Fakeon Prescription in Quantum Field Theory

• Fakeon Prescription is a quantization rule that defines amplitudes via average continuation, preserving unitarity and Lorentz invariance while relinquishing strict analyticity.
• It utilizes a principal-value prescription in propagators to exclude fake degrees of freedom from asymptotic states, ensuring locality of counterterms.
• The approach is applied in advanced quantum gravity and cosmology models, providing a controlled perturbative framework for higher-derivative theories.

Searching arXiv for the cited fakeon-prescription papers to ground the article in current literature. The fakeon prescription is a rule for defining quantum-field-theory amplitudes in the presence of poles off the real axis, including complex-conjugate pairs, such that perturbative unitarity and exact Lorentz invariance are preserved at the price of sacrificing strict analyticity (Anselmi et al., 3 Mar 2025). In the broader fakeon framework, a fakeon is a purely virtual degree of freedom that propagates internally in Feynman diagrams but never appears as an asymptotic external state; its propagator is assigned a principal-value or average prescription rather than the standard Feynman $i\epsilon$ prescription (Anselmi, 2022). Within the comparative analysis of amplitude prescriptions for theories with complex poles, the fakeon prescription is distinguished by preserving unitarity, Lorentz invariance, and locality of counterterms, while giving up analyticity as a single holomorphic dependence on $p^2$ (Anselmi et al., 3 Mar 2025).

1. Definition and conceptual setting

In the fakeon framework, a fakeon is a degree of freedom that propagates internally in Feynman diagrams but never appears as an asymptotic external state (Anselmi, 2022). Unlike a physical particle, whose free propagator in momentum space is

$P_{\rm phys}(p)=\dfrac{i}{p^2-m^2+i\epsilon},$

and unlike a ghost field, which carries a wrong-sign residue with the usual $+i\epsilon$ prescription,

$P_{\rm ghost}(p)=-\,\dfrac{i}{p^2-m^2+i\epsilon},$

a fakeon is assigned the principal-value or average prescription

$$P_{\rm f}(p)=\lim_{\epsilon\to0}\dfrac{i\,(p^2-m^2)}{(p^2-m^2)^2+\epsilon^2} \;=\;\mathcal{P}\!\dfrac{i}{p^2-m^2}.$$

This prescription removes the fakeon from the space of asymptotic states while preserving unitarity for physical processes (Anselmi, 2022).

In theories with complex poles, the fakeon prescription is formulated more generally as the combination of the Lee–Wick contour for the loop energy $k^0$ with a deformation of the spatial-momentum integration domain into a complex path $D_{\bm k}\subset\mathbb{C}^{D-1}$ (Anselmi et al., 3 Mar 2025). An equivalent formulation is to compute the Euclidean amplitude and then perform the average continuation across each branch cut. The corresponding amplitude is

$$M_{\rm AP}(p^2) \;=\;\tfrac12\Bigl[M(p^2-i\epsilon)+M(p^2+i\epsilon)\Bigr].$$

This average continuation is the defining nonanalytic step of the prescription (Anselmi et al., 3 Mar 2025).

The same logic extends to theories with fractional powers of the d’Alembert operator. There, the direct fakeon prescription is again an average continuation of the Euclidean function through its cut, implemented by

$$G_{\rm dir}(p^2) \equiv \frac{i}{2}\Bigl[ \tfrac{1}{(-p^2+i\varepsilon)^\alpha} +\tfrac{1}{(-p^2-i\varepsilon)^\alpha} \Bigr] = -\,i\,\Re\!\Bigl[\tfrac{1}{(-p^2-i\varepsilon)^\alpha}\Bigr],$$

with the would-be branch cut at real $p^2$0 treated as a fakeon threshold (Anselmi, 25 Apr 2026). This suggests that the fakeon prescription is not restricted to isolated poles, but can also be used to define Minkowskian continuations of nonlocal or fractional kinetic terms.

