensional unitarity and for 5- and 6-particles in [57, 55] using the leading singularity technique developed in [58, 57]. Also using the leading singularity technique, the 5-point 3-loop integrand was presented in . Combining -dimensional unitarity with a generalization of quadruple cuts to higher loop order , a method called maximal cuts was introduced in  and used for the computation of the 4-point 5-loop integrand. The 4-point amplitude integrand at loop-level were computed in , , and , respectively. The method to be used in this paper is, however, very different both in philosophy and in practice from the leading singularity or generalized unitarity approaches.
In this paper, we give a brief and quite telegraphic outline of our arguments and results; we will present a much more detailed account of our methods and further elaborate on many of the themes presented here in upcoming work . In section 2, we describe a number of canonical operations on Yangian invariants—adding and removing particles, fusing invariants—that generate a variety of important physical objects in our story. In section 3 we describe the origin of Yangian-invariant loop integrals as arising from the “hidden entanglement” of pairs of removed particles. In section 4 we describe the main result of the paper: a generalization of the BCFW recursion relation to all loop amplitudes in the theory, and discuss some of its salient features through simple 1-loop examples. In section 5 we set the stage for presenting loop amplitudes in a manifestly local form by describing the most natural way of doing this in momentum-twistor space. In section 6 we present a number of new multi-loop integrands computed using the recursion relation and translated into local form for the convenience of comparing with known results where they are available. We conclude in section 7 with a discussion of a number of directions for future work. We discuss indications that not only the integrands but also the loop integrals should be “simple”. The idea of determining the loop integrand for planar amplitudes is a general one that can generalize well beyond maximally supersymmetric theories with Yangian symmetry, and we also very briefly discuss these prospects.
2 Canonical Operations on Yangian Invariants
As a first step towards the construction of the all-loop integrand for SYM in manifestly Yangian form, we study simple operations that can map Yangian invariants to other Yangian invariants. In this discussion it will not matter whether the ’s represent variables in twistor-space or momentum-twistor space; we will simply be describing mathematical operations that mapping between invariants. Combining these operations in various ways yields many objects of physical significance . The same physical object will arise from different combinations of these operations in twistor-space vs. momentum-twistor space; we will content ourselves here by presenting mostly the momentum-twistor space representations.
As mentioned in the introduction, understanding these operations is not strictly necessary if we simply aim to find a formula for the integrand. The reason is that the BCFW recursion relations we introduce in section 4 can be developed independently for theories with less supersymmetry, which do not enjoy a Yangian symmetry. Our insistence in keeping the Yangian manifest however will pay off in two ways. The first is conceptual: the Yangian-invariant formulation will introduce a new physical picture for meaning of loops. The second is computational: the Yangian-invariant formulation gives a powerful way to compute the novel “forward-limit” terms in the BCFW recursions in momentum-twistor space, using the Grassmannian language.
We will begin by discussing how to add and remove particles in a Yangian-invariant way. One motivation is an unusual feature of the Grassmannian integral–the space of integration depends on the number of particles. It is natural to try and connect different ’s by choosing a contour of integration that allows a “particle interpretation”, by which we mean simply that the variety defining the contour for the scattering amplitudes of particles differs from the one for particles only by specifying the extra constraints associated with the new particle . Following this “add one particle at a time”-guideline completely specifies the contour for all tree amplitudes [44, 47], along the way exposing a remarkable connection with twistor string theory [3, 64, 46, 65, 66]. As we will see in this paper, loops are associated with interesting “entangled” ways of removing particles from higher-point amplitudes. We will then move on to discuss how to “fuse” two invariants together. Using these operations we demonstrate the Yangian invariance of all leading singularities, and discuss the important special case of the “BCFW bridge” in some detail.
2.1 Adding Particles
Let us start with a general Yangian-invariant object
We will first describe operations that will add a particle to lower-point invariants to get higher-point invariants known as applying “inverse soft factors” , which are so named because taking the usual soft limit of the resulting object returns the original object. This can be done preserving or increasing . We can discuss these in both twistor- and momentum-twistor space; for the purposes of this paper we will describe these inverse-soft factor operations in momentum-twistor space.
