From Six to Four and More: Massless and Massive Maximal Super Yang-Mills Amplitudes in 6d and 4d and their Hidden Symmetries
A self-consistent exposition of the theory of tree-level superamplitudes of the 4d and 6d maximally supersymmetric Yang-Mills theories is provided. In 4d we work in non-chiral superspace and construct the superconformal and dual superconformal symmetry generators of the SYM theory using the non-chiral BCFW recursion to prove the latter. In 6d we provide a complete derivation of the standard and hidden symmetries of the tree-level superamplitudes of SYM theory, again using the BCFW recursion to prove the dual conformal symmetry. Furthermore, we demonstrate that compact analytical formulae for tree-superamplitudes in SYM can be obtained from a numerical implementation of the supersymmetric BCFW recursion relation. We derive compact manifestly dual conformal representations of the five- and six-point superamplitudes as well as arbitrary multiplicity formulae valid for certain classes of superamplitudes related to ultra-helicity-violating massive amplitudes in 4d. We study massive tree superamplitudes on the Coulomb branch of the SYM theory from dimensional reduction of the massless superamplitudes of the six-dimensional SYM theory. We exploit this correspondence to construct the super-Poincaré and enhanced dual conformal symmetries of massive tree superamplitudes in SYM theory which are shown to close into a finite dimensional algebra of Yangian type. Finally, we address the fascinating possibility of uplifting massless 4d superamplitudes to 6d massless superamplitudes proposed by Huang. We confirm the uplift for multiplicities up to eight but show that finding the uplift is highly non-trivial and in fact not of a practical use for multiplicities larger than five.
Scattering amplitudes of maximally supersymmetric Yang-Mills theories in 3, 4, 6 and 10 dimensions possess remarkable properties. Next to their constitutional maximally extended super-Poincaré symmetries they all enjoy a hidden dual conformal symmetry – at least at the tree-level Lipstein:2012kd (); Drummond:2008vq (); Brandhuber:2008pf (); Dennen:2010dh (); CaronHuot:2010rj (). The four dimensional super Yang-Mills (SYM) theory is distinguished in this series as it also has superconformal symmetry in the standard sense. The standard superconformal symmetry then further enhances the dual conformal symmetry to a dual superconformal symmetry Drummond:2008vq (); Brandhuber:2008pf (). On top the closure of the two sets of superconformal symmetry algebras leads to an infinite dimensional symmetry algebra of Yangian type Drummond:2009fd (). It is the manifestation of an underlying integrable structure in planar SYM. The key to the discoveries of these rich symmetry structures of maximally supersymmetric Yang-Mills theories in various dimensions is the use of a suitable on-shell superspace formalism along with spinor helicity variables to package the component field amplitudes into superamplitudes, which was pioneered in 4d in Nair:1988bq (). In this work we shall focus on the four and six dimensional maximally theories: The 4d SYM and the 6d SYM models.
While the massless tree amplitudes of 4d SYM are very well studied and in fact known analytically Drummond:2008cr (), not so much is known about the massive amplitudes on the Coulomb branch of this theory. These amplitudes are obtained by giving a vacuum expectation value to the scalar fields and yield – arguably – the simplest massive amplitudes in four dimensions. Alternatively, these massive amplitudes arise from the amplitudes of the maximally supersymmetric 6d SYM theory upon dimensional reduction, where the higher dimensional momenta yield the masses in the 4d theory. Indeed, compact arbitrary multiplicity amplitudes for particular subclasses of Coulomb branch amplitudes have been obtained in Craig:2011ws () by making use of modern on-shell techniques. The massive 4d SYM amplitudes are invariant under a dual conformal symmetry which is inherited from the 6d SYM theory as shown in Dennen:2010dh (). Moreover, this symmetry remains intact also at loop-level if one restricts the loop-momentum integrations to a four-dimensional subspace. This prescription is equivalent to the Higgs regularization for infrared divergences in 4d proposed in Alday:2009zm (), where such an extended dual conformal invariance was conjectured and tested at the one-loop four-point level. The dimensional reduction of 6d SYM to four dimensions yields superamplitudes expressed on a non-chiral superspace Huang:2011um () which is distinct to the usual chiral superspace of Nair:1988bq (). In this work we explicitly construct all generators of the standard and dual (super) conformal symmetry generators acting in the non-chiral on-shell superspace as well as in the on-shell superspace. We also determine the standard and dual symmetries of massive amplitudes as they are induced from an enhanced super-Poincaré and enhanced dual conformal symmetry of the 6d SYM theory.
