1.05
MITCTP4042
Dual conformal symmetry of 1loop
[2.5mm] NMHV amplitudes in SYM theory
[2.5mm] Henriette Elvang, Daniel Z. Freedman, Michael Kiermaier
School of Natural Sciences
Institute for Advanced Study
Princeton, NJ 08540, USA
[5mm] Center for Theoretical Physics
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139, USA
[5mm] , ,
We prove that 1loop point NMHV superamplitudes in SYM theory are dual conformal covariant for all numbers of external particles (after regularization and subtraction of IR divergences). This property was previously established for in arXiv:0808.0491. We derive an explicit representation of these superamplitudes in terms of dual conformal crossratios. We also show that all the 1loop ‘box coefficients’ obtained from maximal cuts of NMHV point functions are covariant under dual conformal transformations.
Contents
 1 Introduction
 2 NMHV tree superamplitudes and superconformal invariants
 3 Oneloop superamplitudes and box coefficients
 4 NMHV oneloop superamplitudes
 5 Proof of dual conformal invariance
 6 Superconformal covariance of box coefficients for all oneloop NMHV superamplitudes
 7 Discussion and Conclusions
 A The ratio for
 B Scalar Box Integrals
1 Introduction
Dual conformal symmetry is a newly discovered symmetry of onshell scattering amplitudes in SYM theory. We will survey the development of the subject briefly below, but first we state the particular focus and result of this technical note.
We focus on the proposed [1] dual conformal (and dual superconformal) symmetries of colorordered planar point superamplitudes These encode the values of the gluon amplitudes with negative and positive helicity gluons, plus all amplitudes of the theory related to these gluon amplitudes by conventional Poincaré supersymmetry. Each external particle is described within the superamplitude by a null 4momentum , the associated Weyl spinors and a Grassmann bookkeeping variable with . We use Dirac ‘kets’ to denote spinor variables, e.g. and . Dual superconformal transformations act most simply on the dual bosonic and fermionic ‘zone variables’ and defined by
(1.1) 
It is a bit complicated to describe the action of this symmetry on loop amplitudes because these have infrared divergences, which break conformal properties. The MHV superamplitude is distinguished by its simplicity in the tree approximation and by the fact that, in a sense which we describe momentarily, it captures the ‘universal’ external momentum dependence of the IR divergence of all loop amplitudes. For these reasons it is useful to discuss general NMHV processes in terms of the ratio of superamplitudes:
(1.2) 
with We use , to denote the loop contributions. It is known that in tree approximation all transform covariantly under dual superconformal transformations [1, 2] and that the treelevel ratio is dual superconformal invariant and can be expressed [3] in terms of a set of superconformal invariants. These invariants were defined at the NMHV level in [1] and for general in [3].
It is a consequence of the Ward identities of conventional Poincaré supersymmetry that at 1loop order (and beyond), is the product of the tree factor times an IR divergent function of commuting variables only, i.e. no . It is the IR finite quantity which apparently enjoys simple properties under dual conformal symmetry. It was conjectured [1, 4] that is dual conformal invariant to all orders and for all .^{1}^{1}1Concerning dual superconformal invariance, see brief discussion in section 7. At 1loop order this conjecture was proven for NMHV amplitudes for or external particles. Explicit representations in terms of dual invariant crossratio variables were given for In this note we prove that the 1loop ratio is dual conformal invariant for all at 1loop order, and we obtain explicit expressions in terms of dual conformal crossratios.
Our approach is a generalization to all of the methods developed in [4]. Initially, we obtain a representation of the ratio that contains . Each invariant is multiplied by a linear combination of scalar box integrals, which is neither IR finite nor dual conformal invariant! The invariants are not independent, however, and we recursively solve the linear relationships among them to eliminate the terms of invariants from the ratio. The final form we obtain is then both IR finite and dual conformal invariant, and we express it in terms of dual conformal crossratios. terms, namely one for each dual superconformal invariant
Two strands of investigation have led to the present understanding of dual conformal symmetry. At weak coupling, there were early hints of dual conformal behavior in multiloop diagrams for the offshell 4point functions analyzed in [5]. Then the AdS/CFT correspondence was applied in [6] (see also [7, 8, 9, 10, 11]) to obtain the strong coupling limit of the planar 4gluon amplitude. The amplitude was found to agree with the expectation value of the Wilson loop evaluated on a closed polygonal path with vertices at , and with identified with the null momenta of the gluons. A polygonal Wilson loop embodies [12] an anomalous form of conformal symmetry due to the cusps at the vertices. Dual conformal symmetry of the scattering amplitude is simply conventional conformal symmetry of the Wilson loop.
The second line of investigation concerns dual conformal symmetry of onshell amplitudes at weak coupling. The symmetry was an important aspect of studies [13, 14, 15, 12, 16, 17, 18, 19] of the relation between higherloop MHV amplitudes and polygonal Wilson loops which were motivated by the BDS conjecture [20]. The extension to dual superconformal symmetry was proposed as a symmetry realized on superamplitudes in [1] and proven for MHV and NMHV tree approximation superamplitudes. Using recursion relations for superamplitudes [21, 2, 22], the dual superconformal covariance of NMHV tree superamplitudes was then proven in [2], and an explicit construction of in terms of dual conformal invariants was established in [3]. It is also interesting to note that conventional and dual superconformal symmetries combine into a Yangian structure [23].
