Local Spacetime Physics from the Grassmannian
Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo, Jaroslav Trnka
PPUPT-2326
Local Spacetime Physics from the Grassmannian
N. Arkani-Hamed a , J. Bourjaily a,c , F. Cachazo b,a , J. Trnka a,ca School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA b Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J W29, CA c Department of Physics, Princeton University, Princeton, NJ 08544, USA
Abstract
A duality has recently been conjectured between all leading singularities of n -particleN k − MHV scattering amplitudes in N = 4 SYM and the residues of a contour integralwith a natural measure over the Grassmannian G ( k, n ). In this note we show thata simple contour deformation converts the sum of Grassmannian residues associatedwith the BCFW expansion of NMHV tree amplitudes to the CSW expansion of thesame amplitude. We propose that for general k the same deformation yields the ( k − k − a r X i v : . [ h e p - t h ] D ec N = 4 SYM and the Grassmannian
A dual formulation for the S-Matrix of N = 4 SYM has recently been proposed [1], where theleading singularities of the n -particle N k − MHV amplitudes—to all orders in perturbationtheory—are associated with a remarkably simple integral over the Grassmannian G ( k, n ): L n,k ( W ) = 1vol[GL( k )] (cid:90) d k × n C αa (12 · · · k )(23 · · · k + 1) · · · ( n · · · k − k (cid:89) α =1 δ | ( C αa W a ) . (1)Let us quickly review the notation appearing in (1). First, the Grassmannian is the spaceof k -planes in n dimensions, an element of which can be represented by a collection of kn -vectors in the n -dimensional space whose span specifies the plane. These vectors can beput together into the k × n matrix C αa , where α = 1 , . . . , k and a = 1 , . . . , n . With this, wewrite ( m m · · · m k ) = (cid:15) α ··· α k C α m · · · C α k m k (2)for the minor of the k × n matrix C αa made from the columns ( m , · · · , m k ). Since any k × k linear transformation on these k vectors leaves the k -plane invariant, there is a GL( k ) “gaugesymmetry” C αa (cid:55)→ L βα C βa ; our integral is “gauge-fixed” by dividing by the volume of GL( k ).The amplitude is given in dual twistor space, W a = ( (cid:101) µ a , (cid:101) λ a | (cid:101) η a ), where (cid:101) µ a is the (half-Fouriertransform) conjugate of (cid:101) λ a , and (cid:101) η a is a SUSY Grassmann parameter.This expression can be trivially transformed back to momentum space—the only depen-dence is in the δ | ( C αa W a ) factor, which transforms into δ | ( C αa W a ) → (cid:90) d × k ρ α n (cid:89) a =1 δ ( ρ α C αa − λ a ) × k (cid:89) α =1 δ ( C αa (cid:101) λ a ) × δ ( C αa (cid:101) η a ) . (3)In words, this equation embodies a simple new way of thinking about momentum conserva-tion. The kinematical data is given by specifying n individual λ a ’s and (cid:101) λ a ’s, each of whichhas two Lorentz indices. We can think of each (Lorentz) component as specifying some n -vector in the n -dimensional space of particle labels. Actually, given that the Lorentz groupis SL(2) × SL(2), the Lorentz-invariant statement is that there is a two-plane λ and anothertwo-plane (cid:101) λ ; momentum conservation (cid:80) λ a (cid:101) λ a = 0 is the statement that the two-planes λ and (cid:101) λ are orthogonal. Equation (3) interprets this in a different way, by introducing anauxiliary object—the k -plane C —and forcing C to contain the λ -plane (the first factor) andbe orthogonal to the (cid:101) λ -plane (the second factor).1he final, Grassmann δ -function in equation (3) ensures that the object is invariantunder all of GL( k ) (and not just SL( k )). In fact, we could have motivated the entire con-struction leading to equation (1) from this picture of momentum conservation: the measurein the integral over the Grassmannian is simply the nicest GL( k )-invariant one with manifestcyclic symmetry. Note also that while (1) makes superconformal invariance manifest, themomentum-space form involving (3) makes parity manifest: the action of parity is just theobvious map between G ( k, n ) and G ( n − k, n ). This can be seen explicitly by choosing anatural gauge-fixing of GL( k ), where k of the columns of C are set to an orthonormal basis,corresponding to the “link-representation” [1, 2].The geometric picture of momentum conservation motivates yet another representation of L n,k , which makes dual superconformal invariance manifest [3,4]. Since momentum conserva-tion requires that the C -plane contains the λ two-plane, it is possible to re-write the integralas one over only the space of ( k − D , which are complementary to λ in C . Thiscan be done using a gauge-fixing of GL( k ) which forces the first two rows of the C -matrixto coincide with the λ -plane—thereby manifestly encoding the fact that the Grassmannianincludes the λ -plane. A further linear transformation maps k × k minors to ( k − × ( k − L n,k ( λ, (cid:101) λ, (cid:101) η ) = δ ( (cid:80) a λ a (cid:101) λ a ) δ ( (cid:80) a λ a (cid:101) η a ) (cid:104) (cid:105)(cid:104) (cid:105) · · · (cid:104) n (cid:105) × R