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 $\epsilon$?