The idea is that there are residues in which are trivially related to residues in or . The -preserving operation is particularly simple, being simply the identification
that is, where we have simply added particle as a label (but have not altered the functional form of in any way); thanks to the momentum-twistor variables, momentum conservation is automatically preserved. The -increasing inverse soft factor is slightly more interesting. There is always a residue of which has a -matrix of the form
Here, the non-zero elements in the top row, correspond to particles , and we have generic non-zero entries in the lower matrix. The corresponding residue is easily seen to be associated with
is the basic ‘NMHV’-like -invariant555When two sets of the twistors are consecutive, these “-invariants” are sometimes written These invariants were first introduced in  in dual super-coordinate space. and the are deformed momentum twistor variables. The Bosonic components of the deformed twistors have a very nice interpretation: is simply the intersection of the line with the plane , which we indicate by writing ; and is the intersection of the line with the plane , written . Fully supersymmetrically, we have
2.2 Removing Particles
We can also remove particles to get lower-point Yangian invariants from higher-point ones. This turns out to be more interesting than the inverse-soft factor operation, though physically one might think it is even more straightforward. After all, we can remove a particle simply by taking its soft limit. However, while this is a well-defined operation on e.g. the full tree amplitude, it is not a well-defined operation on the individual residues (BCFW terms) in the tree amplitude. The reason is the presence of spurious poles: each term does not individually have the correct behavior in the soft limit.
Nonetheless, there are completely canonical and simple operations for removing particles in a Yangian-invariant way. One reduces , the other preserves . The -reducing operation removes particle by integrating over its twistor co-ordinate
This gives a Yangian-invariant for any closed contour of integration—meaning that under the Yangian generators for particles , this object transforms into a total derivative with respect to . This statement can be trivially verified by directly examining the action of the level-zero and level-one Yangian generators on the integral. It is also very easy to verify directly from the Grassmannian integral. Note that depending on the contour that is chosen, a given higher-point invariant can in general map to several lower-point invariants.
The -preserving operation “merges” particle with one of its neighbors, or . For example,
The Yangian-invariance of this operation is slightly less obvious to see by simply manipulating Yangian generators, but it can be verified easily from the Grassmannian formula.
We stress again that these operations are perfectly well-defined on any Yangian-invariant object, regardless of whether the standard soft-limits are well defined. Of course, they coincide with the soft limit when acting on e.g. the tree amplitude.
2.3 Fusing Invariants
Finally, we mention a completely trivial way of combining two Yangian invariants to produce a new invariant. Start with two invariants which are functions of a disjoint set of particles, which we can label and . Then, it is easy to see that the simple product
is also Yangian-invariant. Only the vanishing under the level-one generators requires a small comment. Note that the cross terms vanish because the corresponding level-zero generators commute and therefore the level-one generators cleanly splits into the smaller level-one generators.
2.4 Leading Singularities are Yangian Invariant
Combining these operations builds new Yangian invariants from old ones; all of which have nice physical interpretations. An immediate consequence is a simple proof that all leading singularities are Yangian invariant. For this subsection only, we work in ordinary twistor space. Then we take any four Yangian invariants for disjoint sets of particles and we make a new invariant by taking the product of all of them, . We then “merge” and , and , and , and with . We then integrate over . This precisely yields the twistor-space expression for a “1-loop” leading singularity topology [68, 69].
In the figure, a thick black line denotes the merging of the two particles at the ends of the line, and integrating over the remaining variable. The generalization to all leading singularities is obvious; for instance, starting with the “1-loop” leading singularity we have already built, we can use the same merge and integrate operations to build “2-loop” leading singularity topologies such as that shown below.
We conclude that all leading singularities are Yangian invariant. Given that all Yangian invariants are Grassmannian residues, this proves (in passing) the original conjecture in  that all leading singularities can be identified as residues of the Grassmannian integral.
2.5 The BCFW Bridge
A particularly important way of putting together two Yangian invariants to make a third is the “BCFW bridge” [70, 71, 7], associated with the familiar “two-mass hard” leading singularities drawn below in twistor space [70, 71, 72, 73]:
Here, the open and dark circles respectively denote MHV and three-particle amplitudes, respectively. We remark in passing that the inverse-soft factor operations mentioned above are special cases of the BCFW bridge where a given Yangian invariant is bridged with an three-point vertex (for the -preserving case) or an MHV three-point vertex (for the -increasing case).
We will find it useful to also see the bridge expressed as a composition of our basic operations in momentum-twistor space, as
This is a pretty object since it uses all of our basic operations in an interesting way. In the figure, the solid arrows pointing inward indicate that particle-“” is added as an -increasing inverse soft factor on , and is added as a -increasing inverse soft factor on . We are also using the merge operation to identify the repeated “1” and “” labels across the bridge. The internal line, which we label as , is integrated over. The contour of integration is chosen to encircle the -pole from the -piece of the inverse-soft factor on , and the - and -poles from the -piece of the inverse soft factor on . The deformation on induced by the inverse-soft factor adding particle- on is of the form
This is the momentum-twistor space version of the BCFW deformation, which corresponds to deforming in momentum-space. We remind ourselves of this deformation by placing the little arrow pointing from in the figure for the bridge. The momentum-twistor space geometry associated with this object is
which precisely corresponds to the expected BCFW deformation and the corresponding factorization channel.
