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 illdefined 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).

Purely virtual particles versus Lee-Wick ghosts: physical Pauli-Villars fields, finite QED and quantum gravity

“…For most purposes, the muon can be treated as a stable particle, since its width is very small (around 10 −19 GeV). If we resum the muon self-energies into the dressed propagator, as is commonly done for resonances, we find that the theory predicts no muon observation [8]. The reason is that we cannot observe an unstable particle with infinite resolving power on the energy, because this occurrence violates the energy-time uncertainty relation.…”

“…around the peak. The minus sign between ǫ and M gh Γ φ signals that the resummation cannot be trusted close to the peak, as shown in [8], so we have a peak uncertainty. We cannot apply Veltman's projection, because we do not know what dressed propagator we should use inside bigger diagrams.…”

“…The reason is that we cannot observe an unstable particle with infinite resolving power on the energy, because this occurrence violates the energy-time uncertainty relation. Once the energy resolution ∆E of the experimental setup is inserted explicitly, the problem disappears [8]. We can argue in a similar way for every resonance, if we imagine to boost it enough to make it detectable as a particle.…”

“…In ref. [8] it has been shown that the dressed propagators of the iǫ ghosts do not make sense close to the peaks, because the resummation of the perturbative expansion does not exist there. Having no knowledge about the nonperturbative sector of the theory, Veltman's projection cannot be applied to the iǫ ghosts.…”

“…The ǫ prescription is there to show that the corrections proportional to Γ have the same sign. According to the arguments of [8], it is correct to extend the resummation of the self energies from the convergence region to the peak region by means of analyticity.…”