It is well known that 1loop point amplitudes in SYM can be expressed in terms of scalar box integrals [24]. The box integral expansion of 1loop superamplitudes was initiated in [1] and carried out systematically in [4] using the method of maximal cuts [25] to compute the amplitude. The dual conformal invariance of the ratio was established for in [4].
Note Added: After finishing this work, we were made aware (in recent email correspondence with Gabriele Travaglini) of similar work of Brandhuber, Heslop, and Travaglini [26].
2 NMHV tree superamplitudes and superconformal invariants
We briefly review the needed tree level results. We begin by giving the form of NMHV tree superamplitudes^{2}^{2}2We always omit the conventional energymomentum function although superamplitudes are covariant under dual symmetries only if the transformations of these functions are included [1]. which are the ‘input data’ for the 1loop calculations:
(2.1) 
Unless otherwise stated, all line labels are understood (mod ), i.e. . The contain an explicit , which is a polynomial of degree 8 in the , while is a polynomial of degree . The complete polynomial is of order
The MHV () superamplitude is particularly simple, with [27]. Beyond the MHV level, several explicit representation of for arbitrary and are known.^{3}^{3}3The MHV vertex expansion (first introduced in [28]) provides an efficient representation of . The explicit MHV vertex representation for superamplitudes was presented, and its validity proven, in the recent papers [29, 30], generalizing the earlier NMHV level analysis of [31, 32]. A novel version of the MHV vertex expansion based on supersymmetric shift recursion relations was derived in [33]. In addition to these, there are also ambitwistor representations of the tree amplitudes [34, 35, 36]. The most convenient representation for the purposes of this paper, is that of [1, 4, 3] which expresses in terms of dual superconformal invariants. At the NMHV level, this representation is
(2.2)  
(2.3)  
(2.4) 
The are the superconformal invariants of [1] which depend on the differences of zone variables , that are expressed as bispinors above, and on . The and are defined for labels satisfying , , . Spinor indices are suppressed unless needed for clarity. At level NMHV, the are expressed in terms of more complicated superconformal invariants, see [3] and Sec. 2D of [37]. The arguments of [1, 2, 3] show that the transform covariantly under dual superconformal transformations.
Oneloop superamplitudes constructed from unitary cuts contain products of tree subamplitudes which are either NMHV (with ) as discussed above or 3point antiMHV superamplitudes of the form
(2.5) 
which is a polynomial of degree 4. Effectively, a 3point antiMHV subamplitude has , since it is a degree Grassmann polynomial.
3 Oneloop superamplitudes and box coefficients
All 1loop amplitudes of SYM can be expressed in terms of scalar box integrals [24], and the same is true of the superamplitudes in which they are packaged. Our analysis therefore starts with the representation
(3.1) 
Each term in the sum is a contribution of a box diagram
in which the first external line of subamplitude , containing lines, is denoted by , the first external line of subamplitude , containing lines, is denoted by , etc. The conventions are indicated in Fig. 1. The full superamplitude includes the contribution of all box diagrams with inequivalent partitions and, for each of these, a sum over the cyclic permutations of the external lines. We define the ‘external masses’ of the subamplitudes by , etc. Box integrals are classified by the number of nonvanishing external masses, with , which they contain. This classification will be given in detail in the next subsection.
For each box diagram one must integrate over the loop momentum and the variables for the internal particles. The maximal cut method [25] gives an efficient way to separate these tasks. The are simply 1loop Feynman integrals containing four scalar propagators, which are made dimensionless through the inclusion of an overall factor of , defined as^{4}^{4}4Note that and etc.
(3.2) 
The are IR divergent when one or more subamplitudes are 3point vertices. They are tabulated in Appendix B for various configurations of external lines. The box coefficients are obtained from the box diagrams with the four internal lines cut,
(3.3) 
The integrals produce a canceling factor of the ‘Jacobian’ , and the products in (3.1) then do not contain any factors of . The maximal cut conditions actually have two solutions for the momenta of the internal lines [25]. For certain subamplitude configurations only one of them contributes. When both contribute we must modify (3.3) by averaging over .
There is a simple and useful argument based on the order of the polynomials to ascertain which subamplitudes contribute to the 1loop superamplitude , which is an polynomial of order . The internal integrals in (3.3) remove ’s. Thus, if the subamplitudes are NMHV, we must have .
3.1 Box coefficients
In this section we use the results of [4] for internal integrals in the box coefficients (3.3). As first shown in [32], these integrals effectively automate the sum over the various internal particles of the theory which propagate in the loop. For each distinct type of box integral, one must carefully analyze the product of contributing tree subamplitudes. There are two essential properties. First, the Grassmann and factors of these subamplitudes are used to fix the values of the internal . Secondly, at arbitrary NMHV level,^{5}^{5}5We use to denote the factor of subamplitude . the are chosen not to include the Grassmann variables associated with internal lines, i.e. , and . That this is possible can be seen from the explicit representation given in [37] and it is discussed in Sec. 6 below. As a consequence of these two properties, the integrals can be carried out trivially.
It is common in all cases that an overall factor of naturally appears. Also, the various subamplitudes contain spinor products which combine into a factor of and cancel the in the definition of . Let us summarize the results for for the several types of box diagrams, with NMHV subamplitudes for general :