n,k , (4)where R n,k ( Z ) = 1vol[GL( k − (cid:90) d ( k − × n D ˆ αa (12 · · · k − · · · k − · · · ( n · · · k − k − (cid:89) ˆ α =1 δ | ( D ˆ αa Z a ) . (5)Here, the Z a are the “momentum-twistor” variables introduced by Hodges [5], which arethe most natural variables with which to discuss dual superconformal invariance. Externalparticles are associated with points x a in the dual space, with p a = x a +1 − x a . The point x a isassociated with a line in its associated momentum-“twistor space”; and since x a − x a +1 is null,the line in momentum-twistor space associated with x a intersects the line associated with x a +1 . Therefore, we can associate x a with a canonical pair of momentum-twistors ( Z a , Z a − )defined by the intersection of lines. This is illustrated in the figure below. The momentumtwistor Z a is composed of Z a = ( µ a , λ a | η a ), where the variables (cid:101) λ a , (cid:101) η a are determined by µ a , η a . Explicitly, they are given by (cid:101) λ a = (cid:104) a − a (cid:105) µ a +1 + (cid:104) a a + 1 (cid:105) µ a − + (cid:104) a + 1 a − (cid:105) µ a (cid:104) a − a (cid:105)(cid:104) a a + 1 (cid:105) (cid:101) η a = (cid:104) a − a (cid:105) η a +1 + (cid:104) a a + 1 (cid:105) η a − + (cid:104) a + 1 a − (cid:105) η a (cid:104) a − a (cid:105)(cid:104) a a + 1 (cid:105) . (6)2ual superconformal transformations [4–7] are just linear transformations of the Z a , whichis a manifest symmetry of equation (5), just as ordinary superconformal transformations arelinear transformations on W a making them a manifest symmetry of equation (1). Thus,equation (1) makes all the important symmetries of N = 4 SYM amplitudes manifest.The momentum-space formula for L n,k is to be interpreted as a contour integral in( k − × ( n − k −
2) variables, which can be thought of as specifying the unfixed degreesof freedom of a ( k − (cid:101) λ - and λ -planes. In [1], evidence wasgiven that the residues of the integrand are associated with leading singularities up to 2loops, motivating the conjecture that all leading singularities are contained as residues. Thisconjecture carries even more weight given the realization that all the residues are both super-conformal and dual superconformal invariant, which further means they are invariant underthe full Yangian symmetry [7]. Leading singularities are data associated with scatteringamplitudes that are free of IR divergences—at loop level, they can be thought of as beingassociated with loop integrals over compact contours—and should therefore reflect all thesymmetries of the theory. In fact, the residues of our object can be thought of as generat-ing (likely all) Yangian invariants that are algebraic functions of the external spinor-helicityvariables. Furthermore, as emphasized in [1], higher-dimensional residue theorems encodehighly non-trivial relations between these invariants, many of which have striking physicalinterpretations such as loop-level infrared equations.It is clear that there is an enormous amount of fascinating structure to be uncoveredin the properties of the individual residues of L n,k , since they are invariants of the mostremarkable integrable structure we have ever seen in physics! Recent work [8, 9] as well aswork to appear [10] gives strong evidence that infinite classes of all-loop leading singularitiesare indeed contained amongst the residues of L n,k .There is however something even more remarkable than the properties of residues takenindividually: they can be combined in such a way as to produce amplitudes with a localspace-time interpretation. Consider for instance NMHV tree amplitudes ( k = 3). A givenresidue is associated with putting ( k − n − k −
2) = ( n −
5) minors to zero, which can be3abeled as ( m ) · · · ( m n − ), where ( m ) denotes that the minor ( m m + 1 m + 2) has been setto zero. In [1], it was shown that a natural BCFW expansion for the NMHV amplitudes isgiven by a sum of residues M BCFW n, NMHV = (cid:88) ( o )( e )( o ) · · · (cid:124) (cid:123)(cid:122) (cid:125) n − n −
5) alternating odd ( o ) and even ( e )integers; to be explicit the 6-,7- and 8-particle amplitudes are given by M BCFW6 , NMHV = (1) + (3) + (5); M BCFW7 , NMHV = (1)(2) + (1)(4) + (1)(6) + (3)(4) + (3)(6) + (5)(6); M BCFW8 , NMHV = (1)(2)(3) + (1)(2)(5) + (1)(2)(7) + (1)(4)(5) + (1)(4)(7)+ (1)(6)(7) + (3)(4)(5) + (3)(4)(7) + (3)(6)(7) + (5)(6)(7) . (8)We remind the reader of a fact that will be important repeatedly: residues are naturallyalternating in the arguments, so that e.g. ( i )( i ) = − ( i )( i ). The P(BCFW) form of theamplitudes has exactly the same form as BCFW, but switching the role of even and oddintegers: M P(BCFW) n, NMHV = ( − n − (cid:88) ( e )( o )( e ) · · · (cid:124) (cid:123)(cid:122) (cid:125) n − . (9)As shown in [1], the equality M BCFW = M P(BCFW) is a (quite non-trivial) consequence ofglobal residue theorems, which further guarantees the cyclic invariance of the amplitude.This presentation of the NMHV amplitudes makes all of its symmetries manifest, and isstrikingly “combinatorial” in nature. One