2. Position among alternative amplitude prescriptions

The comparative status of the fakeon prescription is most clearly seen by setting it against three inequivalent alternatives for field theories with complex poles (Anselmi et al., 3 Mar 2025). These are the by-the-book Wick rotation, the Lee–Wick–Nakanishi prescription, direct Minkowski integration, and the fakeon prescription itself.

Prescription	Preserved properties	Violated properties
By-the-book	Lorentz invariance, locality of counterterms, analyticity	unitarity
Lee–Wick–Nakanishi	unitarity, locality	Lorentz invariance, analyticity
Direct Minkowski	Lorentz invariance, analyticity	unitarity, locality of counterterms
Fakeon (AP)	Lorentz invariance, unitarity, locality of counterterms	analyticity

The by-the-book prescription computes loop integrals entirely in Euclidean signature and then analytically continues the external momenta below the branch cuts. It preserves Lorentz invariance, locality of counterterms, and analyticity, but violates the optical theorem when purely virtual modes are present (Anselmi et al., 3 Mar 2025). The Lee–Wick–Nakanishi prescription works in Lorentzian signature, integrating $p^2$1 along the Lee–Wick contour while keeping $p^2$2; this enforces the optical theorem but breaks Lorentz invariance and spoils analyticity (Anselmi et al., 3 Mar 2025). Direct Minkowski integration preserves Lorentz invariance and analyticity, but the optical theorem fails and non-local divergences appear, violating the locality of counterterms and BPHZ renormalizability (Anselmi et al., 3 Mar 2025).

The fakeon prescription is singled out because it satisfies unitarity, exact Lorentz invariance, and locality of counterterms, with analyticity as the only sacrificed property (Anselmi et al., 3 Mar 2025). The paper on amplitude prescriptions states that the least physically harmful violation is analyticity, characterized there as a technical rather than conceptual property (Anselmi et al., 3 Mar 2025). A plausible implication is that the fakeon program treats analyticity as subordinate to the conjunction of the optical theorem, Lorentz symmetry, and renormalizability.

This same hierarchy of priorities is visible in the fakeon treatment of purely virtual particles more generally. In the diagrammatic formulation, fakeon cuts vanish identically, so no fakeon threshold contributes to the unitarity sum, while all surviving cuts involve only physical degrees of freedom (Anselmi, 2021, Anselmi, 2022). The result is not a modification of the optical theorem, but a projection of the spectrum entering it.

3. Contours, propagators, and average continuation

A minimal example of a fakeon propagator in a higher-derivative theory with complex-conjugate poles is

$p^2$3

with denominator

$p^2$4

and, in the simplest massless case,

$p^2$5

The four simple poles in $p^2$6 are located at

$p^2$7

with principal-branch conventions (Anselmi et al., 3 Mar 2025).

The loop energy $p^2$8 is integrated along the Lee–Wick contour $p^2$9, described as a path running along the real axis but detouring in finite anticlockwise lassos around non-real poles in the first or third quadrants (Anselmi et al., 3 Mar 2025). The spatial momentum $P_{\rm phys}(p)=\dfrac{i}{p^2-m^2+i\epsilon},$0 is not kept on $P_{\rm phys}(p)=\dfrac{i}{p^2-m^2+i\epsilon},$1, but deformed to a complex domain $P_{\rm phys}(p)=\dfrac{i}{p^2-m^2+i\epsilon},$2. This joint contour is chosen so that no singularity is crossed as the external energy varies, and the amplitude respects both unitarity and Lorentz invariance (Anselmi et al., 3 Mar 2025).

For ordinary fakeons with real mass poles, the propagator is the principal value

$P_{\rm phys}(p)=\dfrac{i}{p^2-m^2+i\epsilon},$3

and all cut propagators of a fakeon leg vanish (Anselmi, 2021). In skeleton language, no $P_{\rm phys}(p)=\dfrac{i}{p^2-m^2+i\epsilon},$4-function appears when such a line is cut (Anselmi, 2021). This is the elementary algebraic mechanism behind the statement that fakeons are purely virtual.

