Contents

MIT-CTP-3943

ROM2F/2008/13

CERN-TH-2008-096

Generating Tree Amplitudes in SYM and SG

Massimo Bianchi, Henriette Elvang and Daniel Z. Freedman

Physics Department, Theory Unit, CERN

CH1211, Geneva 23, Switzerland

[2mm] Dipartimento di Fisica,

Universitá di Roma “Tor Vergata”

I.N.F.N. Sezione di Roma “Tor Vergata”

Via della Ricerca Scientifica, 00133 Roma, Italy

[2mm] Department of Mathematics

Center for Theoretical Physics

Massachusetts Institute of Technology

77 Massachusetts Avenue

Cambridge, MA 02139, USA

[2mm] Department of Applied Mathematics and Theoretical Physics

Centre for Mathematical Sciences

Cambridge University

Cambridge CB3 0WA, United Kingdom

[5mm] Massimo.Bianchi@roma2.infn.it, elvang@lns.mit.edu, dzf@math.mit.edu

We study -point tree amplitudes of super Yang-Mills theory and supergravity for general configurations of external particles of the two theories. We construct generating functions for -point MHV and NMHV amplitudes with general external states. Amplitudes derived from them obey SUSY Ward identities, and the generating functions characterize and count amplitudes in the MHV and NMHV sectors. The MHV generating function provides an efficient way to perform the intermediate state helicity sums required to obtain loop amplitudes from trees. The NMHV generating functions rely on the MHV-vertex expansion obtained from recursion relations associated with a 3-line shift of external momenta involving a reference spinor . The recursion relations remain valid for a subset of supergravity amplitudes which do not vanish asymptotically for all . The MHV-vertex expansion of the -graviton NMHV amplitude for is independent of and exhibits the asymptotic behavior . This presages difficulties for Generating functions show how the symmetries of supergravity can be implemented in the quadratic map between supergravity and gauge theory embodied in the KLT and other similar relations between amplitudes in the two theories.

1 Introduction

Recent calculations and conjectures [1, 2] on the possible ultraviolet finiteness of supergravity theory motivate a search for simplifications of the difficult perturbative calculations needed for further progress.111There are also earlier relevant calculations [3] as well as more recent work [4, 5]. Additional references are given in [1, 2, 4]. Three important techniques used in those calculations are the following:

  1. The integrands of loop diagrams are constructed from tree amplitudes using generalized unitarity cuts. Even when external lines are gravitons, the unitarity sum includes processes involving all possible states of the supergravity theory. New information on these tree amplitudes can be helpful at the loop level.

  2. On-shell tree amplitudes in gauge theory and gravity are best expressed using the spinor helicity formalism and are most easily obtained from the modern form of recursion relations [6, 7, 8, 9] which relate -point amplitudes to those for smaller values of . The simplest expressions appear in the MHV sectors of each theory, but perturbative calculations have reached the point where NMHV amplitudes are required. These have been studied for external gluons and gravitons, but there is less information on amplitudes involving other particles of the theory.

  3. Relatively complicated supergravity trees are constructed from the simpler tree amplitudes of super-Yang-Mills theory using the quadratic relation between gravity and gauge theory embodied in the KLT relations [10]. Implicit in this relation is a map between two copies of the gauge theory and supergravity which we denote by

    (1.1)

    There are 16 distinct particle states in each SYM factor and 256 states in SG.

This paper is motivated by all three issues above. We focus on the construction of -point MHV and NMHV tree amplitudes in SYM and supergravity with general external states. Toward this end we develop and generalize to supergravity the generating function for MHV amplitudes in gauge theory discovered in [11] and further developed and extended to NMHV amplitudes in [12]. The generating functions encode the external state dependence in a compact way and furnish precise characterizations of the MHV and NMHV sectors. To entice the reader we pose three questions to which the formalism gives simple answers. The MHV sector of SYM consists of the -gluon amplitude with two negative helicity gluons plus all amplitudes related by SUSY transformations. Would the reader have guessed that this sector contains the 8-point amplitude with 8 +ve helicity gluini? In supergravity the MHV sector consists of all amplitudes related by SUSY to the -graviton amplitude with two negative helicity gravitons. Would the reader have guessed that there are 186 distinct processes,222For the number of processes is smaller. each with a different set of particles, in this sector? And would the reader have anticipated the external state dependence of -point MHV amplitudes has a simple direct relationship to the properties of -point CFT correlators?

