  In this paper we will show an ultraviolet -infrared connection for ghost-free
infinite derivative field theories where the Lagrangians are made up of
exponentials of entire functions. In particular, for $N$-point amplitudes a new
scale emerges in the infrared from the ultraviolet, i.e. $M_{\rm eff}\sim
M_s/N^\alpha,$ where $M_s$ is the fundamental scale beyond the Standard Model,
and $\alpha>0$ depends on the specific choice of an entire function and on
whether we consider zero or nonzero external momenta. We will illustrate this
by first considering a scalar toy model with a cubic interaction, and
subsequently a scalar toy-model inspired by ghost-free infinite derivative
theories of gravity. We will briefly discuss some phenomenological
implications, such as making the nonlocal region macroscopic in the infrared.