thing that is seemingly not manifest, however, isthat this object has anything whatsoever to do with a local space-time Lagrangian! Each termindividually has “non-local” poles, which magically cancel in the odd/even/odd combinationdefining the amplitude. The cancelation of these non-local poles can be understood indirectlyby the equality M BCFW = M P(BCFW) , since the non-local poles appearing in the two formsturn out to be different. However, this is very far from establishing that this object comesfrom a local Lagrangian, and one would certainly like to see the emergence of space-time ina much more direct and explicit way.In this note, we will argue that the local space-time description of tree scattering ampli-tudes is actually hiding in plain sight in the BCFW sum over residues in the Grassmannian.We will show that a very natural and canonical contour deformation converts the BCFWform of tree scattering amplitudes to the CSW/Risager expansion, which is a direct reflectionof the space-time Lagrangian in light-cone gauge!4
Brief Review of CSW and Risager
To set the stage, let us quickly review the story of the CSW recursion relations [11–13] and thevery closely-related Risager recursion relations [14, 15]. The CSW rules are simply Feynmanrules [16], except that the vertices are off-shell continuations of MHV amplitudes, where the λ ’s for internal lines with momentum P are defined by λ P = P | ζ ] , (10)where ζ is an auxiliary spinor. Note that we use a different notation for this auxiliaryspinor than the usual one in the literature, (cid:101) η , in order to not confuse this object with theSUSY Grassmann parameters. The similarity with usual Feynman rules and the hiddenLorentz invariance of this expansion is not a coincidence: the CSW rules can be derivedfrom the Yang-Mills Lagrangian by going to a more sophisticated version of light-cone gauge[16, 17]; the auxiliary spinor ζ is associated with the light-like direction defining the light-cone gauge. As usual in light-cone gauge, we have only physical degrees of freedom, thetwo polarizations ± of the gluons. There are cubic interactions (+ + − ), ( − − +) and thequartic interaction (+ + −− ). From this, it is possible to make a field redefinition to removethe anti-MHV (+ + − ) interaction; this forces the introduction of an infinite number ofnew MHV vertices, which must—on-shell—reproduce the MHV amplitudes. The resultingLagrangian is precisely the one that gives the CSW rules. The equivalence between the MHVrules in a light-cone gauge and usual Lorentz-invariant formulation of the (super) Yang-MillsLagrangian L = − tr F µν + · · · was nicely established in a different way in [18]. Beginningwith a twistor space action with a large amount of gauge symmetry, one gauge-fixing leadsto the usual manifestly Lorentz-invariant Yang-Mills action, while a different gauge-fixingyields the MHV Lagrangian in light-cone gauge. Thus, the CSW rules should be thought ofas directly reflecting the Yang-Mills Lagrangian in light-cone gauge, encoding local space-timephysics in the most succinct possible way.For future reference, we remind the reader that the terms in the CSW expansion of theN k − MHV amplitude are localized on ( k −
1) intersecting lines in the Z -twistor space: theMHV vertices in the CSW diagrams are associated with lines in twistor space, while theinternal lines are associated with points where these lines intersect. Thus, a general term inthe CSW expansion of NMHV amplitudes with particles m, k, and l of negative helicity islocalized in twistor space as shown below. 5he Risager deformation is closely related, providing an alternate derivation of the CSWrules that closely parallels the logic leading the BCFW recursion relations [19–22]. As withBCFW, it involves a deformation of the spinor helicity variables; specifically, it begins bycanonically deforming the (cid:101) λ i ’s for all the negative helicity particles: (cid:101) λ i → (cid:101) λ i + α i zζ. (11)In order to conserve overall momentum, the α i must satisfy the constraint (cid:88) i α i λ i = 0 . (12)Thus, for k negative helicity gluons, the most general Risager deformation is labeled by( k −
2) parameters. It is possible to show that under this deformation the amplitude vanishesas z → ∞ , so that the familiar BCFW logic leads to recursion relations (see, e.g. [21, 23]).Remarkably, Risager showed that repeated recursive use of this deformation leads to theCSW rules [14].Below we will study the Risager expansion for MHV amplitudes in the split-helicityconfiguration. In this case, the Risager diagrams consist only of ones with a three-point vertexand the lower-point MHV amplitude connected by a propagator. We will find it useful to lookat Risager deformations in momentum-twistor variables µ a , for which the general N k − MHVsplit helicity amplitude A (1 − , − , . . . , ( k − − , k + , . . . , ( n − + , n − ) takes the remarkablysimple form: ˆ µ a = (cid:26) µ a + zβ a ζ for a = 1 , . . . , k − β a arbitrary) µ a for a = k − , . . . , n . (13)Note that this deforms ( k −