Generating functions provide useful answers to a number of questions, and they appear to have practical applications. For example, the unitarity sums over intermediate states required to obtain 1-loop Feynman integrands from the product of tree amplitudes in both gauge theory and supergravity can be done quite efficiently using the generating function.

The generating function for -point amplitudes in gauge theory is an invariant function of the momenta and of Grassmann variables . Here refers to the momentum of each external particle and is the flavor index. The generalization to gravity is straightforward in the MHV sector, in which the generating function is an invariant function of Grassmann variables where is an index. It is very simple to calculate any MHV amplitude from the generating function by applying Grassmann derivatives specific to the external states. All symmetry transformations can be implemented at the level of the generating function as operations involving the and variables, and one can show that all amplitudes automatically satisfy SUSY Ward identities.

The NMHV sector of gauge theory (or respectively, supergravity) consists of all amplitudes linked by SUSY Ward identities to the -gluon amplitude (or respectively, the -graviton ) with 3 negative helicity particles. The construction of a generating function is formally straightforward in the NMHV sector, but its justification is more subtle. There is a different generating function for each diagram in the MHV-vertex expansion of an amplitude. The MHV-vertex expansion was first obtained in () gauge theory in [13] and extended to gravity in [14]. The contribution of each diagram depends on the choice of an arbitrary reference spinor , but the full amplitude, which is the sum of all diagrams, should be independent of .

The simplest justification of the expansion comes from the recursion relation associated with a complex shift of the spinors of the negative helicity lines [15]. The required shift is

(1.2)

where are the cyclic permutations of the momentum labels for a choice of three of the external lines. For pure gluon or graviton amplitudes, these will be the three negative helicity particles. The recursion relation, and therefore the diagrammatic expansion, is valid if the continued amplitude vanishes as This desired property was proven for gluon amplitudes in [13, 15], but was observed in numerical calculation of the graviton amplitudes only for in [14]. It is also known for simpler shifts of two external momenta that the large falloff is slower for amplitudes in which some gluons, or gravitons, are replaced by lower spin particles of the supermultiplet. For these reasons we must be cautious in our applications of the MHV-vertex expansion.

If an amplitude vanishes as for all choices of , Cauchy’s theorem ensures that the sum of MHV-vertex diagrams is independent of . For all NMHV amplitudes in SYM we show that there is always a choice of 3 lines to shift such that the contribution of each diagram falls at least as fast as . We have verified -independence of the expansion numerically for a large number of 6-point NMHV amplitudes. Thus we detect no problems, and the generating function appears to be valid for the whole NMHV sector of the theory.

In gravity the situation is more problematic. For graviton amplitudes we show that the falloff as depends on the number of external legs . Specifically, we have verified numerically for that

(1.3)

This means that the MHV-vertex decomposition of the -graviton NHMV amplitude can be expected to fail for . Indeed for we find that the sum of 1533 MHV-vertex diagrams fails to be independent of .

The evaluation and summation of diagrams is more complicated for general external states in supergravity  so our analysis is limited to 6-point NMHV processes. There are 151 such processes, each with several functionally independent amplitudes obtained by inequivalent assignments of indices to the external particles. For each amplitude there are up to 21 non-vanishing diagrams. Most 6-point amplitudes have the same good properties as those of gauge theory; they vanish under large shifts, and they are constructed correctly using the MHV-vertex expansion with diagrams obtained from the generating function.

The large behavior of individual diagrams for any amplitude can be determined analytically. The result depends on which set of 3 lines are shifted. Our analysis shows that there are processes for which even the best shift contains diagrams with either or behavior at large . Numerical evaluation can then test whether the sum of diagrams depends of . This would indicate that the undesired large behavior persists in the full amplitude, and we have found that it does for a number333Of the total of 151 6-point NMHV processes, we estimate that about half will include amplitudes with asymptotic behavior. of examples. In these cases we recalculate the amplitude using the KLT formula which provides a correct evaluation of any amplitude as a sum of products of SYM amplitudes. The result from KLT can be continued to complex momenta by shifting spinors and the large behavior is then extracted. By this method444We have automated the process by writing a Mathematica code which evaluates the KLT expansion as well as the MHV-vertex decomposition for any 6-point NMHV amplitude of the theory. we have explored about 20 amplitudes whose best shifts give asymptotic behavior. We call these cases “bad” amplitudes, as opposed to “good” amplitudes which vanish asymptotically for one or more 3-line shifts. The large limit of these “bad” amplitudes is a ratio of polynomials in the reference spinor . The amplitude does not vanish asymptotically for all , but it does vanish when is chosen to be a root of the numerator polynomial. The recursion relation becomes valid for these special values of , and the sum of MHV-vertex diagrams then agrees with the KLT evaluation. In this way we have developed a good interpretation, and justification, of the generating function even for “bad” amplitudes.