We leave a detailed derivation of this picture to , but in fact the momentum-twistor structure of the BCFW bridge can be easily understood. Note that have -charge , while has -charge ; given that the decreases the -charge by 1, we must start with and and get objects with -charge and on the left and right. This can be canonically done by acting with -increasing inverse soft factors; the added particle on must be adjacent to in order to affect a deformation on . Finally, the data associated with the “extra” particles introduced by the inverse soft factor must be removed in the only way possible, by using the merge operation. Explicitly, the final result for is
Starting with the tree amplitude 666We remind the reader that we are working in momentum-twistor space, so that what we are calling here is obtained after stripping off the MHV tree-amplitude factor from the full amplitude in momentum space., the BCFW deformation can be used to recursively construct tree amplitudes in the familiar way: by writing,
it is clear that the desired amplitude is obtained by summing-over all the residues of the RHS except the pole at origin . Notice that there is a non-zero pole at infinity in this deformation: as , projectively, and so the tree amplitude gets a contribution from 777Note that here does not correspond to going to infinity in the familiar momentum-space version of BCFW. The pole at infinity in ordinary momentum space here corresponds to a pole involving the infinity twistor . Of course we do not expect such a pole to arise in a dual-conformal invariant theory, not only at tree-level, but at all-loop order, as will be relevant to our subsequent discussion. A direct proof of this fact, not assuming dual conformal invariance, should follow from the “enhanced spin-lorentz symmetry” arguments of .. The pole at corresponds to the term in the usual momentum-space BCFW formula using an three-point vertex bridged with , which simply acts as a -preserving inverse-soft factor The remaining physical poles are of the form . Under , we only access the poles where , and the corresponding residues are computed by the BCFW bridge indicated above, with being the lower-point tree amplitudes.
3 Loops From Hidden Entanglement
Let’s imagine starting with some scattering amplitude or Grassmannian residue, and begin removing particles. The operation that decreases in particular demands a choice for the contour of integration. If we remove particle by integrating over it as , it is natural to choose a -contour of integration for the Bosonic integral and compute a simple residue888Residues of rational functions in complex variables are computed by choosing polynomial factors ’s from the denominator and integrating along a particular -contour, i.e. the product of circles given as the solutions of with and near a common zero of the ’s. See  for more details..
We can then proceed to remove a subsequent particle either by merging, or performing further integrals and so on. In this way we will simply proceed from higher-point Grassmannian residues to lower-point ones. In particular, if these operations are performed on a higher-point tree amplitude, we arrive at lower-point tree amplitudes, and don’t encounter any new objects.
But we can imagine a more interesting way of removing not just one but a pair of particles. Consider removing particle and subsequently removing the adjacent particle . Instead of first integrating-out and then on separate ’s, let’s consider an “entangled” contour of integration, which we will discover to yield, instead of lower-point Grassmannian residue, a loop integral.
Consider as a simple example removing two particles from the 6-particle NMHV = tree amplitude, . Performing the integrals is trivial, and this gives
where we have chosen to label the Bosonic momentum twistors with lower-case ’s for later convenience. As we have claimed, on any closed contour, these integrals should give a Yangian-invariant answer. Indeed, computing the integral by residue on any contour leaves us with
and computing any of the simple residues of this remaining integral gives , which is of course the only Yangian invariant for MHV amplitudes.
We will now see that starting with exactly the same integrand but choosing a different contour of integration yields, instead of “1”, the 4-particle 1-loop amplitude. Geometrically, the points determine a line in momentum-twistor space, which is interpreted as a point in the dual -space, or equivalently, a loop-integral’s four-momentum. We will first integrate over the positions of on the line , and then integrate over all lines .
This contour can be described explicitly by parametrizing as
where will be the loop momentum. The measure is
The integrals will be treated as contour integrals on , while the -integral will be over real points in the (dual) Minkowksi space.
Using that our integral becomes
The factor is linear in the projective variable while the factor is linear in . This implies that there is a unique way to perform the and integrals by contour integration, which gives us
This is precisely the 1-loop MHV amplitude!