For fractional kinetic operators, the direct fakeon prescription yields

$P_{\rm phys}(p)=\dfrac{i}{p^2-m^2+i\epsilon},$5

which is purely imaginary on the real axis and reduces to the principal-value Klein–Gordon propagator when $P_{\rm phys}(p)=\dfrac{i}{p^2-m^2+i\epsilon},$6 (Anselmi, 25 Apr 2026). This suggests a structural continuity between the fakeon prescription for isolated poles and its treatment of branch cuts as fake thresholds.

4. Loop integration and the nonanalytic structure

The construction of amplitudes under the fakeon prescription begins from the Euclidean amplitude. In the prototype one-loop bubble,

$P_{\rm phys}(p)=\dfrac{i}{p^2-m^2+i\epsilon},$7

with all momenta Euclidean (Anselmi et al., 3 Mar 2025). After partial-fraction decomposition, one integrates $P_{\rm phys}(p)=\dfrac{i}{p^2-m^2+i\epsilon},$8 along $P_{\rm phys}(p)=\dfrac{i}{p^2-m^2+i\epsilon},$9, obtaining terms of the form

$+i\epsilon$0

plus complex conjugates and $+i\epsilon$1 mirrors (Anselmi et al., 3 Mar 2025).

The singularity loci in the complex $+i\epsilon$2 plane are the curves $+i\epsilon$3 and $+i\epsilon$4, which for real $+i\epsilon$5 span extended bubble-shaped regions $+i\epsilon$6 and $+i\epsilon$7 (Anselmi et al., 3 Mar 2025). If the spatial domain were kept on $+i\epsilon$8, these regions would break Lorentz invariance. The fakeon prescription instead deforms the spatial domain so that for every real $+i\epsilon$9 outside the normal threshold $P_{\rm ghost}(p)=-\,\dfrac{i}{p^2-m^2+i\epsilon},$0, the regions are pinched onto the real $P_{\rm ghost}(p)=-\,\dfrac{i}{p^2-m^2+i\epsilon},$1 axis, collapsing the extended regions into standard cuts $P_{\rm ghost}(p)=-\,\dfrac{i}{p^2-m^2+i\epsilon},$2 (Anselmi et al., 3 Mar 2025). In $P_{\rm ghost}(p)=-\,\dfrac{i}{p^2-m^2+i\epsilon},$3, the deformation locus can be written as

$P_{\rm ghost}(p)=-\,\dfrac{i}{p^2-m^2+i\epsilon},$4

The resulting amplitude coincides with the average continuation of the Euclidean result,

$P_{\rm ghost}(p)=-\,\dfrac{i}{p^2-m^2+i\epsilon},$5

and is real for real $P_{\rm ghost}(p)=-\,\dfrac{i}{p^2-m^2+i\epsilon},$6, so the optical theorem is satisfied (Anselmi et al., 3 Mar 2025). Analyticity is broken because the amplitude is not given by a single holomorphic function across the cut, but by a piecewise definition obtained by averaging two analytic continuations (Anselmi et al., 3 Mar 2025).

The Euclidean one-loop bubble amplitude in the prototype cubic model is

$P_{\rm ghost}(p)=-\,\dfrac{i}{p^2-m^2+i\epsilon},$7

In practice, one may either compute the Euclidean integral and then replace the ordinary analytic continuation by the average continuation at each threshold, or work directly in Lorentzian signature with the fakeon $P_{\rm ghost}(p)=-\,\dfrac{i}{p^2-m^2+i\epsilon},$8 contour plus the deformed complex-$P_{\rm ghost}(p)=-\,\dfrac{i}{p^2-m^2+i\epsilon},$9 path; both routes yield the same Lorentz- and unitary-invariant answer (Anselmi et al., 3 Mar 2025).

