Fakeon Framework in Quantum Field Theory

• Fakeon Framework is a formulation that defines purely virtual particles, characterizing them with a modified propagator that excludes on‐shell states.
• It employs an average continuation of propagators to eliminate on-shell singularities, thereby preserving both unitarity and renormalizability.
• Its applications span higher-derivative gravity and beyond Standard Model scenarios, offering a consistent method to treat ghost modes in loop processes.

A fakeon (fake particle, purely virtual quantum) is a quantized quantum field degree of freedom that propagates in internal lines of Feynman diagrams, contributing to loop and virtual processes, but is strictly excluded from the spectrum of asymptotic (on-shell) states. The Fakeon Framework introduces a precise nonanalytic quantization prescription ensuring that fakeon modes never correspond to physical particles, ghosts, or instabilities, while preserving both unitarity and renormalizability in quantum field theories—including higher-derivative gravity, Lee–Wick models, and beyond the Standard Model scenarios. Fakeons are defined by replacing the standard Feynman pole with a Cauchy principal value or an "average continuation" across would-be poles, thereby eliminating on-shell singularities and enforcing a characteristic "peak uncertainty" that fundamentally limits energy resolution near their masses. This prescription is uniquely physically viable for propagating would-be ghost or unwanted degrees of freedom, supporting a mathematically consistent perturbative S-matrix, see (Anselmi, 2022, Anselmi et al., 3 Mar 2025, Anselmi, 2020, Anselmi, 2018, Anselmi, 2021).

1. Definition and Motivation

A fakeon, or purely virtual particle, is characterized by a modified propagator that ensures it never goes on shell. Formally, for a scalar field $\chi$ of mass $m_{\mathrm{f}}$, the fakeon propagator in momentum space is

$$P_\chi(p^2) = \lim_{\varepsilon\to0} \frac{i(p^2 - m^2)}{(p^2 - m^2)^2 + \varepsilon^2} = \mathcal{P}\left[\frac{i}{p^2 - m^2}\right],$$

where $\mathcal{P}$ denotes the Cauchy principal value. Unlike a physical particle ($i/(p^2 - m^2 + i\epsilon)$) or a Faddeev–Popov ghost ($-i/(p^2 - m^2 + i\epsilon)$), a fakeon has vanishing on-shell residue and never appears in S-matrix cuts (Anselmi, 2022, Anselmi, 2018).

The central motivation is to realize a field-theoretic framework where degrees of freedom, such as those arising from higher-derivative kinetic terms (e.g., spin-2 ghosts in gravity, Lee–Wick partners), are included in internal lines to ensure renormalizability but removed from the physical spectrum to prevent violation of unitarity. The fakeon prescription achieves this by connecting Euclidean amplitudes to Minkowski-space via a nonanalytic "average continuation" across each fakeon threshold (Anselmi, 2020, Anselmi et al., 3 Mar 2025).

2. Propagator Structure, Quantization, and Feynman Rules

The fakeon prescription operates by modifying the analytic structure of the propagator. After including quantum corrections, the formal resummed fakeon propagator is

$$\widehat{P}_\chi(p^2) = \frac{i(p^2-m^2)}{[p^2 - m^2][p^2 - m^2 - \Sigma(p^2)] + \varepsilon^2},$$

where $\Sigma(p^2)$ is the one-particle-irreducible self-energy (Anselmi, 2022). In contrast to a standard Breit–Wigner form, this propagator lacks analytic continuation across the would-be resonance, reflecting the absence of a physical resonance.

Fakeon diagrammatics is defined by the following modifications:

• For each internal fakeon line, replace the Feynman propagator with the fakeon principal value.
• In multi-loop/cut diagram expansions, omit all contributions from cuts in which any fakeon frequency appears, so the unitarity sum is over physical particles only (Anselmi, 2021, Anselmi, 2018).
• The "average continuation" prescription is operationalized by taking the arithmetic mean of the analytic continuations above and below the real axis across each fakeon threshold:

$$M_\mathrm{fakeon}(p^2) = \frac{1}{2}\bigl[M(p^2 + i0) + M(p^2 - i0)\bigr].$$

This replaces the usual $i\epsilon$ deformation, erasing discontinuities associated with fakeon poles and ensuring no physical or ghostlike on-shell fakeon states (Anselmi et al., 3 Mar 2025, Anselmi, 2022). Interaction vertices are unaffected; only the internal-line prescription is modified.

