Effective scalar field in terms of the scale on which it emerges

Question

Suppose there is the (pseudo)scalar field $\hat{\theta}$ with non-zero VEV $\theta$, which effectively emerges at energy scale $\Lambda$ (for example, the mass of some fermion, the scale of SSB and so on). An example is axion-like field $\theta$, which is present as

$$\int \frac{\theta}{f_{\gamma}}F\wedge F, \quad f_{\gamma} \simeq \alpha_{\text{EM}}^{-1}\Lambda$$
in effective action (with $F$ denoting gauge field strength).

May the quantity

$$\epsilon \equiv \frac{\theta}{\Lambda}$$
be many times larger than one? I.e., does some condition exist, which forbids $\epsilon >> 1$ domain? Or this is the question of our wishes, and there are no restrictions on value of $\epsilon$?

Comments

I don't think that there is a theoretical restriction on the sizes. In practice (comparison with experiment) there is, however, the question of fine-tuning the parameters to match experimental data, which is a very sensitive issue when scales don't behave ''naturally''. See https://en.wikipedia.org/wiki/Naturalness_%28physics%29