The most efficient method to analytically construct tree-level amplitudes is based on an on-shell recursive technique due to Britto, Cachazo, Feng and Witten (BCFW) Britto:2004ap (); Britto:2005fq (). In contrast to the earlier Berends-Giele off-shell recursion relations Berends:1987me (), the BCFW relation uses only on-shell lower-point amplitudes, evaluated at complex, shifted momenta. The BCFW recursion relation is easily generalizable to an on-shell recursion for superamplitudes, as was done for SYM in ArkaniHamed:2009dn () (see also Bianchi:2008pu ()). In fact the knowledge of the dual superconformal invariance of superamplitudes motivates an ansatz in terms of dual conformal invariants. Together with the super BCFW recursion this allowed for the complete analytic solution Drummond:2008cr (). In fact the variant of the BCFW recursion for 4d SYM in non-chiral superspace has not been written down before and we will do so in this work. The BCFW recursion for 6d SYM theory was established in Dennen:2009vk (); Bern:2010qa () and tree-level amplitudes of multiplicities up to five were derived. The one loop corrections were obtained in Brandhuber:2010mm (). In this work we point out how a numerical implementation of the BCFW recursion for SYM amplitudes in combination with a suitable set of dual conformal invariant basis functions may be used to derive compact five and six-point amplitudes as well as arbitrary multiplicity amplitudes for certain subclasses related to the 4d amplitudes with two neighboring massive legs mentioned above Craig:2011ws (). In fact, the method we propose is very general and could be applied to further cases as well.
A very tempting option to obtain massive 4d amplitudes of SYM was introduced by Huang in Huang:2011um (). He indicated that it should be possible to invert the dimensional reduction of to massive by uplifting the massless non-chiral superamplitudes of SYM to six-dimensional superamplitudes of SYM. Non-chiral superamplitudes of SYM are straightforward to obtain using the non-chiral BCFW recursion, resulting in an eminent practical relevance of a potential uplift. It is indeed very surprising that in fact the massive Coulomb branch amplitudes or equivalently the six-dimensional amplitudes might not contain any more information than the massless four-dimensional amplitudes of SYM.
It is the aim of this paper to provide a self-consistent and detailed exposition of the theory of superamplitudes for 4d SYM and 6d SYM. The paper is organized as follows. We discuss the needed spinor helicity formalisms in section 2. Section 3 and 4 are devoted to the on-shell superspaces of both theories and the standard and hidden symmetries of the associated superamplitudes. In section 5 we discuss the dimensional reduction from massless 6d to massive 4d amplitudes and establish the inherited (hidden) symmetries of the 4d amplitudes. Section 6 exposes the on-shell BCFW recursion relations for SYM in non-chiral superspace as well as for SYM. We also provide a proof of dual conformal symmetry of superamplitudes thereby correcting some minor mistakes in the literature. Finally in section 8 we analyze in detail the proposal of Huang for uplifting 4d massless superamplitudes in non-chiral superspace to 6d superamplitudes and point out why this uplift is non-trivial and in fact not of a real practical use for multiplicities larger than five. Notational details and extended formulae are relegated to the appendices.
2 Spinor helicity formalism
2.1 General remarks
Calculating scattering amplitudes of massless particles, the spinor helicity formalism has become a powerful tool in obtaining compact expressions for tree-level and one-loop amplitudes. The basic idea is to use a set of commuting spinor variables instead of the parton momenta . These spinors trivialize the on-shell conditions for the momenta
In what follows we will briefly review the spinor helicity formalism in four and six dimensions. Additional details and conventions can be found in appendix A.
