For nucleon-nucleon interaction please keep in mind that in this low-energy regime pertubative QCD breaks down and reactions are not really calculable. For the specific pion exchange you mention have a look at
http://hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html
as to why this QCD-process can be seen as an exchange of a pion.
In general you can get Lagrangians for effective (i.e. low-energy) theories by integrating out the high momentum degrees of freedom. For example the W-boson propagator
$\frac{-i(g_{\mu\nu}-\frac{q_\mu q_\nu}{m_W})}{q^2-M^2_W}$ in the limit of small $q^2$ becomes $\frac{ig_{\mu\nu}}{M^2_W}$
so instead of the full electro-weak symmetry there would be a new lagrangian with a four point interaction term $\frac{G_F}{\sqrt{2}}J_\mu J^\mu$ where $J_\mu$ is a left-handed Dirac current.
In more general terms this integrating out of high momentum degrees of freedom is the point of view on renormalization taken by Wilson. In this view our current theories are effective theories that result from an unknown fundamental Lagrangian from which all the high momentum d.o.f. have been integrated out. (c.f. renormalization group)
1) As you know,
$$
\tag 1 \theta\epsilon^{\mu \nu \alpha \beta}F_{\mu \nu}F_{\alpha \beta} = \theta\partial_{\mu}K^{\mu},
$$
where $K_{\mu}$ is the so-called Chern-Simons class.
The Feynman diagrams method tells us that the term $(1)$ defines the diagram which corresponds to the two-photon (or two-three-four non-abelian bosons) vertex $V_{A}$ (where the subscript $A$ denotes all of indices) times the delta-function with an argument of difference of ingoing and outgoing momenta times the difference of ingoing and outgoing momenta:
$$
V_{A} \sim \delta (\sum_{i}p_{i} - \sum_{f}p_{f})\times (\sum_{i}p_{i} - \sum_{f}p_{f})\times \text{constant tensor}
$$
This is none but zero, so that the term $\int F\wedge F$ doesn't affect the perturbation theory.
2) Since $\epsilon_{\mu \nu \alpha \beta}$ is the pseudotensor, while $F$ is the tensor (here I would take into an account only the gluon stength tensor $F \equiv G_{\mu \nu}^{a}$, since the effects of corresponding electroweak and electromagnetic terms can be absorbed by chiral rotations of fermion fields), we have that under discrete Lorentz transformations $T$ (with $T_{\mu \nu}x^{\mu} = (-x_{0}, \mathbf x)_{\mu}$) and $P$ (with $P_{\mu \nu}x^{\nu} = (x_{0}, -\mathbf x)_{\mu}$)
$$
\hat{T}\epsilon_{\mu \nu \alpha \beta}G^{\mu \nu}_{a}G^{\alpha \beta}_{a}\hat{T} = -\epsilon_{\mu \nu \alpha \beta}G_{a}^{\mu \nu}G_{a}^{\alpha \beta}, \quad \hat{P}\epsilon_{\mu \nu \alpha \beta}G_{a}^{\mu \nu}G_{a}^{\alpha \beta}\hat{P} = -\epsilon_{\mu \nu \alpha \beta}G_{a}^{\mu \nu}G_{a}^{\alpha \beta}
$$
I.e., $\theta \epsilon_{\mu \nu \alpha \beta}G^{\mu \nu}_{a}G^{\alpha \beta}_{a}$ term is the pseudoscalar.
So that this term, particularly, breaks $CP$ invariance of theory. What is the consequence of such breaking? If you look on the QCD lagrangian in zero temperature confined phase (which is computed nonperturbatively), for example, protons $p$, neutrons $n$ and pions $\pi$ nuclear interaction, this term acts as pseudoscalar (i.e., also $CP$-breaking) $\pi NN$ coupling:
$$
\tag 2 L_{\pi , p, n} = \bar{\Psi}\pi^{a}t_{a}(g_{\pi NN} + \bar{g}_{\pi NN}i\gamma_{5})\Psi , \quad \Psi = \begin{pmatrix} p\\ n \end{pmatrix}
$$
where $\bar{g}_{\pi NN} \sim \theta$. By introducing the nuclear interaction with EM field, you can compute the diagram which defines neutron dipole moment, and see, that it is nonzero due pseudoscalar $CP$-violating coupling (it is proportional to it). How to understand this qualitatively? It is not hard to see this.