For fractional theories, the same pattern appears in one-loop bubbles. The Euclidean result

$$P_{\rm f}(p)=\lim_{\epsilon\to0}\dfrac{i\,(p^2-m^2)}{(p^2-m^2)^2+\epsilon^2} \;=\;\mathcal{P}\!\dfrac{i}{p^2-m^2}.$$0

is average-continued to

$$P_{\rm f}(p)=\lim_{\epsilon\to0}\dfrac{i\,(p^2-m^2)}{(p^2-m^2)^2+\epsilon^2} \;=\;\mathcal{P}\!\dfrac{i}{p^2-m^2}.$$1

which is pure imaginary on the real axis (Anselmi, 25 Apr 2026). The coincidence between purity of the imaginary character and absence of asymptotic fake states is explicit in that formulation.

5. Unitarity, optical identities, and threshold projection

A central feature of the fakeon prescription is that cuts through fakeon lines vanish identically (Anselmi, 2022). In the diagrammatic formulation, this is codified through spectral optical identities that hold threshold by threshold and even at the level of skeleton diagrams, before integrating over spatial loop momenta and phase spaces (Anselmi, 2021). The general identity is

$$P_{\rm f}(p)=\lim_{\epsilon\to0}\dfrac{i\,(p^2-m^2)}{(p^2-m^2)^2+\epsilon^2} \;=\;\mathcal{P}\!\dfrac{i}{p^2-m^2}.$$2

where $$P_{\rm f}(p)=\lim_{\epsilon\to0}\dfrac{i\,(p^2-m^2)}{(p^2-m^2)^2+\epsilon^2} \;=\;\mathcal{P}\!\dfrac{i}{p^2-m^2}.$$3 is the uncut skeleton, $$P_{\rm f}(p)=\lim_{\epsilon\to0}\dfrac{i\,(p^2-m^2)}{(p^2-m^2)^2+\epsilon^2} \;=\;\mathcal{P}\!\dfrac{i}{p^2-m^2}.$$4 its complex conjugate, and $$P_{\rm f}(p)=\lim_{\epsilon\to0}\dfrac{i\,(p^2-m^2)}{(p^2-m^2)^2+\epsilon^2} \;=\;\mathcal{P}\!\dfrac{i}{p^2-m^2}.$$5 the skeletons of its cut versions (Anselmi, 2021).

The fakeon projection is implemented by dropping all terms proportional to $$P_{\rm f}(p)=\lim_{\epsilon\to0}\dfrac{i\,(p^2-m^2)}{(p^2-m^2)^2+\epsilon^2} \;=\;\mathcal{P}\!\dfrac{i}{p^2-m^2}.$$6-factors that involve fakeon frequencies (Anselmi, 2021). For the one-loop bubble with both legs physical,

$$P_{\rm f}(p)=\lim_{\epsilon\to0}\dfrac{i\,(p^2-m^2)}{(p^2-m^2)^2+\epsilon^2} \;=\;\mathcal{P}\!\dfrac{i}{p^2-m^2}.$$7

while if one leg is made a fakeon the threshold terms are dropped and one obtains

$$P_{\rm f}(p)=\lim_{\epsilon\to0}\dfrac{i\,(p^2-m^2)}{(p^2-m^2)^2+\epsilon^2} \;=\;\mathcal{P}\!\dfrac{i}{p^2-m^2}.$$8

with no nontrivial cuts left (Anselmi, 2021). The same threshold-dropping rule extends to triangle, box, pentagon, hexagon, and multi-loop examples (Anselmi, 2021).

This algebraic projection is the mechanism by which fakeons preserve unitarity without appearing as physical states. In the formulation of dressed propagators, the width $$P_{\rm f}(p)=\lim_{\epsilon\to0}\dfrac{i\,(p^2-m^2)}{(p^2-m^2)^2+\epsilon^2} \;=\;\mathcal{P}\!\dfrac{i}{p^2-m^2}.$$9 of a fakeon is positive by the optical theorem, but it is not a conventional decay width into external states; it is an internal absorption parameter governing how internal fakeon exchanges build up the imaginary parts of loop amplitudes (Anselmi, 2022). The dressed propagator is

$k^0$0

where $k^0$1 is the experimental or theoretical energy resolution around the fakeon peak (Anselmi, 2022).

