All OPEs invariant under the infinite symmetry algebra for gluons on the celestial sphere

<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>S</a:mi></a:mrow></a:math> algebra is an infinite-dimensional Lie algebra, which is known to be the symmetry algebra of some gauge theories. It is a “colored version” of the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>w</c:mi><c:mrow><c:mn>1</c:mn><c:mo>+</c:mo><c:mi>∞</c:mi></c:mrow></c:msub></c:math>. In this paper, we write down all possible <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>S</e:mi></e:math>-invariant (celestial) operator product expansions between two positive-helicity outgoing gluons and also find the Knizhnik-Zamolodchikov type null states for these theories. Our analysis hints at the existence of an infinite number of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>S</g:mi></g:math>-invariant gauge theories which include the (tree-level) maximally helicity-violating sector and the self-dual Yang-Mills theory.