What is direct interaction, if exist, between gluons and pions?

Question

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

Instead

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

Answer 1

Having obnunciated any expectations of "direct interactions" or topological concepts you might well be proposing as candidates for insight here, I would steer you to the more mainstream buzzwords "the chiral symmetry breaking transition (or interface)" for further study.

The math is highly elaborate and untypical indirect, and the best experts on it are nuclear-particle physicists.

As you indicated, the nonabelian nature of gluon interactions in QCD induces two incompletely understood mathematically, but qualitatively well described, phenomena: confinement of quarks and gluons inside hadrons, and chiral symmetry breaking (χSB), the weightiest "origin of mass".

They both occur at roughly the same distance scales, ~ a fermi, the radius of hadrons, such as the proton. Equivalently, at energies of a half a GeV. Many experts believe χSB takes place a little inside the confinement radius, so a pion and massive constituent quarks (not QCD, current ones) picture could give insights, in a thin shell between the χSB and confinement scales. This is the last time they stood side-by side--but no gluons, as these have gelled into something nonperturbative and hard to conceive.

Inside the confinement scale, particle physicists use QCD, the theory of quarks and gluons, and lattice simulators stretch our knowledge to the edge of the confinement scale, investigating how and which hadrons form. Here, strong forces increase with distance.

Outside the confinement scale, you only have color-singlet hadrons, and the "cheapest" ones to make are the ones requiring the least energy, the light ones, the pions. These provide the main dynamo behind the binding of nuclei, but the other mesons (vector, scalar, etc...) crucially shape and modify the picture. This is firmly in the ambit of nuclear physics.

The pions are light because of their pivotal role in χSB, so their low mass provides a notional extended "cloud" of them on the edge of the hadron, providing the residual strong nuclear forces studied by strikingly different techniques. (This made my favorite nuclear physicist describe the proton as a "marshmallow".) Such forces decrease with distance.

The above two communities of investigators interface in interesting ways, but there is no complete and stable, let alone elegant!, understanding of the interface. One could argue the millennium problemrelated to confinement hankers for such a mathematical grasp and methodology that would lead to such understanding. It is a very open question.

• A small caveat about your "transformation" vision: pions consist of gluons, quarks and antiquarks, arranged differently than scalar mesons, where it matters (a lot). The scalar mesons are much heavier than the pseudoscalar ones: they are basically as heavy as any other meson type. Particle physicists today cringe at the thought of a "transformation" transitioning among species, as a sought-for "Melosh transformation" in the early 1970s, transitioning between current and constituent quarks turned out to be not only an utterly failed idea, but an insidiously counterproductive one (it basically barked up a wrong tree so aggressively it undermined subsequent understanding).