Clockwork is a new model-building gadget that produces very small couplings starting from a theory with no small numbers at all, in an attempt to solve the hierarchy problem.
To describe it as simply as possible: consider $N$ real scalar fields $\phi_i$ with potential $$V(\phi) = \frac{m^2}{2} \sum_i (\pi_i - q \pi_{i+1})^2$$ where $q > 1$. There is a single mode that remains massless, namely the one with $$\pi_i = \frac{\pi_0}{q^i}.$$ It may be viewed as a Goldstone mode in a simple UV completion which I omit just for brevity. Suppose we are at low energies and see only this mode. If something couples to $\pi_N$ only, then the effect of the coupling is suppressed by $q^N$ since the overlap of the Goldstone mode with $\pi_N$ is small. For order $1$ values of $q$ and $N$, $q^N$ can be very large.
Clockwork theory is extremely hot right now, with a citation occurring about once a week. However, I don't understand the key motivation behind such a potential. The result seems to depend sensitively on the particular pattern of interactions. There is no symmetry that demands the interactions only link $\pi_i$ to $\pi_{i \pm 1}$, and no reason the coefficient $q$ can't depend on $i$. I feel that we have replaced one fine-tuning with $N^2$ tunings (the $\pi_i \pi_j$ interactions), since there is nothing that fixes their values.
I've heard that such models are related to "quiver gauge theories" which are motivated by some complicated stuff in string theory, and known in phenomenology as "moose models". Can somebody explain the motivation here, in a way an ordinary physicist (me) can understand?
