Dressed propagators, fakeon self-energy and peak uncertainty

Abstract

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 ∆E around the peak and argue that a “peak uncertainty” ∆E ≳ ∆E_min ≃ Γ_f/2 around energies E ≃ m_f expresses the impossibility to approach the fakeon too closely, m_f being the fakeon mass and Γ_f being the fakeon width. The introduction of ∆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 ∆E = 0, whenever we separate their observation from the observation of their decay products. We study the regime of large Γ_f, which applies to collider physics (and situations like the one of the Z boson), and the regime of small Γ_f, which applies to quantum gravity (and situations like the one of the muon).