Understanding the prefactor $\frac{\theta g^2}{32\pi^2}$ of the $F\tilde{F}$ term in Yang-Mills theories The most general Yang-Mills (YM) action consistent with Lorentz invariance, gauge invariance and renormalizability should contain a term $$\kappa F_{\mu\nu a}\tilde{F}^{\mu\nu a}\tag{1}$$ where $\kappa$ is a  proportionality constant. Such a term is relevant even if the theory is completely classical. The reason is as is as follows. Though it can be shown that $(1)$ can be reduced to a total divergence term of the form $\partial_\mu K^\mu$, the four-vector $K^\mu$ need not vanish at the boundary making its effects admissible. Therefore, even the classical equation of motion will be modified though $\kappa$ remains undetermined classically. The exact term is, however, $$\frac{\theta g^2}{32\pi^2}F_{\mu\nu a}\tilde{F}^{\mu\nu a}\tag{2}$$ where $\theta$ is a parameter that characterizes the nontrivial vacuum state of YM theory. Apart from the famous factor $32\pi^2$ which is a topological effect, presence of $\theta$ in the proportionality constant reminds us that it is a quantum effect and have to be fixed in the quantum version of the theory.
Question $1$ Can we show that the quantum theory fixes the value of $\kappa$ to be $\frac{\theta g^2}{32\pi^2}$ starting with a classical YM Lagrangian which is already augmented by the term $(1)$?
I was going through Srednicki's book (Pages $598-599$). It doesn't exactly show this. It starts with a classical action $S$ that does not include $\kappa F\tilde{F}$ term. It considers transitions from one $\theta$ vacuum to another (See Eqs. 93.40-93.42) and ultimately shows that this transition amplitude is an integral over all field configurations $A$ weighted by $e^{iS^\prime}$ instead of $e^{iS}$ where $$S^\prime=S-\frac{\theta g^2}{32\pi^2}F\tilde{F}.\tag{3}$$
Question $2$ Why doesn't this calculation start with $S+\kappa F\tilde{F}$ instead of $S$ in which case we would have ended up with $e^{iS^\prime}$ with a different $S^\prime$ than $(3)$.
Srednicki says that he neglected such terms in the classical Lagrangian because it doesn't affect equation of motion which I cannot agree.
 A: The normalization is of course arbitrary, because we can always redefine what we mean by $\theta$. However, given that topological charge is quantized, we know that $\theta$ is periodic. This means it makes sense to define the topological part of the action so that $\theta$ has period $2\pi$, that is
$$
 S = \theta Q_{top}
$$
where $Q_{top}\in Z$ is an integer. Standard QFT textbooks show that $Q_{top} =\frac{g^2}{32\pi}F_{\mu\nu}\tilde{F}^{\mu\nu}$ is an integer.
Regarding question 2: Obviously, Srednicki could have started from the action that includes the $\theta$ term (so this may in part be a pedagogical question). However, it is indeed true that you can determine the tunneling path (the instanton) without including the $\theta$ term. All the $\theta$ term does is give a phase $\exp(\pm i\theta)$ to the tunneling amplitude. More generally, you can compute the QCD partition function at non-zero $\theta$ by computing the partition function at $\theta=0$, and then assigning a factor $\exp(i\theta n)$ to the sector with topological charge $n$. 
A: For your first question, fixing $\kappa$ depends on what our convention is for $\theta$. Our convention is we want $\theta$ to be a parameter such that $\theta$ and $\theta+2\pi$ lead to the same physics. Given that is the case, since $\theta$ appears in the action like $\exp(i\theta \kappa\int d^4x F\bar{F})$ we clearly want to choose $\kappa$ to be such that $\kappa \int d^4x F\bar{F}$ is always an integer. This is a purely classical condition on field configurations in the path integral. Quantum mechanics is only entering through the notion that what matters is $\exp(iS)$ and not the action itself.
For your second question, Srednicki is using the action without a theta term because this is the action we would use to find a transition amplitude between states with definite $n$.
Imagine instead we are considering the ordinary quantum mechanics problem of a particle in a sinusoidal potential. The actual energy eigenstates will be non-localized and have some pseudomomentum corresponding to $\theta$. But it is perfectly well-defined to ask what is the transition amplitude for the particle to go from one particular minimum of the sinusoid to another particular minimum in some finite time. And the action you will use is just the ordinary action in a sinusoidal potential. The analogue of this is what Srednicki is doing with the states of definite $n$.
