Resonances, Infraparticles, and Limits of Particle Language

A resonance and an infraparticle both defeat the ordinary isolated-pole picture, but for different reasons. A resonance is diagnosed by a complex pole reached only after a specified channel function is analytically continued through a specified cut to a specified nonphysical sheet. It is not a complex eigenvalue of the self-adjoint Hamiltonian and not a normalizable unstable one-particle eigenstate. An infraparticle instead arises, under suitable long-range infrared hypotheses, when arbitrarily soft massless quanta make continuous mass-spectrum support reach a charged sector’s nominal lower mass while the isolated mass-shell atom disappears. It need not decay and needs no complex pole or width.

This page develops that physical warning map. It does not extract resonance poles, resum soft radiation, construct dressed states, prove an algebraic infraparticle theorem, or derive LSZ reduction.

Required background. Poles, Cuts, Thresholds, and Stable Particles supplies the physical-sheet pole/cut dictionary, sheet convention, finite-volume qualification, and distinction between a spectral atom and continuum support used here.

Literature status checked through 8 August 2026.

Stable particles, resonances, and infraparticles

Keep the invariant $z=p^2$ and the spectral variable $\sigma$ from the prerequisite page. For a centered Hermitian scalar operator in a positive physical Hilbert space, a stable particle seen by that operator contributes

$$\rho_{\mathcal O}(\mathrm d\sigma) =Z_{\mathcal O}\delta(\sigma-m^2)\,\mathrm d\sigma +\rho_{\mathrm{cont}}(\mathrm d\sigma), \qquad Z_{\mathcal O}>0.$$

If the atom is isolated from the continuum, it gives a real simple pole on the physical sheet. That is the stable pattern developed on the preceding page. The two failure patterns are not small deformations of it:

• A resonance has no physical spectral atom representing an unstable particle. Its defining pole appears in the analytic continuation of a suitable channel amplitude, or sometimes a suitable correlator, across a continuum cut.
• An infraparticle has no isolated mass atom under the declared infrared hypotheses because continuous charged-sector spectrum accumulates at its nominal mass. Its obstruction lives on the real physical spectrum, not at a complex continued-sheet pole.

An ordinary multiparticle threshold is a fourth, more basic object: it marks the endpoint of continuum support. It is neither a resonance nor an infraparticle merely because a cut begins there.

Pattern	Physical spectral datum	Analytic marker	Hilbert-space meaning
Stable particle	Isolated atom at $m^2$	Real physical-sheet pole	Invariant one-particle mass sector
Ordinary threshold	Continuum begins at $s_0$	Physical-sheet branch point and cut in infinite volume	Endpoint of a multiparticle continuum, not a particle
Resonance	No unstable-particle atom	Complex pole on a specified nonphysical sheet	Prepared continuum excitation, not an unstable eigenket
Infraparticle	No atom at the nominal mass; continuous support reaches it	Real threshold singularity whose detailed form is model-dependent	Long-range charged excitation inseparable from soft radiation

This table classifies singularity data. None of its rows alone proves asymptotic completeness or the existence of scattering limits.

Why a normalizable energy eigenstate cannot decay

Let $H$ be self-adjoint and suppose a normalized vector $|\Psi\rangle$ is an exact eigenvector,

$$H|\Psi\rangle=E|\Psi\rangle.$$

Self-adjointness makes $E$ real, and time evolution gives

$$e^{-iHt}|\Psi\rangle=e^{-iEt}|\Psi\rangle, \qquad \left|\langle\Psi|e^{-iHt}|\Psi\rangle\right|^2=1.$$

Only the phase changes. Such a vector cannot lose survival probability into decay products. Relativistic momentum eigenkets are distributional rather than normalizable, so the stable-particle statement is more precisely about an invariant one-particle spectral sector: the mass operator $P^2$ has an atom, and normalizable wave packets can be formed within the associated mass shell.

A physically prepared unstable excitation can certainly be a Hilbert-space vector. It is then a superposition over continuum energies rather than an exact unstable energy or mass eigenstate. If its energy measure is $\nu_\Psi$, its survival amplitude has the form

$$A_\Psi(t) =\langle\Psi|e^{-iHt}|\Psi\rangle =\int_{\mathbb R}e^{-iEt}\,\nu_\Psi(\mathrm dE).$$

A pole approximation may yield approximately exponential behavior over a useful time window, but that approximation does not turn a complex pole into an eigenvalue of $H$. “Gamow vectors” and other generalized resonance states can organize this approximation only in an enlarged dual-space formalism; they are not normalized Hilbert-space kets.

A resonance is a continued-sheet pole

Fix the physical sheet by its physical $+i0$ boundary values and the chosen signs of the channel momenta. In the right-hand-cut-only one-channel schematic used here—and for the two-point function on the prerequisite page—it is also connected to a nonsingular spacelike point $z<0$. Take the continuum cut along $[s_0,\infty)$. Continuing through that cut reverses the corresponding channel-momentum branch and reaches a second sheet. Only after this path has been fixed does the notation $F_{\mathrm{II}}$ have a definite meaning.