3. Unitarity, Renormalizability, and Algebraic Structure

The Fakeon Framework is constructed to preserve the optical theorem and power-counting renormalizability, a property not shared by alternative prescriptions such as Feynman–Wheeler or ordinary Lee–Wick approaches (Anselmi et al., 3 Mar 2025, Anselmi, 2020). Perturbative unitarity is established via algebraic cutting equations, which remain valid when cut-propagators are modified to exclude fakeon cuts (Anselmi, 2021, Anselmi, 2016). For every loop diagram, the Cutkosky rules only acquire nonzero imaginary parts when the cut involves physical (Feynman-prescribed) lines; all fakeon contributions vanish: $m_{\mathrm{f}}$0 (Anselmi, 2020). Thus, the S-matrix is unitary in the physical subspace, and fakeons participate only virtually.

Renormalizability is maintained by working with a Euclidean theory, where the UV structure is unchanged, and continuing to Lorentzian signature using the average continuation prescription. The counterterm structure of the fakeon theory coincides with that of its Euclidean ancestor (Anselmi, 2018, Anselmi et al., 3 Mar 2025).

Table: Comparison of quantization prescriptions for complex poles (Anselmi et al., 3 Mar 2025):

Prescription	Unitarity (Optical Thm)	Lorentz Inv.	Renormalizable	Analyticity
Feynman	✓	✓	✓	✓
Feynman–Wheeler	✗	✓	✗	✓
Lee–Wick–Nakanishi	✓	✗	✓	✗
Fakeon	✓	✓	✓	✗

The fakeon prescription uniquely ensures perturbative unitarity, Lorentz invariance, and local counterterms, sacrificing only global analyticity at fakeon-induced thresholds.

4. Peak Uncertainty and Limitation of Energy Resolution

Resummation of self-energy diagrams for fakeons uncovers a distinctive "peak uncertainty." The geometric series defining the fakeon propagator converges only outside a region $m_{\mathrm{f}}$1 near $m_{\mathrm{f}}$2, where $m_{\mathrm{f}}$3 is the (formally defined) fakeon width (Anselmi, 2022). The region $m_{\mathrm{f}}$4 cannot be probed even in principle, since the fakeon cannot be approached arbitrarily closely by any observable process. This constitutes a form of quantum indeterminacy unique to fakeons, setting a lower bound on the experimental resolution near the would-be resonance.

Formally, for self-energy resummed fakeon propagator near its pole: $m_{\mathrm{f}}$5 with $m_{\mathrm{f}}$6 (Anselmi, 2022).

This peak uncertainty is an experimentally observable prediction: in collider or cosmological settings, no amount of increased energy resolution can resolve a fakeon narrower than its intrinsic half-width. In regimes such as quantum gravity, where $m_{\mathrm{f}}$7, this effect is beyond current observational reach, but for larger widths (e.g., near the $m_{\mathrm{f}}$8 boson in electroweak theory) it sets a measurable limit.

5. Applications in Quantum Gravity and Beyond

A leading application is in constructing a renormalizable, unitary theory of quantum gravity. Starting from a higher-derivative action such as

$m_{\mathrm{f}}$9

the dangerous spin-2 ghost mode is quantized as a fakeon—preserving power-counting renormalizability while eliminating negative-norm ghosts from the S-matrix (Anselmi et al., 2018, Anselmi, 2019, Anselmi, 2019, Anselmi, 2020).

