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Recently two of the authors presented a spinorial extension of the scatteringequations, the `polarized scattering equations' that incorporates spinorpolarization data. These led to new worldsheet amplitude formulae for a varietyof gauge, gravity and brane theories in six dimensions that naturallyincorporate fermions and directly extend to maximal supersymmetry. This paperprovides a number of improvements to the original formulae, together withextended details of the construction, examples and full proofs of some of theformulae by BCFW recursion and factorization. We show how our formulae reduceto corresponding formulae for maximally supersymmetric gauge, gravity and branetheories in five and four dimensions. In four dimensions our frameworknaturally gives the twistorial version of the 4d ambitwistor string, giving newinsights into the nature of the refined and polarized scattering equations theygive rise to, and on the relations between its measure and the CHY measure. Ourformulae exhibit a natural double-copy structure being built from`half-integrands'. We give further discussion of the matrix of theories andformulae to which our half-integrands give rise, including controversialformulae for amplitudes involving Gerbes.

Recursion and worldsheet formulae for 6d superamplitudes