Associativity is enough: an all-orders 2d chiral algebra for 4d form factors

There is a special set of massless four-dimensional gauge theories which admit local and gauge-anomaly-free uplifts to twistor space; we call such theories twistorial. In twistorial theories, generalized towers of soft modes (including states of both helicities) form a 2d chiral algebra even at the quantum level. The 2d OPE limit of this chiral algebra coincides with the holomorphic collinear limit in 4d. This is true, in particular, for self-dual Yang-Mills (SDYM) theory coupled to special choices of matter, the latter being required to make the theory twistorial. Costello and the second author recently proposed that form factors of such twistorial 4d theories could be computed as 2d chiral algebra correlators. In turn, there exist form factors of self-dual theories, with insertions of appropriate local operators, that compute a subclass of observables in full (i.e. non-self-dual) QCD, coupled to appropriate matter. For example, the n-point two-loop all + QCD amplitude has recently been computed using the 1-loop chiral algebra OPEs determined thus far, but higher orders of the quantum-deformed chiral algebra must be determined to continue the ``chiral algebra bootstrap'' program for higher-loop-level form factors of these twistorial theories. In this paper, using only elementary constraints from symmetries and associativity, we obtain closed-form expressions for the extended chiral algebra OPEs to arbitrary loop-order. This can be viewed as providing an all-loop result for a subset of collinear splitting functions in non-supersymmetric, massless QCD coupled to special choices of matter.