You’re gonna need something to break this down first “perturbation expansion for a scattering matrix”. I’m a semi literate layman and have no idea what you’re referring to here.
The elements of the scattering matrix determine how two (or more) colliding particles will interact and emit zero or more particles as a result. These elements are defined by an integral that has no closed form. For certain fields (electromagnetic being the prominent example) the integral can be approximated by the sum of terms in an infinite series.
Each term in the series includes a multiplicative factor of the coupling constant (typically denoted by alpha which is approximately equal to 1/137) raised to a certain power. Since this number is quite small, the higher the power, the less the term contributes to the overall sum.
Additionally, there is a bidirectional mapping of each term in the series to a Feynman diagram (you are likely familiar with examples of these diagrams). The number of vertices in the diagram correspond to the power of the constant mentioned above. So, the terms that dominate the integral are the ones that have a small number of vertices, but you can keep going, adding more and more vertices to get a more accurate sum.
These additional vertices can be added as long as you satisfy the rules of the diagram for the particular field you are considering (different fields have different constraints on invariants that must hold at each vertex). For EM, you can add additional photon/electron vertices as long as the overall electric charge is preserved.
These additional photon edges in the diagram are "virtual" particles. Nothing more than a pictorial representation of a term in an infinite series that approximates an integral.
(First order response might regular 6DOF motions (motion spectra) in response to wave spectra)
Here the analogies are pretty fun, because you are scattering and radiating waves in the water. Very physical!
But the idea is: the scattering matrix (https://en.wikipedia.org/wiki/S-matrix) is a matrix that says for each possible input and output state, what the amplitude of that transition happening.
The perturbation expansion is roughly a Taylor series in the number of interactions that occur in a scattering problem. So: no interaction is order-0, exchanging one particle is order-1, etc.
The first dozen terms or so then tell you the dozen or so highest-amplitude processes that occur in a given interaction. Most of these will involve virtual particles.