Supersymmetric Yang-Mills and Supergravity Amplitudes at One Loop

Supersymmetric Yang-Mills and Supergravity Amplitudes at One Loop

Anthony Hall Department of Physics and Astronomy, UCLA
Los Angeles, CA 90095–1547, USA
July 14, 2019

By applying the known expressions for SYM and SUGRA tree amplitudes, we write generating functions for the NNMHV box coefficients of SYM as well as the MHV, NMHV, and NNMHV box coefficients for SUGRA. The all-multiplicity generating functions utilize covariant, on-shell superspace whereby the contribution from arbitrary external states in the supermultiplet can be extracted by Grassmann operators. In support of the relation between dual Wilson loops and SYM scattering amplitudes at weak coupling, the SYM amplitudes are presented in a manifestly dual superconformal form. We introduce ordered box coefficients for calculating SUGRA quadruple cuts and prove that ordered coefficients generate physical cut amplitudes after summing over permutations of the external legs. The ordered box coefficients are produced by sewing ordered subamplitudes, previously used in applying on-shell recursion relations at tree level. We describe our verification of the results against the literature, and a formula for extracting the contributions from external gluons or gravitons to NNMHV superamplitudes is presented.

I Introduction

Supersymmetric gauge theory is profoundly linked to string theory, perturbatively and at strong coupling. The prescient work of Nair nair () recognized the Parke-Taylor scattering amplitudes of SYM as fermion correlators on a sphere. This result was generalized by Witten Witten () to describe a weak-weak coupling duality between SYM and D-instantons of the B-model topological string in supersymmetric twistor space. Other topological, dual descriptions of SYM have been proposed by Berkovits berkovits1 (); berkovits2 (), Neitzke and Vafa neitzkevafa (), and Siegel siegel (). The representation of scattering amplitudes in twistor space has been studied since the inception of twistor theory twistors (). The BCFW on-shell recursion relations for spacetime signature have recently been formulated in twistor space newtwistors (), and a CSW prescription for SYM is presented in Ref. supercsw ().

The AdS/CFT correspondence relates the quantum theories of weakly coupled Type IIB strings in an geometry to strongly coupled SYM on the four-dimensional boundary of . In light of this correspondence, Alday and Maldacena aldaymaldacena () conjectured that the strong-coupling limit of -gluon scattering amplitudes, to all-loop order, in SYM are related to minimal surfaces in . The minimal surface is a polygon with light-like edges , where the dual coordinates are related to the gluon momenta by . This method of evaluating SYM amplitudes is equivalent to calculating a dual Wilson loop along the light-like polygon edges . The scattering amplitude/dual-Wilson loop duality is conjectured to hold for weak and strong coupling, with agreement confirmed up to the six-point, two-loop MHV amplitude drummondfourgluon (); brandhuberwilson (); drummondloops1 (); drummondloops2 (); drummondloops3 ().

In Ref. drummondmagic (), it was observed that the integrals required for the three-loop MHV amplitudes, calculated by Bern, Dixon, and Smirnov bernloopiteration (), are conformally covariant when formulated in the dual -coordinates. The relationship with Wilson loops, which have a conformal symmetry, hinted at the presence of an unexpected dual-conformal symmetry for SYM scattering amplitudes. The conformal symmetry of Wilson loops is manifested as an ultraviolet-anomalous Ward identity. The conformal Ward identity dictates the form of the finite part of up-to-five cusp Wilson loops at weak drummondloops1 (); drummondloops2 () and strong coupling aldaymaldacenanobds (); komargodski (). Aiming to explain why the MHV amplitudes continue to agree with the Wilson loop duality beyond five cusps, the authors of Ref. drummondsym () postulate a new, larger symmetry at work, the superconformal symmetry acting on dual superspace coordinates.