4m: For the 4m box, each subamplitude must be MHV or higher, so this box contributes only at the NMHV level and beyond. We postpone the discussion until Sec. 6.

3m: There is now a “massless” 3point subamplitude which is placed at . It is required by special kinematics to be antiMHV. The box coefficient can be written as
(3.4) One must keep the particle label (in)equalities in mind. All external line labels are defined mod .

2mh: In this case there are massless 3point subamplitudes at and . Thus we have the (in)equalities . There are two contributions which must be added: one with antiMHV and MHV, the other with the opposite arrangement. It is clear that these two boxes are just extreme cases of the 3m boxes. We therefore have
(3.5) 
2me: The massless subamplitudes and must be antiMHV for generic external momenta. The result for the 2me box is
(3.6) with , and .

1m: We take massless. There are two distinct contributions:

and are antiMHV. This is a special case of a 2me (with ).

and are MHV and is antiMHV. This is a special case of a 3m box.
The result is then
(3.7) with , , and .

3.2 Oneloop MHV amplitude
It follows from counting that only the type (a) 1m boxes and the 2me boxes contribute to the oneloop MHV amplitude. The 1m box has , so by (3.6) we have . Hence the 1m contributes + cyclic).
In the 2me box coefficient, , so with and . The 2me contribution to the amplitude is therefore + cyclic). The 1/2 compensates the overcounting due to the symmetry of the 2me box diagram.
The full result for the 1loop amplitude is then
(3.8)  
(3.9) 
Using the results for the box integrals in Appendix B one finds that the IR divergent parts satisfy a nice relationship, namely
(3.10) 
Using (3.10), the IR divergent part of the MHV amplitude can be written very compactly as
(3.11) 
With the results for in appendix B, and exploiting cyclic symmetry, one finds
(3.12) 
which is the ‘universal’ IR divergence of oneloop scattering amplitudes in SYM.
4 NMHV oneloop superamplitudes
We have discussed the fact that oneloop amplitudes are IR divergent and that this spoils dual conformal symmetry. On the other hand, the external state dependent IR singularity is encoded in a tree level factor: it is captured fully by . For this reason, the dual conformal properties of the ratio defined in (1.2) were studied in [4]. Thus we expand the quantities in (1.2) to first order in the ’t Hooft coupling and define:
(4.1)  
(4.2) 
The quantities are infrared divergent, but the ratio
(4.3) 
must be IR finite. The claim is that it is also dual conformal invariant. We will prove this for all .
The subamplitude factors that can occur in the box coefficients of Sec. 3.1 are restricted for oneloop NMHV amplitudes. This follows from the count given below (3.3). For the 3m box, subamplitudes , and are all MHV, so , and hence . The 2me box requires considerably more intricate arguments, since one of the massive subamplitudes is NMHV. As shown very nicely in [4], the 2me coefficient can be written as a certain sum over 3m coefficients. We summarize the results for all needed box coefficients:

with .

with and .