2) terms, which is exactly the number of independent α ’s in (11).There are no constraints on the β a since—by construction—any choice of µ a is guaranteedto produce (cid:101) λ a ’s that satisfy momentum conservation. This choice of β a determines thedeformation of the negative helicity particles α i as α i = (cid:104) i i − (cid:105) β i +1 + (cid:104) i + 1 i (cid:105) β i − + (cid:104) i − i + 1 (cid:105) β i (cid:104) i + 1 i (cid:105)(cid:104) i i − (cid:105) . (14)6 Relaxing δ -functions We now describe the contour deformation that will lead us from the BCFW contour in theGrassmannian to the space-time Lagrangian in light-cone gauge, passing through the CSWand Risager expansions of tree amplitudes. We begin with the form of L n,k in momentumspace. It is most convenient to use the momentum-twistor form, since this explicitly exhibitsthe (super) momentum-conserving δ -functions in the pre-factor, and we can study insteadthe object R n,k .There is something seemingly unnatural in the expression for R n,k : it is a nice, holomor-phic contour integral, but it has explicit δ -function factors! This is not unnatural at all, sincethese are in fact to be thought of “holomorphic” δ -functions, which are properly interpretedas poles. In other words, we may interpret δ ( µ ) as being really δ ( µ ) = 1 µ × µ ; (15)or more generally, introducing a pair of auxiliary spinors χ, ζ , we write δ ( µ ) = [ χ ζ ][ χ µ ][ ζ µ ] (16)where we also demand that the contour of integration enforce the poles where [ χ µ ] = [ ζ µ ] = 0.Note that the expression in equation (16) is not manifestly Lorentz invariant—but of coursethe residue obtained on the pole of both factors is Lorentz invariant. The reason for using thenotation “ δ ( µ )” is to emphasize the Lorentz invariance of the final object. Thus, when wesay that the expression for R n,k is a contour integral in ( k − n − k −
2) variables, we reallymean that we started with a larger ( k − n − k + 2)-dimensional integral and have alreadyfixed part of the contour by specifying that it enforces 4( k −
2) poles associated with theBosonic parts of the δ ( D ˆ αa Z a )-factors. Similarly, what we have been referring to as “the”residues of R n,k are really particular residues in this higher-dimensional integral, evaluatedon 4( k −
2) extra poles, with an extra ( k − n − k −
2) conditions involving the minorsneeded to fully-specify the residue.This way of thinking about the δ -functions explicitly as poles naturally suggests somethingvery remarkable. We can “relax” any one of the δ -functions, using a residue theorem to movethe contour off one of its associated poles, and thereby express a manifestly Lorentz-invariantresidue as a sum over non-Lorentz invariant terms which involve putting an extra minor tozero. Inspired by this, we will take one of the δ -factors and replace it by δ ( µ ) = δ ([ ζ µ ]) × [ χ ζ ][ χ µ ] , (17)where we mean that the pole at [ ζ µ ] = 0 is still being enforced while we allow ourselves thefreedom to deform the contour off the pole at [ χ µ ] = 0. Note that while this expression is7ot Lorentz-invariant away from both poles, it is independent of the choice of χ . The reasonis that on the zero of [ ζ µ ] = 0, µ is proportional to ζ and we may write µ = d × ζ , andso [ χ ζ ] / [ χ µ ] = 1 /d is χ -independent. Thus, relaxing the δ -function in this way expressesa Lorentz-invariant reside as a sum over non-Lorentz invariant terms which are a functionof only a single auxiliary spinor ζ . Concretely, we can do this for one of the δ ( D ˆ αa µ a )factors—e.g. that of ˆ α = 1—by making the replacement δ ( D a µ a ) → δ ( D a [ ζ µ a ]) × [ χ ζ ] D a [ χ µ a ] (18)and deforming the contour off the D a [ χ µ a ] pole.Clearly, this operation can be extended to relax even more δ -functions; but we will see thatrelaxing just one δ -function “blows up” Lorentz-invariant residues into a sum of non-Lorentzinvariant terms with a beautiful physical interpretation. For the NMHV case, we will seethat some of the terms in the sum are precisely the ones that appear in the CSW expansion ofNMHV amplitudes. This is strongly suggested—even without a direct computation—by thelocalization properties of these terms both in the Grassmannian and twistor space, and theprecise equality can be easily verified. Other terms in the sum do not have the appropriatelocalization properties and are not associated with CSW terms. The CSW terms have a localspace-time interpretation and are therefore free of non-local poles, while the others do containnon-local poles. In a sense our δ -relaxing contour deformation has performed a particularlypowerful partial fraction expansion of the residue into a sum over local and non-local pieces.Remarkably, in the sum over residues with the alternating odd/even structure of equation (9),all the non-CSW terms appear precisely twice with opposite signs and cancel in pairs, whilethe remaining terms are exactly the terms of the CSW expansion of the amplitude!For k >