Finally we must mention that our analysis locates two “very bad” 6-point NMHV amplitudes whose KLT evaluations show linear growth in as . There are no values of which validate the MHV-vertex decomposition, so the generating function is not useful. However, the SUSY Ward identities can be used to express each “very bad” amplitude as a sum of two other amplitudes which are constructible via the MHV-vertex expansion and generating function.

Complications with the large behavior in supergravity suggest that it may be difficult to apply the generating function to intermediate state helicity sums involving NMHV amplitudes. It is important to explore this question, but it is beyond the scope of the present paper.

Let’s return to the map (1.1) because another focus of this paper concerns how the supersymmetry and global symmetry of supergravity are implemented in the tensor product of gauge theory states. One question of concern is how the flavor symmetry of the product of gauge theory factors is promoted to the global symmetry of supergravity. The derivation of the KLT relations from string theory does not really settle this question, since only emerges as an accidental symmetry in the limit.

To investigate such questions we write the detailed algebra of the SUSY charges and the annihilation and creation operators of the gauge and supergravity theories and provide a detailed version of the map (1.1) which is compatible with these symmetry operations. In the map, any index on the supergravity side splits into in the left () factor of the gauge theory and in the right () factor. Although not manifested in this split, transformations can be formally implemented on the gauge theory side of the map of states. We take the attitude that the implementation of is better tested on amplitudes, for example through the KLT relations, which read for ,

(1.4)

To apply these to a supergravity process, one places the images of the supergravity operators under the map (1.1) into the gauge factors on the right side of the relations. We will discuss one example, although the notation is not fully described until section 2. Consider the scattering amplitude of two gravitons, and , and two graviphotons, and , with helicities as indicated. The gauge theory images of these operators involve gluons , gluinos , and scalars , and the images of the graviphotons depend on whether the indices lie in the range or . In other words, the helicity-1 particles can decompose either as or as . Using the KLT result (1.4) leads to the formulas

The supergravity amplitude is proportional to the antisymmetric tensor , so the product of two bosonic amplitudes in the first expression must equal (to within a sign) the product of fermion amplitudes in the second. This agreement is not a miracle. It must work because the KLT relations are derived from the low energy limit of superstring theory. Nevertheless we are happy to see the sometimes intricate way it does work in this and several other examples we have studied.

The generating function enables us to go beyond examples and give a simple argument that all supergravity symmetries are consistent with the map (1.1). In the MHV sector the supergravity generating function factors into the product of gauge theory generating functions as

(1.5)

Symmetry transformations of supergravity, written in terms of the variables, automatically work correctly when the split into and , and the transformations applied to the product of gauge theory generating functions on the right side of (1.5). The situation is somewhat more complicated, but very similar in the NMHV sector, where factorization occurs at the level of diagrams.

The plan of the paper is as follows. In section 2 we discuss the algebra of supercharges and the annihilation operators in gauge theory and supergravity and then the operator map. We also discuss the derivation of SUSY Ward identities and their application in the MHV sector. In section 3 we derive the generating functions for the MHV sectors of gauge theory and gravity. An application to the intermediate state helicity sums is presented in section 4. The connection between state dependence of MHV amplitudes and CFT correlators is discussed in section 5. We turn our attention to NMHV amplitudes in section 6. We first discuss recursion relations, especially those derived from (1.2) which lead to the MHV-vertex expansion. Using this we derive the NMHV generating function for gauge theory and discuss its properties. Then we define the NMHV generating function for gravity and discuss the key properties of independence of and behavior as . A discussion section concludes the main text. Our conventions are summarized in appendix A. In appendix B, we derive the solution of the SUSY Ward identities for 6-point NMHV amplitudes.

2 SUSY Ward identities and the operator map

In section 2.1 we set up our notation and present the and SUSY transformation rules for annihilation operators of the bosons and fermions of the gauge and supergravity theories we are concerned with. Further information about our conventions is given in appendix A. In section 2.2 we present the detailed correspondence between the products of pairs of gauge theory annihilators and the 256 annihilation operators in supergravity, and in section 2.3 we show how transformations can be implemented formally in the product space. We discuss SUSY Ward identities for on-shell amplitudes in SYM and supergravity in section 2.4. We show by example how to solve the Ward identities in the MHV sectors of the two theories.

