Exact Green's function for fermions in an external Yang-Mills gauge field

Referee: Abstract and construction: the claim of an exact solution for a non-Abelian SU(N) background is load-bearing for the title and abstract, yet the plane-wave ansatz A_μ = ε_μ f(k·x) T^a (fixed color direction) yields F_μν with vanishing [A,A] term for null waves; this reduces the covariant derivative to an effectively Abelian form, so the Volkov-like phase factor exp(-i g ∫ A) does not demonstrate a genuinely non-Abelian result. The manuscript must explicitly state the color embedding and verify that the Dirac operator does not factorize only because of this reduction.

Authors: We concur that the ansatz with a fixed color direction T^a results in [A_μ, A_ν] = 0, as the commutator of the generator with itself is zero. This causes the Yang-Mills field strength to reduce to its Abelian form, and the background is a solution that does not utilize the non-Abelian self-coupling. The resulting Green's function is obtained by a Volkov-type construction where the phase factor is exp(-i g ∫ A · dx) with A proportional to T^a. This is indeed an effectively Abelian problem within the SU(N) framework. Nevertheless, the derivation provides an exact closed-form expression for the propagator in this background, which is a valid external field for the non-Abelian theory. To address the referee's request, we will revise the manuscript to explicitly describe the color embedding as A_μ = ε_μ f(k·x) T^a where T^a is a fixed generator in the fundamental representation of SU(N). We will also add a paragraph verifying the factorization of the Dirac operator, confirming that it occurs precisely because of the vanishing commutator and the alignment in color space, rather than other properties of the light-cone plane wave. This clarification will be included in the introduction and the section describing the background field. revision: yes