3, it is easy to see that relaxing a single δ -function can not directly produce CSWterms. Nonetheless, such a canonical operation must have a physical meaning, and the onlynatural candidate for a non-manifestly Lorentz invariant form of amplitudes depending ona single auxiliary spinor is the Risager expansion. This raises a puzzle, however, since theRisager expansion is not unique, but is labeled by ( k −
2) degrees of freedom. We establishthe precise equivalence and understand the origin of these degrees of freedom for the case ofsplit-helicity MHV amplitudes, where the ( k −
2) free parameters of the Risager deformationare seen to be quite non-trivially determined by the degrees of freedom associated with theGL( k −
2) “gauge symmetry” of the momentum-twistor formula.As was shown by Risager [14], a recursive application of the Risager recursion eventuallyyields the CSW expansion for general amplitudes. Although we won’t pursue this directionfurther in this note, this strongly suggests that the CSW expansion for general amplitudescan be directly obtained by recursively relaxing many δ -function factors.8 NMHV and CSW from δ -Relaxation Let us work in the momentum-twistor picture, where L n, = M MHV × (cid:90) d n − D a (1)(2) · · · ( n ) δ | ( D a Z a ) . (19)Here the 1 × j ) are of course just single variables D j ; we remind the reader that thelinear transformation from the G ( k, n ) to the G ( k − , n ) picture makes the ( k − × ( k − · · · k − D proportional to the k × k minor (1 2 · · · k ) C , so that e.g. the minor (2)in the momentum-twistor picture is proportional to the minor (1 2 3) in the G (3 , n ) picture.For convenience we will denote the elements of the 1 × n matrix D ˆ αa as( D , D , . . . , D n ) . (20)In other words, we remove the index ˆ α when k = 3 since it takes a single value.A given residue is associated with setting ( n −
5) of the minors to zero as is obvious: aftergauge-fixing any one of the D a , setting ( n −
5) of the D a ’s to zero allows us to use the Bosonic δ -function to solve for the remaining four D ’s. We denote this residue as ( a )( a )( a )( a )( a ),which instructs us to write all minors in cyclic order starting from (1), with ( a ) , . . . , ( a )left off. As an example with n = 8, (2)(3)(4)(6)(7) denotes the residue (1)(5)(8) where theminors (1) , (5) , (8) are set to zero. We remind the reader once again that residues of functionsin several complex variables are antisymmetric objects, so that the order in which the minorsare presented matters, and e.g., (1)(5)(8) = − (5)(1)(8).We will be looking at explicit gluon amplitudes in what follows, so we need to integrateover the SUSY Grassmann parameters to extract these. This is a completely straightforwardexercise. We set the gluons with a ∈ I to have negative helicity, strip-off the ordinarymomentum-conserving δ -function, and we write L n,k = δ ( (cid:80) a λ a (cid:101) λ a ) L n,k with L n,k = 1vol[GL( k − (cid:90) d ( k − × n D ˆ αa (1 2 · · · k − · · · k − · · · ( n · · · k −
3) (det (cid:101) D ) × δ ( D ˆ αa Z a )(21)where (cid:101) D is a k × k matrix (cid:101) D αI = (cid:32) λ αI ˆ D αI (cid:33) with ˆ D αI = Θ( I − n (cid:88) a = I +1 D αa (cid:104) I a (cid:105) . (22)Here, Θ( x ) is 1 for x > L n,k agreeing on the supportof the δ -functions.Returning to the k = 3 case, a general residue is explicitly given by( a )( a )( a )( a )( a ) = (cid:90) dD a · · · dD a D a · · · D a (det (cid:101) D ) δ ( D a Z a + · · · + D a Z a ); (23)9e can relax the δ -function for the µ -term by making the replacement δ ( D a µ a ) → δ ( D a [ µ a ζ ]) × [ χ ζ ]( D a [ µ a χ ]) ≡ d δ ( D a [ µ a ζ ]) . (24)Then, we can use a residue theorem to deform the contour off D a [ χ µ a ] = 0, or equivalentlyoff d = 0, and write( a )( a )( a )( a )( a ) = (cid:88) σ ∈ Z (cid:104) ( a σ (1) )( a σ (2) )( a σ (3) )( a σ (4) ) d ( a σ (5) ) (cid:105) , (25)where the sum is over cyclic permutations of { , , , , } . For example, (cid:104) ( a )( a )( a )( a ) d ( a ) (cid:105) is given by (cid:90) D a =0 dD a · · · dD a D a · · · D a (det (cid:101) D ) d δ ( D a λ a + · · · + D a λ a ) δ ( D a [ ζ µ a ] + · · · + D a [ ζ µ a ]) . (26) Before we demonstrate the complete equivalence of the CSW expansion and the terms gener-ated by “blowing-up” each residue of the NMHV contour, it is worthwhile to give an intuitiveunderstanding of why this should work.One of the strongest hints that there should be a direct connection between the CSWexpansion and L n,k is how the localization in twistor-space implied by CSW is mirrored by a localization within the Grassmannian itself . We can see this directly by Fourier-transformingthe kinematical δ -function δ | ( C αa W a ) from the W -twistor variables to their (ordinary) dualtwistor-space variables Z : k (cid:89) α =1 δ | ( C αa W a ) → (cid:90) d | z α k (cid:89) α =1 δ | ( Z a − C αa z α ) . (27)(These twistors Z a are ordinary twistors, which are the duals of W a , and should not beconfused with momentum-twistors.)If we think of each column of G ( k, n ) as projectively defining a point in CP k − , then thevanishing of a minor of G ( k, n )—consecutive or otherwise—is equivalent to some localizationcondition among these points in CP k − . The first nontrivial example of this can be easilyseen for G (3 , n ), where a minor ( i j k ) = 0 if and only if the corresponding points i, j, and k are collinear in CP . It is not hard to see that the twistor-space “collinearity operator” (cid:15) IJKL Z Ii Z Jj Z Kk , which vanishes whenever the (Bosonic parts of the) twistors Z i , Z j , and Z k are collinear [24], manifestly annihilates any residue of the Grassmannian supported wherethe minor ( i j k ) vanishes. Similarly, for k = 4, the “coplanarity operator” (cid:15) IJKL Z Ii Z Jj Z Kk Z Ll which test whether Z i , . . . , Z l are coplanar, will annihilate any residue for which the minor( i j k l ) = 0. (Although beyond the scope of the present discussion, there are many reasonsto suspect that localization in the Grassmannian is very natural and fundamental [25].)10he simplest example to begin with is the 5-point NMHV(=MHV) amplitude. Of course,this amplitude is entirely fixed by the δ -functions, and ordinarily no residue would be chosenat all. Therefore, the contour deformation corresponding to relaxing the δ -function gives riseto a sum over each of the 5 minors M , NMHV = (cid:104) (2)(3)(4)(5) d (1) (cid:105) + (cid:104) (1)(3)(4)(5) d (2) (cid:105) + . . . ≡ (cid:88) j =1 [( j j + 1 j + 2)] . (28)From our discussion above, it is clear that the term in the expansion setting (1 2 3) = 0forces the points 1 ,
2, 3 to be collinear in twistor space; it is trivial that NMHV amplitudesare all localized on a CP inside the CP of twistor space, so the line connecting 4 , ,
2, 3 and thus, this term has the localization properties we expect of aCSW diagram. This is true for all the terms in (28), and we can make an association withthe terms setting the minors to zero and each of the CSW diagrams illustrated above.Before showing the computation that establishes the precise equivalence with the CSWterms, let us understand this localization picture for general NMHV amplitudes, startingwith the 6-particle case. A given residue ( j j + 1 j + 2) is blown-up into the sum of 5 terms,( j j + 1 j + 2) → (cid:88) k (cid:54) = j [( j j + 1 j + 2)( k k + 1 k + 2)] ≡ (cid:88) k (cid:54) = j [( j )( k )] (29)where the term [( j )( j )] vanishes due to antisymmetry (or said another way, because it is a dou-ble pole with vanishing residue). Although we are choosing to write ( j j + 1 j + 2) ≡ ( j ) forconvenience, these should not be confused with minors in themomentum-twistor picture. Let us look at the 5 terms in the blow-up of the residue (1 2 3);these terms have the the following localizations structure in twistor space:11ote that while the terms [(1)(2)] , [(1)(4)] , [(1)(6)] do have CSW localization properties,the terms [(1)(3)] and [(1)(5)] do not . Similarly, the terms [(3)(1)] and [(3)(5)] in the blow-upof (3), and the terms [(5)(1)] , [(5)(3)] in the blow-up of (5) do not have CSW localization.However, and quite remarkably, these 6 non-local terms cancel each other in pairs due tothe antisymmetric property of the residues, as e.g. [(1)(3)] + [(3)(1)] = 0. The 9 remainingterms all have CSW localization and are indeed in perfect correspondence with the 9 CSWdiagrams for this amplitude!This pattern holds for all NMHV amplitudes. It is easiest to see this pictorially: let thesum over residues giving the BCFW form of the amplitude be represented as follows,where each term represents ( i − i )( j − j )( n ), i.e., the open circles correspond to theminors that are not being set to zero.Now, when we blow up each residue with our contour deformation, we have a sum overterms setting an extra minor tacked-on at the end of the chain to zero, which can be repre-sented in the picture by summing over terms “coloring-in” one of the white dots, leaving uswith 4 minors that are not set to zero. Each of these has some localization properties, but itis easy to see that the only ones that have CSW localization are the ones of the form:Now let us see what we get from coloring-in a white dot in a general term of our NMHVsum. The ones where ( n ) is colored in automatically has good CSW properties; these give asubset of CSW diagrams, where the white circles do not include ( n ):But in addition to these good terms, there are dangerous terms which do not have CSWlocalization properties, arising from coloring-in ( i − i ) is colored in, and they cancel in pairs due to antisymmetry of the residues:—with the obviously symmetrical statements holding for coloring-in ( j ). There are also thediagrams where we color-in ( i ) which cancel in pairs with the one where i → i −
1, exceptfor the case where i − i − j = n −
