  We study the resummation of self-energy diagrams into dressed propagators in
the case of purely virtual particles and compare the results with those
obtained for physical particles and ghosts. The three geometric series differ
by infinitely many contact terms, which do not admit well-defined sums. The
peak region, which is outside the convergence domain, can only be reached in
the case of physical particles, thanks to analyticity. In the other cases,
nonperturbative effects become important. To clarify the matter, we introduce
the energy resolution $\Delta E$ around the peak and argue that a "peak
uncertainty" $\Delta E\gtrsim \Delta E_{\text{min}}\simeq \Gamma _{\text{f}}/2$
around energies $ E\simeq m_{\text{f}}$ expresses the impossibility to approach
the fakeon too closely, $m_{\text{f}}$ being the fakeon mass and $\Gamma
_{\text{f}}$ being the fakeon width. The introduction of $\Delta E$ is also
crucial to explain the observation of unstable long-lived particles, like the
muon. Indeed, by the common energy-time uncertainty relation, such particles
are also affected by ill-defined sums at $\Delta E=0$, whenever we separate
their observation from the observation of their decay products. We study the
regime of large $\Gamma _{\text{f}}$, which applies to collider physics (and
situations like the one of the $Z$ boson), and the regime of small $\Gamma
_{\text{f}}$, which applies to quantum gravity (and situations like the one of
the muon).