The interaction hamiltonian of uncharged particle with spin $\mathbf S$ and EM field $\mathbf E, \mathbf B$ is
$$
\tag 3 H = -\mu \left(\mathbf B \cdot \frac{\mathbf S}{|\mathbf S|}\right) - d\left( \mathbf E \cdot \frac{\mathbf S}{S}\right)
$$
Since the magnetic field $\mathbf B$ is pseudovector, the electric field is vector and the spin $\mathbf S$ is pseudovector, we have that under $P$ transformation
$$
\hat{P}\left(\mathbf B \cdot \frac{\mathbf S}{|\mathbf S|}\right)\hat{P} = \left(\mathbf B \cdot \frac{\mathbf S}{|\mathbf S|}\right),
$$
$$
\hat{P}\left(\mathbf E \cdot \frac{\mathbf S}{|\mathbf S|}\right)\hat{P} = -\left(\mathbf E \cdot \frac{\mathbf S}{|\mathbf S|}\right)
$$
So that the presence of nonzero dipole moment in $(3)$ directly violates $CP$ invariance of theory,
$$
\hat{P}H\hat{P} \neq H
$$
Formally it is computed in the way which I've described above: the gluon $\theta$ term $(1)$ generates pseudoscalar nuclear coupling $(2)$, while this coupling generates the vertex between the neutron and EM field which defines dipole moment.
An effect of $\theta$ term doesn't arise in classical physics, since it affects on physics only through quantum loops.
3) When nonperturbative effects arise
Formally the information about all of nonperturbative effects is contained in the path integral of theory, which is equal to the Green function in Heisenberg representation. It is lost, however, in cases when we try to use completely perturbative approach at all scales of theory.
Note that there are three important facts which are connected with "tiny" couplings expansion (QED, QCD etc.).
The first one is related to the fact that in interacting theories couplings are running ones, i.e. they are different on the different scales. For example, QED coupling is small up to very large energies, but there are scale where it grows fast (near the so-called Landau pole), so that the perturbation theory is unapplicable; the one more example: QCD coupling quickly grows with decreasing the scale to $\sim \Lambda_{QCD}$, below which the perturbative theory becomes unapplicable.
The second one is related to the changing of the expansion coupling constants near infrared zones of momentum. Two examples:
Electron-proton interaction. You know that electron-proton scattering matrix element in momentum representation must have the pole in the point $p_{0}^{\text{pole}} = m_{P} + m_{e} - 13.6 \text{ eV}$. But there are no perturbation theory term which generates such pole; we need to take into an account the sum of all diagrams near the given pole. The reason of such pole is not hard to understand if we take into an account the diagram, for which in CM frame electron and proton momentums are small, $|\mathbf p| << m_{e}$, while their intermediate state of scattering is characterized by the slightly different momentums. It can be shown that such diagram has effective coupling constant $\frac{e^{2}m_{e}}{q^{2}}$, which is large for $q^{2} < e^{2}m_{e}$. Such scale is the scale of the bound state. In general, the each bound state is invisible in terms of perturbation theory, since the renormalization constant $Z$ is zero for them.
EW phase transition. When we compute effective action of EW theory near the point of the phase transition, we can't use perturbative approach based on the naive expansion on $\sim \frac{m(\langle H \rangle)}{T}$, since the temperature partition function of bosons are large at small energies (in infrared zone). We have instead of naive thinking that the effective coupling constant is $\sim \frac{T}{m(\langle H \rangle )}$.