The unitarity method bernunitarity (); bernfusing () supplies the technology we use for manifestly on-shell calculations of loop-diagram quantum corrections to scattering amplitudes in quantum field theory. In a generalized unitary approach berngenunitarity (), the coefficients of loop integrals are obtained by “cutting” multiple virtual particles, exposing the loop-integral coefficients as products of on-shell tree amplitudes. Quadruple cuts BCFUnitarity () yield integral coefficients which are the product of four on-shell trees, and quadruple cuts freeze the remaining Lorentz-invariant phase-space integrals to a finite set of solutions for the on-shell loop momenta. The “no-triangle property” of SYM and SUGRA arkanihamed (); berngraviton (); berntwistorgravity (); gravitynmhv (); notriangle (); trianglecancel () allows one-loop amplitudes to be completely specified by quadruple cuts. The first cut calculation of SUGRA amplitudes by exploiting the KLT relations between gravity and squared gauge-theory tree amplitudes were carried out in Ref. berngaugegravity (). Multi-leg results for SUGRA box coefficients were first presented in Ref. berngraviton ().

On-shell recursion relations for amplitudes in on-shell superspace freedman (); brandhubersuperconformal (); arkanihamed (); freedmanrecursion () allowed the authors of Ref. drummondtrees () to expose SYM tree amplitudes in a manifestly dual superconformal form. Although infrared divergences spoil the dual conformal properties at one loop, in Ref. drummondloop () the authors develop a supersymmetric version of generalized unitarity. The ratio between the NMHV and MHV one-loop superamplitude is a dual conformal invariant drummondloop (); brandhuberdualconf (); freedmandualconf (). The use of covariant, on-shell superspace allows the supersymmetric sums over states crossing unitarity cuts to be written as Grassmann integrals drummondsym (). Diagrammatic methods for directly computing such sums are presented in Ref. bernsupersums (). In the present paper, we apply generalized unitarity in on-shell superspace to calculate the NNMHV amplitudes for SYM at one loop.

The on-shell superspace description of maximally supersymmetric Yang-Mills requires only minor modifications to be applied for SUGRA. The contributions of MHV and non-MHV amplitudes to SUGRA scattering are likewise classified as coefficients of Grassmann-valued polynomials. In Ref. drummondsugra () the authors invented “ordered subamplitudes” for SUGRA tree amplitudes. The subamplitudes are added together with permutations of of the external legs to yield a physical amplitude. The use of ordered subamplitudes allowed efficient application of on-shell recursion relations, and the authors present the MHV, NMHV and NNMHV contributions to SUGRA tree amplitudes. Echoes of the intriguing squaring relationship between gauge theory and gravity berngaugegravity (); gaugegravity (); freedmanmhv () are observed, since the on-shell recursion relations are seeded with MHV and amplitudes which are both proportional to squared SYM tree amplitudes.

In this paper we present the planar, one-loop contributions to -point NNMHV scattering amplitudes in Super Yang-Mills (SYM) and Supergravity (SUGRA) theories. Generalized unitarity allows us to utilize the compact representations of tree level scattering amplitudes obtained previously through the use of on-shell recursion relations drummondtrees (); drummondsugra (). Scattering amplitudes for SYM at weak and strong coupling are conjectured to possess dual superconformal symmetry, a new symmetry beyond the familiar supersymmetry and conformal invariance. Our results for the one-loop amplitudes of SYM confirm that the NNMHV box coefficients are covariant under dual superconformal transformations.

We prove that ordered tree amplitudes for SUGRA may be sewn together to yield “ordered box coefficients” via generalized unitarity. The ordered box coefficients yield physical box coefficients after adding the permutations of all external legs. We calculate explicit expressions for the ordered box coefficients which contribute to SUGRA at one loop.

This paper is organized as follows. We begin with a review of the on-shell, covariant superspace formalism for describing scattering amplitudes in SYM and SUGRA. In this framework the contributions to SUSY amplitudes of Grassmann degree generate the scattering amplitudes. Next we describe the tree-level SYM amplitudes we require and review generalized unitarity in the construction of supersymmetric box coefficients. We calculate the NNMHV box diagrams for SYM and present the box coefficients. Our results for supersymmetric scattering amplitudes are expressed as Grassmann-valued generating functions, exploiting the on-shell superspace formulated by Drummond, Henn, Korchemsky, and Sokatchev drummondsym ().