2.1 Transformation rules of annihilation operators

We focus on annihilation operators because we adopt the common convention that all particle momenta in an -point process are viewed as outgoing. An amplitude, such as the -gluon MHV amplitude, can therefore be viewed as a string of annihilation operators acting to the left on the “out” vacuum. Thus if and are annihilation operators for gluons of energy-momentum and helicity , we can represent the color-ordered amplitude as

(2.1)

In general the amplitudes are regarded as functions of complex null energy-momentum vectors which may be continued to the physical region. If the energy-momentum is physical, i.e. a positive real null vector, then the operator (or ) describes a particle in the final state of a physical process. If is real, but negative null, then the operator corresponds to the anti-particle of opposite helicity in the initial state, which carries physical momentum .

The bosons and fermions of SYM theory are described by the following annihilators, which are listed in order of descending helicity:

(2.2)

The scalar particles are complex, and satisfy the indicated self-duality condition with . The gauge group of the theory is with all particles in the adjoint representation. Notation for the “color” degree of freedom is omitted, and we consider only “color-ordered” amplitudes.

The global symmetry group is , and we use upper and lower indices to distinguish the two inequivalent conjugate four-dimensional representations. To achieve an covariant notation, we separate the left and right chiral components of the supercharges and write them as and respectively. We then define and , where is the anti-commuting parameter of SUSY transformations. (See appendix A for details.) Note that .

We now state the independent commutation rules for the operators and with the various annihilators:

(2.3)

Note that raises the helicity of all operators and involves the spinor angle bracket in which is the dotted spinor for a particle of momentum . Similarly, lowers the helicity and spinor square brackets appear. Commutators with and are related by self-duality. The and operators generate independent Ward identities for -point amplitudes. We will primarily be concerned with those for .

For distinct SUSY parameters and , we define and . For any operator above, the SUSY algebra reads

(2.4)

Next we proceed in a similar fashion to discuss the transformation rules of supergravity, in which the annihilation operators for the 128 bosons and 128 fermions are

(2.5)

The 70 scalars satisfy an self-duality condition with . The notation is redundant, since the information on particle type and helicity is determined by the number and position of the global symmetry indices.

There are chiral spinor supercharges and which transform in the inequivalent 8 and representations. We contract these charges with a SUSY Grassmann parameter and define and . It is then straightforward to write covariant commutators with annihilation operators:

(2.6)

The supersymmetry generators satisfy (2.4) for any operator above.

Supercharge commutators with creation operators can be obtained as the adjoints of the relations given in (2.3) and (2.6). Phases in these commutators have been fixed to be compatible with crossing. Crossing symmetry relates an S-matrix element containing a particle with physical (positive null) momentum in the initial state to the amplitude containing its anti-particle with opposite helicity and unphysical (negative null) momentum in the final state. Thus the SUSY transformation of any creator must agree with that of the annihilator multiplied by the conventional [16] crossing phase of helicity amplitudes (which has the value only for negative helicity fermions). Note that spinors for negative null momenta satisfy .

2.2 The operator map

The precise operator map between is presented in Table 1. Operators in the gauge theory are dressed with tildes whereas the operators of the factor are undecorated. The entries in the map are determined, up to signs, by matching the helicity and global symmetry properties of supergravity operators with products of gauge theory operators. Unfixed signs are then determined by compatibility with the scalar self-duality conditions and especially by the consistent action of the supercharges of the and theories.

To discuss the implementation of the SUSY commutators we denote a generic annihilation operator by in supergravity and by and in the and copies of the gauge theory. The image of any under the map (1.1) is a specific product . A supercharge component from the first sector acts non-trivially only on , while from the second sector acts non-trivially only on . Thus we have the scheme

(2.7)

with similar definitions of the action of and . We then require that the left and right sides of (2.7) still map properly when the transformation rules of section 2.1 are used. This determines the signs of entries in Table 1.

Here are two examples, interesting because the two sectors mix. The first example is

(2.8)
(2.9)

This is compatible with the supersymmetry algebras because the right sides are images under the map (2.13). The other example is

(2.10)
(2.11)

After multiplication of the second equation by , we see that the map works properly. We have checked explicitly that all entries in the map are consistent with the transformation rules.