2.2 Four dimensions
The starting point of the spinor helicity formalism in four dimensions DeCausmaecker:1981bg (); Berends:1981uq (); Kleiss:1985yh (); Xu:1986xb (), which we briefly review here, is to express all momenta by matrices
where we take and with being the Pauli matrices. Raising and lowering of the and indices may be conveniently defined by left multiplication with the antisymmetric symbol for which we choose the following conventions
Besides being related by , these matrices satisfy , and . Hence, the matrices and have rank one for massless momenta, implying the existence of chiral spinors and anti-chiral spinors solving the massless Weyl equations
These spinors can be normalized such that
For complex momenta the spinors and are independent. However, for real momenta we have the reality condition , implying for some . Hence, the spinors can be normalized such that
An explicit representation is
Obviously, eq. 5 is invariant under the little group transformations
Labeling the external particles by , each parton momentum is invariant under its own little group transformation . The simplest Lorentz invariant and little group covariant objects that can be built out of the chiral and anti-chiral spinors are the anti-symmetric spinor products
The little group invariant scalar products of massless momenta are then given by a product of two spinor brackets
The spinor helicity formalism allows for a compact treatment of polarizations. Each external gluon carries helicity and a momentum specified by the spinors and . Given this data the associated polarization vectors are
where are auxiliary light-like momenta reflecting the freedom of on-shell gauge transformations. It is straightforward to verify that the polarization vectors fulfill
as well as the completeness relation
A summary of all our conventions for four dimensional spinors can be found in appendix A.
2.3 Six dimensions
Similar to four dimensions, the six-dimensional spinor-helicity formalism Cheung:2009dc () provides a solution to the on-shell condition for massless momenta by expressing them in terms of spinors. As a first step one uses the six-dimensional analog of the Pauli matrices and to represent a six-dimensional vector by an antisymmetric matrix
Besides being related by , these matrices satisfy and . Hence, for massless momenta, and have rank 2 and therefore the chiral and anti-chiral part of the Dirac equation
have two independent solutions, labeled by their little group indices and respectively. Raising and lowering of the little group indices may be conveniently defined by contraction with the antisymmetric tensors and
The anti-symmetry of and together with the on-shell condition yields the bispinor representation
An explicit representation of the chiral and anti-chiral spinors is given by
As a consequence of the properties of the six-dimensional Pauli matrices, the spinors are subject to the constraint
It is convenient to introduce the bra-ket notation
By fully contracting all Lorentz indices it is possible to construct little group covariant and Lorentz invariant objects. The simplest Lorentz invariants are the products of chiral and anti-chiral spinors
These little group covariant spinor products are related to the little group invariant scalar products by
The spinor products are matrices whose inverse is
Each set of four linear independent spinors labeled by , , , can be contracted with the antisymmetric tensor, to give the Lorentz invariant four brackets
Note that in the above expressions the 4x4 matrix appearing in the determinants is defined through its four columns vectors and similarly for the second expression.