We have thus seen that, removing a pair of particles with this “entangled” contour of integration, where we first integrate over the position of two points along the line joining them and then integrate over all lines, naturally produces objects that look like loop integrals.
There is a nicer way of characterizing this “entangled” contour that is also more convenient for doing calculations, let us describe it in detail. Given , a general GL(2)-transformation on the 2-vector moves along the line . Thus, in integrating over , we’d like to “do the GL(2)-part of the integral first” to leave us with an integral that only depends on the line :
We can do this explicitly by writing
and our integral becomes—this time writing it out fully:
The integral is naturally performed on a contour ‘encircling’ , yielding . More generally, if “234” and “123” in the definitions of were to be replaced by arbitrary “” and “”, where is the line corresponding to the intersection of the planes and . We are then left with
where the integration region is such that the line corresponds to a real point in the (dual) Minkowski space-time. We recognize this object as the 1-loop MHV amplitude, exactly as above.
We can clearly perform this operation starting with any Yangian invariant object , which we will graphically denote as:
and write as
This object is formally Yangian-invariant, in the precise sense that the integrand will transform into a total derivative under the action of the Yangian generators for the external particles. Of course, such integrals may have IR-divergences along some contours of integration, which is how Yangian-invariance is broken in practice.
The usual way of writing the loop amplitudes as “leading singularity scalar integral” ensures that the leading singularities of the individual terms are Yangian-invariant, but the factorized form seems very un-natural, and there is no obvious action of the symmetry generators on the integrand. By contrast, the loop integrals we have defined, as we will see, will not take the artificial “residue integral” form, but of course their leading singularities are automatically Grassmannian residues. The reason is that a leading singularity of the -integral can be computed as a simple residue of the underlying integral, which is free of IR divergences and guaranteed to be Yangian-invariant.
4 Recursion Relations For Arbitrary Loop Amplitudes
Having familiarized ourselves with the basic operations on Yangian invariants, we are ready to discuss the recursion relations for loops in the most transparent way. The loop integrand is a rational function of both the loop integration variables and the external kinematical variables. Just as the BCFW recursion relations allow us to compute a rational function from its poles under a simple deformation, the loop integrand can be determined in the same way. Consider the -loop integrand , and consider again the (supersymmetric) momentum-twistor deformation
and we sum over all the residues of the RHS away from the origin, all of which can be determined from lower-point/lower-loop amplitudes. This recursion relation can be derived in a large class of theories and is not directly tied to SYM or Yangian-invariance. However our experience with building Yangian-invariant objects will help us to understand (and compute) the terms in the recursion relations in a transparent way, and also easily recognize them as manifestly Yangian-invariant objects.
As in our discussion of the BCFW bridge at tree-level, the pole at infinity is simply the lower-point integrand with particle removed. All the rest of the poles in also have a simple interpretation: in general, all the poles arise either from or , where denotes the line in momentum twistor space associated with the loop-variable. The first type of pole simply corresponds to factorization channels, and the corresponding residue is computed by the BCFW bridges between lower-loop/lower-point amplitudes:
where , , . Note that we treat all the poles (including the pole at infinity) on an equal footing by declaring the term with to be given by the -preserving inverse soft-factor acting on lower-point amplitude.
This is the most obvious generalization of the BCFW recursion relation from trees to loops, but it clearly can’t be the whole story, since it would allow us to recursively reduce loop amplitudes to the 3-particle loop amplitude, which vanishes! Obviously, at loop-level, a “source” term is needed for the recursive formula.
4.1 Single-Cuts and the Forward-Limit
This source term is clearly provided by the second set of poles, arising from . For simplicity of discussion let’s first consider the 1-loop amplitude. This pole corresponds to cutting the loop momentum running between and , and is therefore given by a tree-amplitude with two additional particles sandwiched-between , with momenta , summing-over the multiplet of states running around the loop. These single-cuts associated with “forward-limits” of lower-loop integrands are precisely the objects that make an appearance in the context of the Feynman tree theorem . The geometry of the forward limit is shown below for both in the dual -space and momentum-twistor space:
Here, between particles and , we have particles with momenta , where is a null vector. In momentum-twistor space, the null condition means that the line intersects , while in the forward limit both and approach the intersection point .
In a generic gauge theory, the forward limits of tree amplitudes suffer from collinear divergences and are not obviously well-defined. However remarkably, as pointed out in , in supersymmetric theories the sum over the full multiplet makes these objects completely well-defined and equal to single-cuts!