There is a choice of the scalar self-duality phases , , and in the supergravity theory and in the two SYM factors. It turns out that consistency of the map with the commutator algebras requires that

(2.12)

We leave , , and arbitrary in the map in Table 1, but in applications below we will often set .

(2.13)
Table 1: Operator map for annihilators of . Indices and refer to the splitting of into the two separate factors.

2.3 symmetry and the operator map

The generators of the fundamental representation of are the set of traceless matrices:

(2.14)

in which denote the Lie algebra element, and are row and column indices. The commutators are

(2.15)

The algebra decomposes with respect to the subgroup . We use indices ( and ). After a minor rearrangement of the basis of (2.14), we obtain a set of 63 matrices whose non-vanishing elements are

(2.16)

We now define the action of the corresponding Hilbert space operators on the states of the operator map. The generators and have the usual matrix action of , defined in (2.16), on gauge theory operators. Nothing special is required. Examples make things clear:

(2.17)

The remaining generators are more subtle, but very simple. They have no well defined action on single operators of the gauge theory, but we define their action on tensor products of gauge theory operators to match the appropriate supergravity states. The generator is diagonal on all states. Thus, for example,

(2.18)

For the mixed generators the definitions require changes from boson to fermion operators in each gauge theory factor. Hence

(2.19)

The consistency test for any claimed realization of is that the commutation relations (2.15) are satisfied. But this is certainly true here, by explicit construction, since our definitions simply track the conventional implementation of in supergravity.

This implementation of in the operator map is correct but formal. The acid test is that supergravity amplitudes constructed from gauge theory transform correctly. This requires the kind of non-miracle discussed in the introduction. The dynamical parts of products of very different gauge theory amplitudes must agree, and so must their group theory factors. To show that this non-miracle happens, we will use SUSY Ward identities.

2.4 SUSY Ward identities for on-shell amplitudes

To begin the discussion, we use the generic notation of [17]. An annihilation operator of SYM or supergravity is denoted either by or . The subscript indicates particle momentum, while helicity and global symmetry indices are suppressed. For a pair of supercharges of SYM with fixed index, an operator is defined as one for which is non-vanishing, and a operator is one for which is non-vanishing. It is clear that and . The division into - and -operators depends on the index choice . For example, the operators for the supercharge pair are

(2.20)
(2.21)

where . The definition of operators in supergravity is entirely analogous.

The basic Ward identities are simply the statements that, since supercharges annihilate the vacuum,

(2.22)

By adding and subtracting terms, we convert (2.22) into a sum of commutators or . We can then rewrite (2.22) as

(2.23)
(2.24)

Since the spinors have two components, the analytic and anti-analytic expressions each contain two independent constraints on the amplitudes. To obtain a useful identity one must start with a string of operators in (2.22) which contains an odd number of fermion annihilators. Then the individual amplitudes which appear in the constraints contain an even number of fermions. Otherwise they vanish trivially. The ordering of operators is relevant in gauge theory because amplitudes are color ordered, but it has no significance in supergravity.

Let’s consider the two cases in which the initial string of operators in (2.22) contains only one or two operators, respectively. Then the Ward identities read

(2.25)
(2.26)

We now exploit the freedom to choose the two-component spinor . We can choose it so that . Then (2.25) tells us that any amplitude which contains only operators must vanish. To exploit the information in (2.26) we choose, in turn, and then . We learn that any amplitude with operators and one operator must vanish. By similar arguments, we can use the Ward identity to show that any amplitude containing at most one operator must vanish. These statements comprise the well known helicity conservation rules for -point functions. For amplitudes containing only gluons, they read and .

Relations between different amplitudes are obtained when the initial string contains operators plus operators. The case of exactly three operators is particularly easy to analyze and very useful. The analytic Ward identity reads

(2.27)

As stated above this equation contains two independent relations among the three amplitudes involved. By choosing and then , we obtain

(2.28)
(2.29)

An example of these relations is the Ward identity in gauge theory. The two constraints above then become

(2.30)
(2.31)

Thus an amplitude containing a pair of opposite helicity gluinos is related to the well known MHV -gluon amplitude. For this reason the set of amplitudes with two operators and operators is called the MHV sector of the theory. Note that the gluinos can be placed in any positions by change in the placement of the three initial operators.

As another example of an MHV Ward identity in the gauge theory, consider

and use (2.31) to simplify the terms. With or we get two Ward identities: