Gluons mediate the strong force between quarks. Pions mediate the nuclear force or nucleon-nucleon interaction or residual strong force.
I had thought of some scalar bosons for my idea because if I'm not mistaken between the gluon and the pion it should occur a particular transformation properties under Lorentz transformation that is, instead, a scalar boson.
I remember that force between quarks is non-abelian force (whose substrate can be thought of in terms of non-abelian Lie groups) but I'm not sure.
I want to know math model or if exist something that explain a possible direct interaction between gluons and pions because I want to understand what kind of mathematical surface/manifold/variety could describe the "surface" of the proton because proton surface remember me a sort interface and separation between gluons and pions a sort of interphase or phase transition but
topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.
Just for an idea
In the physical sciences, an interface is the boundary between two spatial regions occupied by different matter, or by matter in different physical states. The interface between matter and air, or matter and vacuum, is called a surface
the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S