Indeed, we can go further and express this single-cut “forward limit” term in a manifestly Yangian-invariant way. It turns out to to be a beautiful object, combining the entangled removal of two particles with the “merge” operation:
Here a particle is added adjacent to as a -increasing inverse soft factor, then are removed by entangled integration. The GL(2)-contour is chosen to encircle points where both points on the line are located at the intersection of the line with the plane . Note that there is no actual integral to be done here; the GL(2) integral is done on residues and is computed purely algebraically. Finally, the added particle is merged with .
As in our discussion of the BCFW bridge, this form can be easily understood by looking at the deformations induced by the inverse soft factors; the associated momentum-twistor geometry turns out to be
exactly as needed. The picture is the same for taking the single cut of any Yangian-invariant object.
Note that we were able to identify the BCFW terms in a straightforward way since the residues of the poles of the integrand have obvious “factorization” and “cut” interpretations. This is another significant advantage of working with the integrand, since as is well known, the full loop amplitudes (after integration) have more complicated factorization properties . This is due to the IR divergences which occur when the loop momenta becomes collinear to external particles, when the integration is performed.
4.2 BCFW For All Loop Amplitudes
Putting the pieces together, we can give the recursive definition for all loop integrands in planar SYM as
To be fully explicit, the recursion relation is
where , , and the shifted momentum (super-)twistors that enter are
Beyond 1-loop, it is understood that this expression is to be fully-symmetrized with equal weight in all the loop-integration variables ; it is easy to see that this correctly captures the recursive combinatorics. Recall again that GL(2)-integral is done on simple residues and is thus computed purely algebraically; the contour is chosen so that the points are sent to . As we will show in , recursively using the BCFW form for the lower-loop amplitudes appearing in the forward limit allows us to carry out the GL(2)-integral completely explicitly, but the form we have given here will suffice for this paper.
4.3 Simple Examples
In , we will describe the loop-level BCFW computations in detail; here we will just highlight some of the results for some simple cases, to illustrate some of the important properties of the recursion and the amplitudes that result. We start by giving the BCFW formula for all one-loop MHV amplitudes.
In this case the second line in the above formula vanishes, and the recursion relation trivially reduces to a single sum. To compute the NMHV tree amplitudes which enters through the third line, it is convenient to use the tree BCFW deformation which leads to
where the omitted terms are independent of and vanish upon Fermionic-integration. The GL(2) and Fermion integrals are readily evaluated, as explained above, reducing this to
This is the full one-loop integrand for MHV amplitudes.
As expected on general grounds from Yangian-invariance, and also as familiar from BCFW recursion at tree-level, the individual terms in this formula contain both local and non-local poles. We will graphically denote a factor in the denominator by drawing a line ; the numerators of tensor integrals (required by dual conformal invariance) will be denoted by wavy- and dashed-lines—the precise meaning of which will be explained shortly. In this notation, this result is
Notice that all the terms have 6 factors in the denominator, and hence by dual conformal invariance we must have two factors containing in the numerators. These are denoted by the wavy lines: the numerator is , where the power of 2 has been indicated by the line’s multiplicity.
Notice that when , the numerator cancels the two factors in the denominator: by a simple use of the Schouten identity it is easy to see that
In general, all of these terms contain both physical as well as spurious poles. Physical poles are denominator factors of the form and while spurious poles are all other denominator factors. We often refer to physical poles as local poles and to spurious poles as non-local. A small explanation for the “non-local” terminology is in order. Consider the 5-particle amplitude as an example, where there are three terms in the integrand. These three terms are
The spurious poles are and . The line defined by and corresponds to a complex point, but what makes non-local? The reason is that in field theory could only come from a loop integration, e.g. it is generated by a local one-loop integral of the form
(This is also the secret origin of the non-local poles in BCFW at tree-level.)
Back to the 5-particle example, and occur each in two of the three terms and they cancel in pairs. Indeed upon collecting denominators we find, after repeated uses of the Schouten identity, the result for the sum
This is furthermore cyclically-invariant, albeit in a nontrivial way involving Schouten identities.
Let us also briefly discuss the 6-particle NMHV amplitude at 1-loop. The full integrand has 16 terms which differs even more sharply from familiar local forms of writing the amplitude. As we will review in the next section, the usual box decomposition of 1-loop amplitudes does not match the full integrand (only the “parity-even” part of the integrand); even so, there is a natural generalization of the basis of integrals that can be used to match the full integrand in a manifestly dual conformal invariant form. Any such representation, however, will have the familiar form “leading singularity/Grassmannian residue loop integral”. However, this is not the form we encounter with loop-level BCFW. Instead, the supersymmetric -variables are entangled with the loop integration variables in an interesting way. For instance, one of the terms from the forward limit contribution to the 6-particle NMHV amplitude integrand is the following,
The full expression is given in appendix A. Note the presence of the explicit -dependence in the argument of the Fermionic -function. Seemingly miraculously, when the residues of this integral are computed on its leading singularities, the -dependence precisely reproduces the standard NMHV -invariants. Of course this miracle is guaranteed by our general arguments about the Yangian-invariance of these objects.