The four brackets are related to the spinor products by
where , are multi indices labeling the spinors. Finally, it is convenient to define the following Lorentz invariant objects
Similar to the four dimensional case, the polarization vectors of the gluons can be expressed in terms of spinors by introducing some light-like reference momentum with , where denotes the gluon momentum. The four polarization states are labeled by little group indices and can be defined as
It is straightforward to verify the properties
as well as the completeness relation
3 Four-dimensional SYM theory
3.1 On-shell superspaces and superamplitudes
Dealing with scattering amplitudes of supersymmetric gauge theories is most conveniently done using appropriate on-shell superspaces. Most common for treating super Yang-Mills theory are Nair:1988bq (); Witten:2003nn (); Georgiou:2004by ()
|chiral superspace:||anti-chiral superspace:||(33)|
The Grassmann variables , transform in the fundamental, anti-fundamental representation of and can be assigned the helicities
with denoting the helicity operator acting on leg . With their help it is possible to decode the sixteen on-shell states
into a chiral or an anti-chiral superfield , , defined by
As a consequence of eq. 34 the super fields carry the helicities
The chiral and anti-chiral superfield are related by a Grassmann Fourier transformation
Chiral and anti-chiral color ordered superamplitudes can be defined as functions of the respective superfields
Due to eq. 39 both superamplitudes are related by a Grassmann Fourier transformation
The superamplitudes are inhomogeneous polynomials in the Grassmann odd variables , , whose coefficients are given by the color ordered component amplitudes. A particular component amplitude can be extracted by projecting upon the relevant term in the expansion of the super-amplitude via
and similar in anti-chiral superspace. By construction the chiral and anti-chiral superamplitudes have a manifest symmetry. The only invariants are contractions with the epsilon tensor
Consequently the appearing powers of the Grassmann variables within the superamplitudes need to be multiples of four. As a consequence of supersymmetry the superamplitudes are proportional to the supermomentum conserving delta function
with the chiral or anti-chiral conserved supermomentum . Since the Grassmann variables carry helicity, eq. 34, their powers keep track of the amount of helicity violation present in the component amplitudes. Hence, decomposing the superamplitudes into homogeneous polynomials is equivalent to categorizing the component amplitudes according to their degree of helicity violation
The highest amount of helicity violation is present in the maximally helicity violating (MHV) superamplitude or in the superamplitude in anti-chiral superspace. In general, and are the (Next to) MHV and the (Next to) superamplitudes . The complexity of the amplitudes is increasing with the degree of helicity violation, the simplest being the MHV superamplitude in chiral superspace Nair:1988bq ()
and the superamplitude in anti-chiral superspace
which are supersymmetric versions of the well known Parke-Taylor formula Parke:1986gb (). The increasingly complicated formulae for the amplitudes have been obtained in reference Drummond:2008cr (). Plugging the MHV decomposition, eq. 46, into eq. 41 we obtain the relation
simply stating that and contain the same component amplitudes. Depending on whether or it is therefore more convenient to use the chiral or the anti-chiral description of the amplitudes, e. g. the amplitudes are complicated in chiral superspace whereas they are trivial in anti-chiral superspace. Hence the most complicated amplitudes appearing in an point chiral or anti-chiral superamplitude are the helicity amplitudes of degree , called minimal helicity violating (minHV) amplitudes .
3.2 Non-chiral superspace
Besides the well studied chiral and anti-chiral superspaces there is as well the non-chiral superspace
which is more natural from the perspective of the massive amplitudes and the six dimensional parent theory that we are interested in. Here the indices of the fields get split into two indices and according to
Note that the due antisymmetry the fields and represent only one scalar field respectively, whereas the account for the four remaining scalars. If raising and lowering of the indices are defined by left multiplication with and , the non-chiral superfield reads
with the abbreviations , . The non-chiral superfield is a scalar and has zero helicity. Obviously, the non-chiral superamplitudes will not have a symmetry, but rather will be invariant under transformations. With the convention , the non-chiral superfield is related to the chiral and anti-chiral superfield by the half Grassmann Fourier transformations
As a consequence of supersymmetry, the superamplitudes are proportional to the supermomentum conserving delta functions
with the conserved supermomenta and . Since we additionally have , the non-chiral superamplitudes have the general form
It should be stressed that the dependence of only on the momenta is distinct to the situation for the chiral or anti-chiral superamplitudes, where we have a dependence on the super-spinors or . Analyzing the half Fourier transform (55) relating the superfields we see that the non-chiral superamplitudes are homogeneous polynomials in the variables and of degree and the MHV decomposition (46) of the chiral superamplitudes translates to a MHV decomposition of the non-chiral superamplitudes
where the NMHV sector corresponds to a fixed degree in the variables and
This reflects the chiral nature of SYM theory.
Each of the three superspaces presented above has an associated dual superspace. In general, dual superspaces naturally arise when studying dual conformal properties of color ordered scattering amplitudes. Part of the spinor variables get replaced by the region momenta , which are related to the ordinary momenta of the external legs by
and a new set of dual fermionic variables or is introduced, related to the fermionic momenta by