1) has CSW localizationand provide the missing CSW terms with white circles covering ( n ), giving us the sum overall CSW terms We finally prove that each of the remaining residues in the sum above precisely correspondsto the corresponding term in the CSW expansion of the NMHV amplitude. To begin with,it is convenient to introduce the following notation { a b c } = µ a (cid:104) b c (cid:105) + µ b (cid:104) c a (cid:105) + µ c (cid:104) a b (cid:105) (30)so that, e.g., (cid:101) λ i = { i + 1 i i − }(cid:104) i + 1 i (cid:105)(cid:104) i i − (cid:105) . (31)13et us compute each of the residues ( i )( i + 1)( j )( j + 1) d , corresponding to the vanishingof all D ’s except D i , D i +1 , D j , D j +1 and d .Recall that we have three delta functions to impose: δ ( D i λ i + D i +1 λ i +1 + D j λ j + D j +1 λ j +1 ) δ ( D i [ µ i ζ ]+ D i +1 [ µ i +1 ζ ]+ D j [ µ j ζ ]+ D j +1 [ µ j +1 ζ ]) . (32)Using GL(1) to fix D i = 1, it is easy to solve explicitly for the rest of the D ’s D i +1 = [ { i j j + 1 } ζ ][ { i + 1 j j + 1 } ζ ] , D j = [ { i i + 1 j + 1 } ζ ][ { i + 1 j j + 1 } ζ ] and D j +1 = [ { i i + 1 j } ζ ][ { i + 1 j j + 1 } ζ ] . (33)Here [ { a b c } ζ ] means the Lorentz invariant contraction of spinors.The three δ -functions in (32) yield a Jacobian J = 1[ { i + 1 j j + 1 } ζ ] (34)while the product of D ’s in the denominator of the residue becomes1 D i D i +1 D j D j +1 = [ { i + 1 j j + 1 } ζ ] [ { i + 1 i j + 1 } ζ ][ { i + 1 i j } ζ ][ { i j j + 1 } ζ ] . (35)Finally, d = (cid:104) Z i Z i +1 Z j Z j +1 (cid:105) (36)where (cid:104) Z i Z i +1 Z j Z j +1 (cid:105) = (cid:15) IJKL Z i,I Z i +1 ,J Z j,K Z j +1 ,L is the dual conformal invariant inner prod-uct of four momentum-twistors. In fact, this particular combination has a special meaning, (cid:104) Z j Z j − Z i Z i − (cid:105)(cid:104) j j − (cid:105)(cid:104) i i − (cid:105) = ( x j − x i ) = ( p i + p i +1 + · · · + p j − ) (37)which is nothing but the propagator in the corresponding CSW diagram!In this computation we are taking as the minus-helicity particles gluons k, l and m .Therefore, the helicity-factor (det (cid:101) D ) has the form(det (cid:101) D ) = (cid:12)(cid:12)(cid:12)(cid:12) λ m λ k λ l ˆ D m ˆ D k ˆ D l (cid:12)(cid:12)(cid:12)(cid:12) . (38)14n the case where particle m is on the right-side and k , l on the left-side as in the figureabove, referring to equation (22), we can writeˆ D m = D j (cid:104) m j (cid:105) + D j +1 (cid:104) m j + 1 (cid:105) = [ { j + 1 i i − } ζ ] (cid:104) m j (cid:105) − [ { j + 1 i i − } ζ ] (cid:104) m j (cid:105) [ { i + 1 j j + 1 } ζ ] (39)while ˆ D k = ˆ D l = 0. Then (det (cid:101) D ) = (cid:104) k l (cid:105) ˆ D m .The residue ( i )( i + 1)( j )( j + 1) d , which equals J (det (cid:101) D ) / ( d D i D i +1 D j D j +1 ), becomes([ { j + 1 i + 1 i } ζ ] (cid:104) m j (cid:105) − [ { j i + 1 i } ζ ] (cid:104) m j + 1 (cid:105) ) (cid:104) k l (cid:105) (cid:104) Z j +1 Z j Z i +1 Z i (cid:105) [ { j + 1 i + 1 i } ζ ][ { j + 1 j i + 1 } ζ ][ { i j + 1 j } ζ ][ { j i + 1 i } ζ ] . (40)A simple computation using, e.g,( p j + · · · + p i +1 ) | i (cid:105) = { j + 1 j i }(cid:104) j + 1 j (cid:105) , (41) (cid:104) j + 1 | ( p j + · · · + p i ) = { j + 1 i i − }(cid:104) i i − (cid:105) , (42)reveals that equation (40) precisely reproduces the CSW contribution associated to the cor-responding diagram. δ -Relaxation For k >
3, it is easy to see that relaxing a single δ -function does not directly lead to the CSWexpansion. This is obvious since localization in the Grassmannian associated with putting k × k minors to zero for k > k −
2) degrees of freedomof the Risager deformation are reflected in the Grassmannian picture–exactly which Risagerexpansion are we landing on? In this section we establish the correspondence with Risager,and also understand the origin of the Risager degrees of freedom, by examining MHV am-plitudes. This will determine precisely which Risager expansion must be associated with ourcontour deformation for general ( n, k ).The only Risager diagrams that contribute involve the points i, i + 1 and the internal line P on one side, connected with a propagator to the lower-point MHV amplitude on the otherside 15hich can be nicely simplified to the form A Risager i = [ k l ] [ˆ1 ˆ2] . . . [ (cid:91) i − i ][ i i + 1][ (cid:91) i + 1 (cid:91) i + 2] . . . [ˆ n ˆ1] . (43)Here, the deformation parameter z is evaluated where P ( z ∗ ) = 0. We will now see thatthis expansion is reproduced for the first non-trivial case of the split-helicity 6-particle MHVamplitude A (1 − , − , − , + , + , − ). The D -matrix in the momentum twistor form of theGrassmannian is D = (cid:18) D D D D D D D D D D D D (cid:19) . (44)As before, we will be relaxing one of the δ ( D a µ a )-factors. Our strategy is to use four δ -function constraints for the second row, and to solve for D , . . . , D in terms of D and D , and to use the remaining three δ -functions to solve for D , . . . , D in terms of D , D ,and D . Now, in deforming the contour, we will get a sum over terms where a given minor( j ) is set to zero. Here, we use the notation ( j ) to refer to the minor ( j j + 1 · · · j + k − D and plug it backinto our equations for D , . . . , D . Notice that we can gauge-fix the GL(2) so that e.g. D , D , D , D are anything we like, but we will leave them arbitrary for now. The reasonis that while the sum over all the terms will be GL(2)-invariant, each individual term willnot, and as we will see the dependence