A characteristic consequence is the peak-uncertainty bound

$k^0$2

which expresses an intrinsic impossibility to resolve the fakeon peak more sharply than its own width allows (Anselmi, 2022). Below this minimum resolution, the perturbative series ceases to converge near the would-be pole and nonperturbative effects tied to the nonanalytic structure of the fakeon prescription become unavoidable (Anselmi, 2022). This does not alter the statement that cuts through fakeon lines vanish; rather, it constrains how closely one can probe the internal fakeon exchange.

6. Applications and extensions

The amplitude analysis of field theories with complex poles concludes that only the fakeon prescription is physically viable and can have applications to quantum gravity (Anselmi et al., 3 Mar 2025). In higher-derivative gravity, the mode generated by the $k^0$3 term would be a ghost under the ordinary Feynman prescription, but is quantized as a fakeon by replacing

$k^0$4

or, equivalently, by dropping any absorptive part proportional to $k^0$5 in loop diagrams (Anselmi, 2020). This enforces that the spin-2 mode $k^0$6 never appears as an external, on-shell state, yet propagates internally (Anselmi, 2020).

In the renormalizable theory $k^0$7, the fakeon prescription is used to compute inflationary perturbation spectra to next-to-next-to-leading log order (Anselmi, 2020). The tensor fluctuations receive contributions from the spin-2 fakeon at every order in the expansion in powers of $k^0$8, while the dependence of the scalar spectrum on the fakeon mass $k^0$9 starts from the $D_{\bm k}\subset\mathbb{C}^{D-1}$0 corrections (Anselmi, 2020). The stated theoretical errors range from $D_{\bm k}\subset\mathbb{C}^{D-1}$1 to $D_{\bm k}\subset\mathbb{C}^{D-1}$2, depending on the observable (Anselmi, 2020). These results place the fakeon prescription within a concrete perturbative cosmological program.

The same conceptual framework extends to nonlocal quantum gravity, fractional quantum gravity, and Lee–Wick extensions of the Standard Model, which are listed as applications of the fakeon prescription in the amplitude study (Anselmi et al., 3 Mar 2025). In fractional field theories, the direct average-continuation prescription and the decomposition prescription into a continuum of ordinary fakeons produce inequivalent Minkowskian theories with the same Euclidean counterpart (Anselmi, 25 Apr 2026). Both, however, satisfy Ward and Cutkosky identities (Anselmi, 25 Apr 2026). This suggests that the fakeon prescription is not a unique Minkowskian completion of a Euclidean model in every generalized setting, although it remains a controlled route to perturbative unitarity.

A recurring misconception is that fakeons are merely ghosts with a modified contour choice. The cited literature distinguishes them sharply: ghosts carry negative norm with the standard $D_{\bm k}\subset\mathbb{C}^{D-1}$3 prescription and violate unitarity, whereas fakeons have positive metric, obey a principal-value or average prescription, never appear on external legs, and preserve unitarity for physical processes (Anselmi, 2022). Another misconception is that the fakeon prescription simply reproduces ordinary analyticity by a nonstandard derivation. The amplitude analysis states the contrary: analyticity is genuinely lost, since the amplitude is defined by averaging across cuts rather than by a single holomorphic branch (Anselmi et al., 3 Mar 2025).

The resulting picture is a distinct quantization rule for unwanted or purely virtual degrees of freedom. It preserves the optical theorem, exact Lorentz invariance, and local counterterms, while replacing standard analyticity with average continuation and threshold projection (Anselmi et al., 3 Mar 2025, Anselmi, 2021).