The third one arises when we try to summing all of the diagrams of perturbation theory. Since perturbation theory series (on coupling constant) has zero convergence radius (expansion terms grow as $n!$), we need to use some technics like Borel resummation to restore the full result. However, in such cases it is unapplicable. The first important case is the classical solutions of equations of motion of the theory called instantons. They are stationar points of the quantity $\int D\varphi \oint dg g^{-n-1}e^{-S[g, \varphi]}$, where $n$ is an integer quantity. Corresponding Borel series usually has the negative sign pole (as in the QCD). The second case is infrared renormalon, which arises in operator expansion, for example, of all of the diagrams which contain two 4-currents in QCD. Corresponding set of diagrams generate positive pole, which makes using of perturbation theory completely inapplicable.
How to calculate nonperturbative effects
So that, because of existence of many cases where perturbative theory fails, there are also exist many approaches to calculate corresponding nonperturbative results.
If you want to calculate the poles associated with the bound states, you have to solve the EOM for the one-particle state which is one of constituent particles, in which the effects of other constituents (i.e., their operators in EOM) are replaced by external potential (for example, for electron-proton task the proton effect is the Coulomb central potential). Then for such range of energies you may introduce elementary Hydrogen atom field in the lagrangian by Hubbard-Stratonovich transformation of the path integral (an example given in Weinberg's QFT Vol. 1, paragraph 14).
Also famous exactly solvable case is the spontaneous symmetry breaking (in the result of which $p^{2} = 0$ poles arise), for which you have to do following:
- determine the unbroken group;
- extract the Goldstone degrees of freedom (the number of which coincide with the number of broken generators of the symmetry group) by their parametrization of coordinates of elements of broken group;
- then by explicit calculations to construct invariant forms from these elements, which determine the lagrangian of effective degrees of freedom after symmetry breaking.
Obtained theory contains elementary fields correspons to bound states of underlined theory.
- If you want to calculate the effects of solutions of classical EOM in quantum theory (you have to take into an account these solutions because the cluster decomposition principle of the S-matrix fails when you haven't, which is well explained in Weinberg's QFT vol. 2) you have to find the homogopy group of the symmetry group of the given theory, then find nontivial representations of the homotopy classes of such groups (i.e., solutions $\varphi_{\text{classical}}$ of EOM which correspond to different values of Maurer-Cartan invariant), then to find the number of collective degrees of freedom of the solution, and, finally, by having such results, to calculate coefficient factor near $e^{-S[\varphi_{\text{classical}}]}$ in path integral (an example given in Weinberg's QFT Vol. 2, paragraph 23.5).
Best Answer
If the theory is Borel summable, you can recover non-perturbative information from the perturbative series. This can be shown explicitly for example by calculating the exact effective action in the presence of a constant electromagnetic field, à la Schwinger. You can find a very clear exposition in A.R.Bogojevi's lecture notes, chapters 24&25. For other examples of Borel summable results, see Heisenberg-Euler Effective Lagrangians: Basics and Extensions, by G. V. Dunne. See also Why is the Borel summation relevant for asymptotic series of physical observables?, from Physics Overflow, for other relevant references.
Another well-known example where you can extract non-perturbative information from the perturbative series is zero-dimensional QFT. See e.g. A Non-perturbative Solution of the Zero-Dimensional $\lambda\phi^4$ Field Theory, by Malbouisson, Portugal, Svaiter. The calculation in $d=0$ is particularly illuminating because we can obtain explicit expressions, but in principle the analysis holds for higher $d$ (cf. A. Neumaier's answer in the post on Overflow).
Moreover, it is usually the case that objects that are protected by topological considerations, perturbative calculations are in fact exact. Such is the case of the axial anomaly, which can be analised non-perturbatively (by means of, say, a change of variables in the path integral as Fujikawa taught us to), and the result agrees with a one-loop calculation. You can find a nice discussion in the PSE post Instantons, anomalies, and 1-loop effects.
Finally, let me mention that for objects protected by symmetries, such as BPS states, you can usually extract reliable information valid in the strongly coupled regime by studying the same object in the weakly coupled regime. The philosophy behind this is that the symmetries are sometimes so constraining that the properties of the object evolve rigidly from one regime to the other. In the same vein, in theories related by dualities you can obtain results valid in the strongly coupled theory by studying the weakly coupled one. Whether this qualifies as extracting non-perturbative information from a perturbative result is a matter of opinion, so this may be taken as a more contrived examples than those from before.