The framework supports:

• Nonlocal, microcausal violations confined to scales $$P_\chi(p^2) = \lim_{\varepsilon\to0} \frac{i(p^2 - m^2)}{(p^2 - m^2)^2 + \varepsilon^2} = \mathcal{P}\left[\frac{i}{p^2 - m^2}\right],$$0, with macrocausality preserved (Anselmi et al., 2018, Anselmi, 2019).
• A classical limit (classicization), where fakeon-induced nonlocality arises in effective field equations (e.g., through Green's function averages), leading to only two initial data and no spurious solutions (Anselmi, 2019).
• Well-controlled phenomenology: fakeons do not yield missing energy signals or appear as direct resonance peaks, evading standard experimental bounds and opening new search strategies (Anselmi, 2022).

In inflationary cosmology, quantum gravity with fakeons predicts a sharply constrained tensor-to-scalar ratio, $$P_\chi(p^2) = \lim_{\varepsilon\to0} \frac{i(p^2 - m^2)}{(p^2 - m^2)^2 + \varepsilon^2} = \mathcal{P}\left[\frac{i}{p^2 - m^2}\right],$$1, robust against the specific fakeon mass scale (Anselmi et al., 2020, Anselmi, 2020, Anselmi, 2022).

6. Extensions, Boundary Conditions, and Limitations

The Fakeon Framework encompasses a broad class of modifications, including:

• Fractional powers of the d'Alembertian ($$P_\chi(p^2) = \lim_{\varepsilon\to0} \frac{i(p^2 - m^2)}{(p^2 - m^2)^2 + \varepsilon^2} = \mathcal{P}\left[\frac{i}{p^2 - m^2}\right],$$2), leading to nonlocal but unitary and classically well-posed theories when treated by average continuation (Anselmi, 25 Apr 2026).
• Complex-conjugate-pole (Lee–Wick-like) theories and spectral decompositions into a continuum of fakeons (Anselmi, 2018, Anselmi, 25 Apr 2026).

It fails for tachyonic degrees of freedom (masses-squared negative), as such "virtual tachyons" lead to unavoidable violations of Lorentz covariance, non-invariant commutators under boosts, and inconsistencies in the support of propagators, rendering a covariant, unitary, interacting theory impossible (Jodłowski, 24 Feb 2026).

In all versions, the choice of fakeon projection (average continuation) must precede any spectral decomposition when nonperturbative or ambiguous analytic continuations may arise. Multiple inequivalent Minkowski real-time theories can share the same Euclidean correlators; thus, physical requirements must dictate the correct prescription (Anselmi, 25 Apr 2026, Anselmi et al., 3 Mar 2025).

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References for the above discussion:

• (Anselmi, 2022): Dressed propagators, fakeon self-energy and peak uncertainty
• (Anselmi et al., 3 Mar 2025): Amplitude prescriptions in field theories with complex poles
• (Anselmi, 2020): The quest for purely virtual quanta: fakeons versus Feynman-Wheeler particles
• (Anselmi, 2018): Fakeons And Lee-Wick Models
• (Anselmi, 2021): Diagrammar of physical and fake particles and spectral optical theorem
• (Anselmi et al., 2018): Quantum Gravity, Fakeons And Microcausality
• (Anselmi, 2019): Fakeons and the classicization of quantum gravity: the FLRW metric
• (Piva, 2019): On The Behavior Of Gravitational Force At Small Scales
• (Anselmi et al., 2020): Predictions of quantum gravity in inflationary cosmology: effects of the Weyl-squared term
• (Anselmi, 2022): Purely Virtual Particles in Quantum Gravity, Inflationary Cosmology and Collider Physics
• (Anselmi, 2020): High-order corrections to inflationary perturbation spectra in quantum gravity
• (Anselmi, 2019): Fakeons, quantum gravity and the correspondence principle
• (Anselmi, 25 Apr 2026): Fractions and Fakeons in Quantum Field Theory
• (Jodłowski, 24 Feb 2026): Is a covariant virtual tachyon viable?
• (Anselmi, 2018): On the nature of the Higgs boson
• (Lubo et al., 2023): The Higgs Boson and The Fakeon Hypothesis
• (Anselmi, 2016): Algebraic cutting equations