^{1}

^{2}

^{3}.

We introduce a spinorial version of the scattering equations, the polarized scattering equations, that incorporates spinor polarization data. They underpin new formulas for tree-level scattering amplitudes in six dimensions that directly extend to maximal supersymmetry. We find new ingredients for integrands for super Yang-Mills theory, gravity,

Six dimensions has proved to be a fertile arena both for constructing and unifying four-dimensional field theories via dimensional reduction and for studying nontrivial lower dimensional consequences of

Another approach initiated in four dimensions in Ref.

In six dimensions, the spin group in the complex is

For massless particles, the little group is

A Maxwell field strength is represented by

We take

It is easy to see that Eq.

The polarized scattering equations enhance the

We will express 6D scattering amplitudes as

In six dimensions,

A key example is (1,1) super Yang-Mills theory, with the supermultiplet

To construct an on-shell superspace, half of the generators need to be selected as supermomenta. Two mechanisms have been discussed in the literature

In the context of the polarized scattering equations,

In general, given a scattering amplitude of the form

We now construct the integrands

On the polarized scattering equations, the determinant

To see this, we extend the argument of Ref.

The reduced determinant

Another important building block, relevant for the D5 and

The final two ingredients are constructed exclusively using the variables

At this point, we have all the ingredients to present the integrands of

It is easily seen that the resulting superamplitudes are

In the following, we develop a twistorial 6d ambitwistor string—a chiral 2D CFT whose target space is ambitwistor space

We focus here on the model using a chiral representation of super ambitwistor space. However, amplitudes in ambidextrous theories require both the chiral and antichiral supertwistors. Since these are alternative coordinates on the same space, they are related by nontrivial constraints. Implementing these constraints at the level of the worldsheet model, as well as adding “worldsheet matter” systems giving rise to the integrands

In six dimensions

In the superambitwistor string, the supertwistors are taken to be worldsheet spinors, with the chiral action

Vertex operators in the ambitwistor string are built from ambitwistor representatives for space-time fields, which are Dolbeault cohomology classes in

This general form for the vertex operator is sufficient to derive the polarized scattering equations and the supersymmetry factor

The 6D (1,1) SYM amplitudes

For 4D

Denote 4D little group contractions by round brackets, e.g.,

The matrix

Combinations of half-integrands.

Our new formulae are represented in double-copy form in Table

There are many directions for future investigation. A natural task is to find suitable worldsheet matter systems to provide integrands for the bare ambitwistor-string models described here. Provided that the constraints relating chiral and antichiral supertwistors have been implemented in the worldsheet model, there is a natural candidate for the matter system giving rise to the

A different question concerns what other theories might be constructed using the polarized scattering equations. One immediate observation is that new formulas can be constructed by combining the hal-integrands introduced in this Letter given in the last column of Table

We would like to thank the authors of Ref.

In the following, we suppress the particle index

For scalar amplitudes, this corresponds to a partition into two non-self-interacting scalar sectors.

This is equivalent to the action in a vector representation