4.4 Unitarity as a Residue Theorem
The BCFW construction of tree-level amplitudes make Yangian-invariance manifest, but are not manifestly cyclic-invariant. The statement of cyclic-invariance is then a remarkable identity between rational functions. Of course one could say that the field theory derivation of the recursion relation gives a proof of these identities, but this is quite a circuitous argument. One of the initial striking features of the Grassmannian picture for tree amplitudes was that these identities were instead a direct consequence of the global residue theorem applied to the Grassmannian integral. This observation ultimately led to the “particle interpretation” picture for the tree contour, giving a completely autonomous and deeper understanding of tree amplitudes, removed from the crutch of their field theory origin.
In complete analogy with BCFW at tree-level, the BCFW construction of the loop integrand is not manifestly cyclically-invariant. Again cyclic-invariance is a remarkable identity between rational functions, and again this identity can be thought of as a consequence of the field theory derivation of the recursion relation. But of course we strongly suspect that there is an extension of the “particle interpretation” picture that gives a completely autonomous and deeper understanding of loop amplitudes, independent of any field theoretic derivation.
Just as at tree-level, a first step in this direction is to find a new understanding of the cyclic-invariance identities. To whit, we have understood how the cyclic-identity for the 1-loop MHV amplitude can be understood as a residue theorem; we very briefly outline the argument here, deferring a detailed explanation to . The idea is to identify the terms appearing in the MHV 1-loop formulas as the residues of a new Grassmannian integral. All the terms in the MHV 1-loop formula can actually be thought of as arising from , where is computed from the Grassmannian integral. Note that appear in the delta functions of the integral in the combination , so the GL(2)-action on also acts on . Performing the and GL(2)-integrals leaves us with a new Grassmannian integral:
By construction, this integral has a GL(2)-invariance acting on columns and , and hence all of is residues are only a function of the line . In particular all terms appearing in the MHV 1-loop formula, after GL(2) integration, are particular residues of this Grassmannian integral.
As we will discuss at greater length in , the equality of cyclically-related BCFW expressions of the 1-loop amplitude follows from a residue theorem applied to this integral. In fact, it can be shown that the only combination of these residues that is free of spurious poles is the physical 1-loop amplitude.
At tree level, the cyclic-identity applied to e.g. NMHV amplitudes ensures the absence of spurious poles. The same is true at 1-loop level. Since the BCFW formula manifestly guarantees that one of the single cuts is correctly reproduced, cyclicity guarantees that all the single cuts are correct. Having all correct single cuts, automatically ensures that all higher cuts—and in particular unitarity cuts—are correctly reproduced. Unitarity then finds a deeper origin in this residue theorem.
5 The Loop Integrand in Local Form
We have seen that the loop integrand produced by BCFW consists of a sum over non-local terms. In order to present the results in a more familiar form, and also as a powerful check on the formalism, it is interesting to instead re-write the integrand in a manifestly local way (which will of course spoil the Yangian-invariance of each term). We will do this for a number of multi-loop examples in the next section, but first we must describe a new basis of local loop integrals which differs in significant ways from the standard scalar integrals, but which will greatly simplify the results and make the physics much more transparent.
Loop amplitudes are normally written as scalar integrals999Here we abuse terminology and use the term “scalar”, which is appropriate at one-loop, to refer to possibly tensor integrals at higher-loop order where the tensor structure is the product of “local” factors, i.e., of the form and . with rational coefficients. Obviously this form can not match the full loop integrand, since scalar integrals are even under parity but the amplitude is chiral. Let’s consider one-loop integrals to begin the discussion. In the usual way of discussing the integral reduction procedure, manipulations at the level of the integrand reduces integrals down to pentagons . The final reduction to the familiar boxes uses the fact that the parity-odd parts of the integrand integrate to zero.