.
This takes to be NMHV and to be MHV. The reverse arrangement is related by cyclic symmetry and is thus accounted for after we sum over all cyclic permutations. Note that the sum over includes only the external legs on the subamplitude . 
,
i.e. with , and .
Every box coefficient is a linear combination of the invariants. The oneloop superamplitude is a sum over all contributing boxes, . We find that defined in (4.2) is
(4.4)  
It is useful to reorder the sums so that each individual is multiplied by linear combinations of box integrals, to wit
(4.5) 
When in the first line, the sum over is empty and we set it to zero.
Finally we need to subtract the universal IR divergence in order to obtain the ratio as defined in (4.3). The result is a compact representation of the ratio:
(4.6) 
with
(4.7) 
Unfortunately, none of the coefficients are dual conformal invariant, and all except are also infrared divergent.^{6}^{6}6IR finiteness of follows from (3.10). Thus it is not clear from the representation (4.6) that is dual conformal invariant — or even IR finite! We now proceed to bring (4.6) to a manifestly dual conformal invariant form. The IR finite coefficient will play a special role in our analysis.
5 Proof of dual conformal invariance
The main tools for our manipulations of are cyclic symmetry and the fact that there exist linear relations among the invariants. Our strategy, which is similar to that in [4], is to use these properties to eliminate certain from the sum in (4.6) and thus obtain a new representation of in which the combinations of box integrals that appear are both IR finite and dual conformal invariant. Two steps are required to find this representation. First, we use the cyclic sum in (4.6) to make one term in the sum manifestly cyclically invariant. Then we systematically use linear relations between the to reduce the sum to a smaller set of contributing invariants.
5.1 Using cyclicity
The overall cyclic symmetry of (4.6) allows us to eliminate in favor of the cyclically invariant , defined as
(5.1) 
We choose to eliminate because its coefficient is already IR finite. This step leads to the modified representation
(5.2) 
Note that now no longer appears in because its coefficient vanishes.^{7}^{7}7 For the special case , there is no in (4.6), and (5.2) is then to be understood with . Although it is far from obvious, is our first dual conformal invariant combination of box integrals. It will be expressed in terms of dual crossratio variables in (5.23) below.
5.2 Eliminating ‘dependent’
We now eliminate all with and all with plus their cyclically related companions. We have already replaced and its cyclic companions by , which eliminates additional invariants. So in total, we eliminate distinct . We first eliminate , then in a separate step .
To eliminate the , we use the identity [1, 4]
(5.3) 
which implies
(5.4) 
The operator is defined as the shift of all indices of the object immediately to its right. Applying (5.4) to the ratio (5.2), we obtain
(5.5) 
Here, we have used the overall cyclic sum to convert shift operators acting on invariants into shift operators acting on scalar box integrals in the coefficients .
The steps taken so far are essentially all that is needed to establish dual conformal symmetry for the cases . In Appendix A we show that the representation (5.5) agrees with the forms given in [4]. For the rest of this section we assume that .
For , the coefficients in the representation (5.5) of the ratio function are not all finite and dual conformal invariant. For example, all invariants with have coefficients with IR singularities:
(5.6) 
Fortunately, we can also eliminate all these with IRdivergent coefficients from the ratio (5.5). Indeed, the identities^{8}^{8}8Considerable evidence for this identity was given in [4]. It was proven in the very recent [26].
(5.7) 
can be used in conjunction with (5.4) to eliminate all () without reintroducing the already eliminated invariants or . To see this, we proceed as follows. For , we solve (5.7) for and obtain
(5.8) 
For , we solve (5.7) for and use (5.8) to eliminate . This gives
(5.9) 
This approach can be generalized to all . We solve (5.7) for , and eliminate all with using the previously obtained identities. An inductive argument then shows that, for the full range ,
(5.10) 
We note that none of the (cyclic permutations of) or that we want to eliminate appear on the right hand side of (5.10).
5.3 Manifestly dual conformal invariant amplitude
Using (5.8) and (5.10) to eliminate all with from in (5.5), we obtain our final form for the ratio:
(5.11) 
with
(5.12) 
In all coefficients , empty sums are understood to vanish.
All coefficients in (5.12) are finite and dual conformal invariant. To verify this we checked that the infrared divergences cancel in all coefficients, and we then showed that each of them is invariant under the conformal inversion, which acts on zone variables or invariant squares of their differences as
(5.13) 
Inversion symmetry guarantees that all coefficients can be expressed as functions of the dual conformal invariant crossratios:
(5.14) 
In the next section we list these expressions and indicate briefly how they were obtained. In particular, we use the dilog identity
(5.15) 
5.4 Coefficients in terms of dual conformal crossratios
To express the coefficients with in terms of dual conformal crossratios, we first compute
(5.16) 
which can then be summed to obtain
(5.17)  
A short computation shows that can be expressed in terms of dual conformal crossratios as