In order to efficiently calculate SUGRA box coefficients, we introduce “ordered box coefficients.” The ordered box coefficients are formed by fusing ordered tree-level subamplitudes via unitarity. After summing over external leg permutations, the ordered box coefficients yield physical quadruple-cut coefficients. We proceed then to write the MHV, NMHV, NNMHV box coefficients for SUGRA in the on-shell superspace language. We then present a simple formula for extracting gluon and graviton scattering amplitudes from the NNMHV superamplitudes. Finally we describe the checks we have performed in comparison with amplitudes in the literature.

Ii Preliminaries

ii.1 Spinor helicity formalism

In order to efficiently utilize the four-dimensional polarization and momenta data for a scattering process, we describe amplitudes in the spinor helicity formalism for massless particles. In this formalism we write the Weyl spinors and for a particle with complex and null momentum as


suppressing the spinor indices. The convention we use is that and have helicity weights of , respectively. This helicity assignment is consistent with the polarization vectors for an on-shell particle with momentum , with reference spinors and ,


The null momentum is written in this formalism as a bi-spinor,


The Lorentz invariant spinor inner-products between spinors for the particles labeled and are denoted


In this formalism the spinor products for strings of momenta and spinors are expressed as, for example,


ii.2 Covariant description of on-shell superspace

Now we discuss the manifestly Lorentz covariant description of the multiplet of massless states as formulated by Drummond, Henn, Korchemsky, and Sokatchev drummondsym (). The covariant description utilizes the bi-spinor representation of a complex and null momentum in four dimensions,


Then the supersymmetry algebra generated by and for is written as


In reference to the null momentum , the spinor components of can be decomposed into two linearly independent spinors, , one parallel and one orthogonal to the spinor . The parallel component satisfies , and one defines the operator through the relation . Then the operator-valued part of is chosen to be . Similar considerations apply for the decomposition of relative to the spinor . Forming the spinor product with and on both sides of the supersymmetry algebra eqn. (7), one finds the algebra


Then we can identify a maximal, mutually anticommuting set of operators (i.e. annihilation operators) to be either or .

The vacuum state is defined as the state annihilated by the chosen set of annihilation operators and with helicity with respect to the null vector . Following the convention established in Ref. drummondsym (), we choose the set of annihilation operators . The remaining operators are creation operators whose action changes the helicity of a state by . Then must carry helicity to be consistent with the relations eqn. (II.2). With a vacuum state of helicity , the states created by repeated application of have helicities ranging from and produce the multiplet of massless states, self-conjugate under CPT.

The algebra eqn. (II.2) is conveniently realized by Grassmann variables satisfying anticommutation relations . The operators are identified with


The Grassmann variables transform according to the fundamental representation of the global -symmetry group of SYM.

All the component states of the on-shell supermultiplet can be assembled into a single super-wavefunction,


The different states in the supermultiplet are obtained from the super-wavefunction by the action of . For example, the positive-helicity gluon state is , and the negative-helicity gluon state is given by . Each component particle of the supermultiplet is distinguished by a unique power of . Because the Grassmann variables have helicity , each term in the super-wavefunction of eqn. (II.2) has total helicity . The momentum is chosen by convention to be an outgoing momentum for scattering processes. A physical particle’s momentum is null and future pointing.

An exercise in Grassmann integration yields an important identity, the single-particle completeness relation,


We will apply this identity to write the sum over each on-shell state in the multiplet as a Grassmann integral. Thus the discrete sum over particle states which cross an on-shell line in unitarity cuts and on-shell recursion relations is replaced by integration.

The SYM scattering amplitudes for external superparticles are denoted


The superamplitude is a generating function for the scattering of all particles in the multiplet, since the contribution of various component fields to a superamplitude are distinguished by the particular Grassmann-valued coefficients appearing in the superfield, eqn. (II.2). For example, a gluon MHV amplitude appears as


As discussed in Refs. freedmanmhv (); freedman (), the component particle scattering amplitudes are obtained by applying Grassmann-variable derivatives to a superamplitude. Equivalently, as in Ref. drummondtrees (), Grassmann integrations can be used to isolate a component scattering amplitude. Noting that


and referring to the component fields in eqn. (II.2), negative-helicity gluon contributions, for example, to an amplitude are selected by the Grassmann-integration while positive-helicity gluons are indicated by a factor of unity. Thus MHV, NMHV, and NNMHV gluon amplitudes are given, respectively, by


Because the multiplet is CPT self-conjugate, the same on-shell supermultiplet could have been obtained by using the triplet of annihilation operators with the creation operators . In that case we would write


The Grassmann variables transform in the anti-fundamental representation of . The conjugate super-wavefunction is


For complex momentum, the super-wavefunctions and are related by a Grassman-variable Fourier transform,


The conjugate description of a superamplitude may likewise be used, where the conjugate is obtained by the replacements and ,


A pair of conjugate superamplitudes are related by a Grassmann Fourier transform,


All scattering amplitudes in this paper are given in the “holomorphic” description, where every particle is described by the super-wavefunction .

An -point Super Yang-Mills amplitude is invariant under the supersymmetry algebra generated by


where and the total momentum is . The superamplitude can be expressed, for , as


where -supersymmetry requires that . For superamplitudes, invariance under -supersymmetry is manifest because of the delta function. The exceptional three-point amplitudes are shown in detail below.

The -symmetry for SYM implies that the superamplitude is an singlet. Then is expanded in a series of -invariant, homogeneous polynomials of degree in the ’s,


The invariance under the supersymmetry can be used to set to zero two of the variables, corresponding to two external particles in , so that the total Grassmann degree of the polynomial is . Thus the Grassmann degree of the superamplitude, including the factor of , is .

Next we will identify the scattering amplitudes of the component particles in the supermultiplet with the Grassmann polynomials . With each superparticle carrying total helicity , the total helicity of an -point superamplitude is . The scattering amplitude in eqn. (22) is the product of a momentum delta function with zero Grassmann-variable and spinor helicity, and the supercharge delta function has Grassmann-variable helicity of and spinor helicity of . Since the momentum and supercharge delta functions in eqn. (22) carry total helicity zero, each Grassmann polynomial has Grassmann-variable helicity and thus spinor helicity .

The scattering amplitudes for the component particles of the supermultiplet are obtained as coefficients of Grassmann-polynomial factors in the superamplitude. According to the above helicity count, the component-particle amplitudes arising from in this way have spinor helicity . In other words, the Grassmann polynomial yields a generating function for the amplitudes,


We will use the following less-concise but simpler notation,


to make the relationship between Grassmann polynomials and component amplitudes explicit.

Momentum conservation for three-point vertices, places exceptional constraints on the particles’ spinors. By contracting the momentum conservation condition with, say, , we have


and similarly for contractions with and . Instead contracting with we find


and similarly for contractions with and . Both sets of conditions taken together amount to the trivial solution where all momenta vanish. This is because eqn. (26) and its companions imply that , , and are all proportional, which, considered together with eqn. (27) and its companions, would imply the vanishing of , , and . Therefore we must choose one set of solutions, eqn. (26) and its companions or eqn. (27) and its companions, at a three-point vertex.

Iii Scattering Amplitudes for Sym

iii.1 SYM tree amplitudes

The tree-level MHV amplitudes of SYM are given by the generating function of Nair, presented in Ref. nair (). The corresponding Grassmann polynomial of degree zero is


The MHV amplitudes have obvious -supersymmetry because of the factor, and the -supersymmetry follows from momentum conservation. MHV three-point tree amplitudes are well-defined only for the kinematics of eqn. (27).

The exceptional three-point vertex, the Grassmann-variable Fourier transform of the conjugate three-point MHV vertex, has a Grassmann degree of four,


The vertex requires using the kinematic constraints of eqn. (26). By virtue of these constraints,


which annihilates eqn. (29) to ensure -supersymmetry. The -supersymmetry follows from applying the Schouten identity,


We will be using the formulas for tree amplitudes deduced from the application of on-shell recursion relations. Here we review the SUSY generalization brandhubersuperconformal (); arkanihamed () of the BCFW recursion relations bcf (); bcfw (). A superamplitude becomes a meromorphic function of the complex variable under the shift of external-particle spinors and Grassmann variables,


This complex shift is chosen to preserve overall momentum and the supercharge ,


The Feynman diagram representation of scattering amplitudes implies that has simple poles at the values of which yield an internal line with on-shell momentum ,


The values of the shift parameter which yield the multi-particle poles are denoted .

Both SYM and SUGRA amplitudes have the remarkable ultraviolet behavior that vanishes as , which implies, for the contour at infinity,


Then we apply Cauchy’s Theorem, deforming the contour to the origin to yield residues for the the multiparticle poles at and at . Scattering amplitudes factorize at multiparticle poles, and the residue at is simply the desired, unshifted scattering amplitude. Then we arrive at the BCFW recursion relations in their supersymmetric form,


The sum over intermediate particle states has been written as an integration over the internal particle’s Grassmann variable.

In Refs. drummondtrees (); drummondsugra (), Drummond et al. use the supersymmetric BCFW recursion relations to develop a graphical algorithm for writing the SYM tree amplitudes. All SYM amplitudes are dual superconformal invariant, depending on the dual conformal invariant functions




The dual-superspace variables and are related to superspace momenta and spinors by


In an -point superamplitude the subscript indices of the dual conformal invariants which appear range over the values . When the index attains the lower limit of its range, the spinor is to be modified according to the superscript indices ,


Dual conformal invariants with no superscripts present require no modification. We note for later that the dual conformal invariants (and their modified versions) depend on only through the spinor and have phase weight zero in .

For our present purposes we need only the NMHV and NNMHV tree amplitudes. The dual conformal invariants which appear in these amplitudes are given explicitly by


where . The NMHV and NNMHV tree amplitudes in SYM are generated by the Grassmann-valued polynomials


We are using a convention for double summations where it is understood that in a sum .

In order to carry out the Grassmann integrals appearing in unitarity cuts, it will be important to notice that the dual conformal invariant factors in the SYM tree amplitudes are independent of the Grassmann variables and . The dual conformal invariant functions all share the property that they depend on the Grassmann variables through the dual superspace coordinates , where the indices range only over the values . In this form of presenting the amplitudes, therefore, the Grassmann variables for the external particles and appear only in the overall supersymmetric delta function.

iii.2 SYM one-loop amplitudes

Figure 1: The kinematics used for the box coefficients. All external legs are outgoing and the loop momenta point counterclockwise. Gray blobs indicate on-shell tree amplitudes.

The planar, color-ordered, one-loop amplitudes in SYM can be decomposed onto a basis of scalar box integrals bernunitarity (); bernfusing () with Grassmann-valued box coefficients,


The summation runs over all possible distributions of the color-ordered external particles, and the dimensionally-regularized scalar box integrals are


As illustrated in Fig. 1, the are sums of the momenta leaving each corner of the box. The four-mass () integrals correspond to for all four corners; three-mass () integrals have for exactly one corner; two adjacent have in the two-mass hard () integral, and two opposite corners have vanishing in the two-mass easy () integral; one-mass () integrals have for three corners of the box. We will frequently use a slight abuse of notation, using the symbol to indicate both the total momentum leaving the -th corner and also the set of external-particle labels for that corner. For example, we write or depending on the context.

The Grassmann-valued box coefficients are given by the quadruple cuts of the superamplitude ,


Each is a SYM tree amplitude, with external particles in the cluster , stripped of its momentum-conserving delta function. The unitarity cut coefficient contains a sum over all the component particles of the supermultiplet which cross each cut loop momentum, distinguished by the spin of each particle.

There are two solutions for the complex momenta satisfying the on-shell conditions for the cut loop momenta, . The pair of general solutions for each loop momentum are given in BCFUnitarity (). When one corner of the box is massless, the two solutions are given in a simple form by the authors of Ref. blackhat (). For numerically checking our results against seven-point gluon and six-point graviton amplitudes the unitarity cuts with at least one massless corner are sufficient. Considering the routing of momenta we use in Fig. 1, the solutions are expressed in terms of the spinors and for the massless corner ,


Here we use the spinor notation . Noticing that the two solutions are distinguished according to whether or , we see that the kinematic solution is applicable when is an vertex and is used for a MHV three-vertex. In the equations for box coefficients that follow, we leave implicit the sum over appropriate loop momenta solutions.

The contribution from each on-shell particle in the supermultiplet which crosses a unitarity cut is conveniently calculated by an integral over the Grassmann variable of each superparticle, as indicated by the completeness relation of eqn. (II.2). Then the unitarity cuts contributing to a box coefficient take the general form


In the case of superamplitudes as given by eqn. (22), the loop Grassmann variables will appear in delta functions of the form for the cluster of external legs at the corner of the box. To carry out the Grassmann integrations which appear in a box coefficient, eqn. (47), we will apply the identity


The pair of Grassmann delta functions simply freezes the value of the loop variables and .

Certain configurations of on-shell three-vertices which could appear in the unitarity cuts are forbidden because of the kinematic constraints of eqns. (26) and (27). If two on-shell MHV, or , three vertices are adjacent and thus share a common particle line, the special kinematic constraints would require that the pair of external particles at these vertices must have spinors , or , respectively, which are proportional. General kinematics does not allow such a restriction. Quadruple cut diagrams with an and vertex at opposite corners also vanish for kinematic reasons.

Figure 2: The quadruple-cut diagrams which contribute to NNMHV box coefficients. tree amplitudes are labeled with the number , and and MHV tree amplitudes are indicated by black and white blobs, respectively. The diagrams are indicated in the text with the labels , , and , respectively. The last pair of diagrams, which differ by a simple relabeling, are collectively denoted .

iii.3 NNMHV box coefficients for SYM

Now we describe all the unitarity cuts which contribute to a NNMHV box coefficient, as in eqn. (47). The NNMHV superamplitudes have total Grassmann degree of , whereas the Grassmann degree of vertices, MHV amplitudes, and NMHV amplitudes are , , and , respectively. The Grassmann loop integrations for the unitarity cuts each reduce the total Grassmann degree by four, for a net contribution of . Thus to have a NNMHV one-loop superamplitude we require the box coefficients to be built from tree amplitudes with total Grassmann degree of . This is achieved in four different ways; there can be four MHV tree amplitudes, one NMHV with two MHV and one tree amplitude, one NNMHV with one MHV and two tree amplitudes, or two NMHV and two tree amplitudes. The kinematic restriction on vertices which share a common particle forbids the case with three vertices and a tree amplitude. Fig. 2 illustrates all the box diagrams required for the NNMHV one-loop superamplitudes.

iii.3.1 All-MHV cut contributions to the NNMHV box coefficients

The unitarity cuts built from four MHV tree amplitudes have Grassmann degree and thus contribute to the NNMHV one-loop SYM amplitude. The result for this cut coefficient is presented in Ref. drummondloop (). After calculating the four-mass, all-MHV cut contribution we can immediately obtain the all-MHV cut contributions to the three-, two-, and one-mass box coefficients by simply restricting the number of external particles at each tree. We write to denote the only contribution to the four-mass box coefficient, where , , , and , and we have


The sum of the delta-functions’ arguments yields the supersymmetric delta function . We replace the argument of the first delta function with and set the first delta function aside. Now we use eqn. (III.2) to factor the three remaining functions into pairs of functions to find that their product is