We are instead interested in the full integrand, however, and since the amplitudes aren’t parity symmetric, there is no natural division between “parity-odd” and “parity-even”. In fact, for the purpose of writing recursion relations, it is crucial to know both. Furthermore, the BCFW recursion relation guarantees that the loop integrand is dual conformally invariant and thus most usefully discussed in momentum-twistor space. We are then led to construct a novel basis of naturally chiral integrals written directly in momentum-twistor space, as we now briefly describe. These issues will be discussed at much greater length in .
Let’s look at a few quick examples of local integrals written in momentum-twistor space. We have encountered the simplest example already; the zero mass integral at 1-loop
Henceforth, we will drop the integration measure and only write the integrand. The most general -loop integrand is of the form
where are general antisymmetric matrices or ‘bitwistors’; with 6 independent components. Momentum-twistors make integral reduction trivial. Suppose there are 6 or more local propagator factors including in the denominator. We can always expand all the ’s in a basis of the 6 bitwistors . Inserting this expansion into the integrand, each term knocks-out a propagator from the denominator. Thus we can reduce any integral down to pentagons.
These will contain 5 factors in the denominator and a single factor in the numerator. In the literature, -space loop integrals are written with numerator factors like , which in momentum-twistor space correspond to . However, we will find more general numerators to be more natural. For instance, a typical pentagon integrand we consider takes the form
We can trivially translate this integral into -space; the numerator is proportional to , where is a complex point associated with the line in momentum-twistor space; specifically, the pentagon-integral (40) is given by
The complex point is null-separated from and ; the second point sharing this property is its parity conjugate which will be described shortly. These complex points play an important role in the story, and it is most convenient to discuss them on an equal footing with the rest of the points by working directly with momentum-twistor space integrands.
Notably, unlike standard scalar integrals, this pentagon integral is chiral. Like any pentagon integral, it has 5 quadruple cuts and twice as many leading singularities. But unlike a generic pentagon integral, with this special numerator, half of the leading singularities vanish, and the others are all equal up to sign—hence, we say that this integral has “unit leading singularities”. All of the local integrals we consider have this quite remarkable feature.
Local momentum-twistor space integrals can be drawn in exactly the same way as familiar planar integrals in -space; we introduce a new bit of notation to denote the numerator factors. The pentagon integral we just discussed is drawn as,
where the dashed line connecting denotes the numerator factor . We will also have recourse to use the parity conjugates of these lines. The point in momentum twistor space is naturally paired with its projective-dual plane , and the parity conjugate of a line is the line which is the intersection of the corresponding planes . The numerator factor will be denoted by a wavy-line connecting .
With this notation we can nicely write the integrand for -particle 1-loop MHV amplitudes as
In this expression we sum over all cyclic integrands, including duplicates, which is related to the presence of the pre-factor.
For definiteness, we have indicated the numerator factor beneath the corresponding picture. Recall the familiar form of the MHV amplitude as a sum over all 2-mass easy boxes; it is amusing that in our form the only boxes are 2-mass hard. The algorithm by which this form was deduced will be explained shortly.
We pause to point out that the full integrand for some MHV amplitudes have been computed in the literature, in the context of using the leading singularity method to determine the integrand . A peculiarity in these papers was that the set of integrals that were used to match all the leading singularities did not appear to be manifestly dual conformal invariant—which is particularly ironic, given that the leading singularities themselves are fully Yangian-invariant! This led some authors to the conclusion that the parity-odd parts of the amplitude are somehow irrelevant, since they not only integrate to zero on the real contour but are also not dual conformal invariant. Of course, nothing could be further from the truth: we have seen very clearly that the full integrand is determined recursively and exhibits the Yangian symmetry of the theory; the decomposition into parity even and odd parts is artificial. The problem is quite simple, the basis of scalar integrals has only parity even elements! Therefore, one is trying to model the full integrand with a very inappropriate basis.
From the momentum-twistor viewpoint, the source of the previous difficulties can be seen quite explicitly. We have seen that all 1-loop integrals can be reduced to pentagons, but these are tensor pentagons, i.e. with factors of in the numerator. Now, it is possible to further reduce a pentagon with numerator , with corresponding to a real line or not, to a scalar pentagon integral, by expanding in a basis of the 5 bitwistors appearing in the denominators, together with the infinity twistor . But this breaks manifest dual conformal invariance! Thus the integrands obtained in [57, 55, 59] are indeed dual conformal invariant, but the symmetry was obscured by insistence to use scalar integrals.