on gauge degrees of freedom will precisely mirror thefreedom in the Risager deformations.A somewhat lengthy computation yields a lovely result for the term where the minor ( j )is set to zero; we find that it precisely corresponds to a term in the Risager expansion[( j )] = A Risager j +3 (45)where the Risager deformation is particularly simple and is given in terms of the followingdeformation on momentum twistor variables ˆ µ i = µ i + β i zζ with β = D , and β = D . (46)That is, as advertised, the degrees of the freedom in the Risager expansion are contained inthe GL(2) freedom of the momentum-twistor Grassmannian formula!Moving on to the 7-point amplitude A (1 − , − , − , − , + , + , − ) we find exactly the samepattern: we find that the sum over terms setting a minor to zero precisely matches theRisager expansion of the amplitude, with the β -deformations now with β = M , β = M , and β = M , (47)where the M ij are determined by the GL(3) gauge degrees of freedom as M i,j = (cid:12)(cid:12)(cid:12)(cid:12) D i D j D i D j (cid:12)(cid:12)(cid:12)(cid:12) . (48)16he case for general split-helicity amplitudes follows the same pattern. We use the D ij , i, j = 1 , . . . , n −
4, as free gauge-fixing parameters. We solve for D ij , i = 2 , . . . , n − j = n − , . . . , n in terms of gauge-fixed parameters D ij , j = 1 , . . . , n −
4, and then solvefor the D j , j = n − , n − , n in terms of gauge fixing parameters D ij , j = 1 , . . . , n − D n − . Then, for each individual residue characterized by some vanishing minor ( j ), wedetermine D n − , and substitute it back into other D j . We can then calculate all minors andJacobian factors, and compare with the Risager expansion. Remarkably the two expressionsagree using a Risager shift most nicely given in terms of a deformations of µ ’s: β j = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D , . . . D ,j − D ,j +1 . . . D ,n − ... ... ... ... ... ... D n − , . . . D n − ,j − D n − ,j +1 . . . D n − ,n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (49)Again, the general pattern is that the deformations are constructed just from gauge-fixing pa-rameters. This just demonstrates the fact that the freedom in choosing Risager deformations β j is included in the GL( k −
2) redundancy in the Grassmannian.
We have argued that a simple and canonical “ δ -relaxing” contour deformation takes us fromthe Grassmannian formulation of BCFW tree amplitudes—which has a remarkably “combi-natorial” form making all symmetries manifest—to the CSW expansion, which manifests thelocal space-time Lagrangian in light-cone gauge. Relaxing a single δ -function already yieldsthe full CSW expansion for NMHV amplitudes, and must lead to the Risager expansionfor general k as we established for the MHV case. It would be interesting to see this moreexplicitly, and also to understand whether the recursive application of the Risager expansionleading to the CSW expansion has a natural interpretation in terms of relaxing multiple δ -functions.The operation we have found gives a natural way of “blowing up” residues into com-ponents, separating pieces with a local space-time interpretation from the non-local ones.This allows us to give the sum over Grassmannian residues corresponding to the tree contoura “particle interpretation” in space-time. As we will see in [25], there is a second naturaloperation on the sum over residues—rather than blowing each residue up into many pieces,we can instead unify them together as the zero set of a single map. This manifests an evenmore surprising feature than a particle interpretation in space-time—the integral localizes onconfigurations with a “particle interpretation” in the Grassmannian , allowing us to constructhigher-point tree amplitudes by “adding one particle at a time” to lower-point ones. Fur-thermore, a natural deformation not simply of the contour but of the integrand itself directlyconnects our Grassmannian picture with the connected prescription [26] of Witten’s twistorstring theory [24, 27–29]. 17e find it remarkable that almost all the concepts surrounding perturbative scatteringamplitudes in this decade—the twistor string theory, CSW, BCFW and Risager recursion re-lations, infrared equations, leading singularities and dual superconformal invariance—are uni-fied in the Grassmannian integral we have been exploring. The only important object that hasyet to make a direct appearance in this story is the light-like Wilson loop (see e.g. [30–36])—making this connection will surely tell us how to extract loop-level information beyond theall-loop leading singularities that are already clearly present in the Grassmannian. Acknowledgments
We thank Louise Dolan, Peter Goddard and Edward Witten for stimulating discussions.N.A.-H. is supported by the DOE under grant DE-FG02-91ER40654, F.C. was supportedin part by the NSERC of Canada, MEDT of Ontario and by The Ambrose Monell Founda-tion. J.T. is supported by the U.S. Department of State through a Fulbright Science andTechnology Award.
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