Let’s give an example of an interesting two-loop integrand using our notation:
which we draw as
At two-loops, there are generally 4 solutions to cutting any eight propagators, and so this integral has different (non-composite) leading singularities. However, the integral is maximally chiral: putting any choice of eight propagators on shell will have only one solution with a non-vanishing residue. Moreover, the non-vanishing residues are equal up to a sign. This non-trivial fact can be understood as following from the global residue theorem applied to the integral. All the tensor integrals we write in this paper are chiral in this sense, and the overall normalization of each has been chosen so that all its non-vanishing leading singularities are equal to .
These chiral momentum-twistor integrals have another remarkable feature: they are less IR-divergent than generic loop integrals; indeed, many of them are completely IR-finite. Infrared divergences arise when the loop momenta become collinear with the external momenta . In the dual co-ordinate space, this happens when a loop-integration variable lies on the line connecting and . In momentum-twistor space, this corresponds to configurations where the associated line passes through the point while lying in the plane . An integral is IR-finite if the numerator factors have a zero in the dangerous configurations. There are an infinite class of IR-finite integrals at any loop order; for instance, it is easy to see that the two-loop example above is IR-finite. Further discussion of these objects and their role in determining IR-finite parts of amplitudes like the remainder  and ratio  functions will be carried out in . Of course we expect that IR finite quantities, such as the ratio function, are manifestly finite already at the level of the integrand.
It is interesting that the naively “hardest” multi-loop integrands can be reduced to finite integrals plus simpler integrals. Consider for instance a general double pentagon integrand for six particles, of the form
We can expand in terms of the 6 bitwistors as well as the bitwistors corresponding to and its parity conjugate . Similarly we can expand in terms of as well as and . Doing this reduces the integral to finite double-pentagon integrals, plus simpler pentagon-box and double-box integrals.
Finally, let us describe the general algorithm which we used to find local forms of the loop integrands. The first step is to construct an algebraic basis of dual conformal-invariant integrals, over which the integrand is to be expanded. It turns out, quite remarkably, that for at least 1- and 2-loops an (over-complete) algebraic basis can be constructed which contains exclusively integrals with unit leading singularities, in the sense just defined. We have explicitly constructed such a bases at 1- and 2-loops and arbitrary . The second step is to match the integrand as generated by equation (29) with a linear combination of the basis integrals. Since the loop integrand is a well-defined function of external momenta and loop momenta, this can be done by simply evaluating it at sufficiently many random points. Numerical evaluation of the integrand is itself quite fast. Finally, this procedure is greatly facilitated by the fact that, when using our particular integral basis, the coefficients are guaranteed to be pure numbers (or multiple of leading singularities, for arbitrary NMHV), as opposed to arbitrary rational functions of the external momenta.
6 Multi-Loop Examples
The recursion relation for loops gives a completely systematic way of determining the integrand for amplitudes with any . All the required operations are completely algebraic and can be easily automated. In this section we use the recursion relation to present a number of multi-loop results.
As we have stressed repeatedly, the individual terms in the BCFW expansion of the loop integrand have spurious poles and are also not manifestly cyclically-invariant; thus as a very strong consistency check on our results, necessary for a local form to exist, we verify that the integrand is free of all spurious poles: the only poles in the integrand should be of the form . We also explicitly check cyclic-invariance. Recall that the absence of spurious poles and cyclicity guarantees that all single-cuts of the amplitude are reproduced, and thus all cuts are automatically correctly matched. While preparing this paper we have explicitly checked that our recursive determination of the integrand passes these checks up to 14 pt NMHV amplitudes at 1-loop, 22-pt MHV amplitudes at 2-loops, 8 pt NMHV amplitudes at 2-loops and 5-pt MHV amplitude at 3 loops.
We can expand the integral in a local basis of chiral momentum-space integrals with unit leading singularities using the algorithm briefly described in the previous section. While the BCFW form of the integrand is almost always more concise than the local form, the local form is more familiar, so we will present the results in this way. Indeed, the (modestly) non-trivial work here is only in determining the natural basis for local integrands. While this is a straightforward exercise using momentum-twistor machinery, the result is non-trivial, yielding a canonical basis of multi-loop integrals, which we have constructed explicitly for all up to 2-loops. In order to present a tree-loop result, we also found the 5pt basis at three-loops, deferring a complete discussion to . Given the basis of local integrals with unit leading singularities, generating the integrand and finding its expansion in the basis is not difficult. The natural basis is over-complete and so the results can be expressed in a number of equivalent forms. We will choose the forms that seem canonical and reveal patterns. As we will see, somewhat surprisingly, the local forms are also often remarkably simple.
6.1 All 2-loop MHV Amplitudes
The two-loop amplitude for 4- and 5-particles is given by, respectively,