Near an isolated simple resonance pole on that sheet,

$$F_{\mathrm{II}}(z) =\frac{r_R}{z-z_R}+F_{\mathrm{reg}}(z), \qquad \operatorname{Im}z_R<0$$

in the usual causal convention. If one chooses the square-root branch with positive real part and negative imaginary part, the pole parameters may be defined by

$$\sqrt{z_R}=M_R-\frac{i}{2}\Gamma_R, \qquad M_R>0, \qquad \Gamma_R>0.$$

Thus $z_R=(M_R-i\Gamma_R/2)^2$. This is an isolated-pole convention, not a universal Breit–Wigner fit on the real axis. The residue $r_R$ is generally complex and channel-dependent; it is not a positive probability like the weight of a physical spectral atom.

Particle Data Group 2025, “Resonances,” § 50.2, p. 10, Eqs. (50.20)–(50.24) (PDF) defines the pole mass and width from $\sqrt{z_R}$ and separates them from real-axis Breit–Wigner parameters and background choices.

In a coupled-channel problem, every channel momentum and threshold introduces its own branch choice. There may be several adjacent sheets, and which one borders the physical region depends on the energy. The phrase “the second sheet” is therefore ambiguous unless the channel and sign convention are named. A real pole below threshold on the physical sheet is a bound-state pattern; a real pole on a nonphysical sheet can instead be a virtual-state pattern. Neither should be relabeled a resonance merely because it is a pole. Detailed continuation, pole extraction, residue factorization, and broad or overlapping resonances belong to the later scattering treatment.

Real-axis data provide clues, not the definition. Threshold cusps, kinematic singularities, and interference with backgrounds can make bumps without resonance poles, while a genuine pole may produce no obvious bump in a particular observable. The 2025 Particle Data Group resonance review emphasizes both facts and treats the pole only after the relevant sheets and channels are specified Particle Data Group 2025, “Resonances,” § 50.1.1, pp. 4–5 (PDF).

Long-range fields can remove an isolated mass shell

An infraparticle is most cleanly described through the physical mass spectrum rather than through a gauge-fixed charged-field propagator. Let $\mathcal H_q$ be the charged sector, let $E_{P^2}$ be the projection-valued spectral measure of the self-adjoint mass-squared operator there, and define

$$m_*^2=\inf\operatorname{spec}\!\left(P^2\big|_{\mathcal H_q}\right).$$

A schematic sharp lower edge without a mass eigenstate is then

$$E_{P^2}(\{m_*^2\})=0, \qquad E_{P^2}\!\left((m_*^2,m_*^2+\epsilon)\right)\ne0 \quad\text{for every }\epsilon>0.$$

The first condition removes the atom. The second says that continuous spectrum comes arbitrarily close to the nominal mass. It deliberately does not assume that a density exists or that the threshold obeys a universal power law.

In a long-range electromagnetic charged sector, Gauss’ law ties charge to electric flux at arbitrarily large distance. A charged excitation must carry a correlated electromagnetic field, and photons of arbitrarily low energy can modify that field without opening a positive mass gap above the charged excitation. Buchholz assumes that the spacelike asymptotic electromagnetic field has convergent expectation values and bounded fluctuations, so its nonzero flux determines the charge. Under those hypotheses, the charged vector cannot be an eigenvector of the mass operator Buchholz 1986, DESY 86-035, pp. 1, 6–7 (Open PDF). The soft cloud is correlated with the charge’s motion; separating it as a finite collection of ordinary Fock photons is precisely what fails.

The hypotheses matter. The modern review by Duch and Dybalski distinguishes the usual infraparticle representations, in which an asymptotic flux is defined, from infravacuum representations whose large-distance field fluctuations can obstruct that flux limit. It also stresses that full relativistic QED is not available as a constructed nonperturbative model, so rigorous results use algebraic assumptions, controlled models, or perturbation theory Duch and Dybalski 2023, §§ 1 and 2.2, pp. 1–4 (Open manuscript PDF). This alternative does not by itself restore a sharp mass atom; it shows why the Buchholz conclusion must not be detached from its representation and asymptotic-field assumptions.

Several boundaries follow immediately:

• Not every theory containing a massless field has infraparticles.
• A stable neutral particle can retain an isolated mass shell in a theory that also contains massless quanta.
• An infraparticle need not have a decay channel or a width. Charge may be exactly conserved while the sharp-mass Wigner-particle description fails.
• A gauge-fixed electron propagator lives in an auxiliary, gauge-dependent setting; its spectral function does not automatically inherit the positive scalar measure used on the prerequisite page.
• Gauge-invariant charged fields are necessarily nonlocal in the relevant Gauss-law sense, so a local neutral scalar operator is not a universal template for the charged sector.

A second-sheet pole versus an infraparticle threshold

The first QFT contrast can be made without performing either specialist calculation.

For a four-dimensional scalar model, treat

$$\mathcal L_{\mathrm{int}}=-\frac{g}{2}\chi\phi^2$$

as a weak-coupling perturbative toy interaction inside a theory whose full potential is stabilized. Suppose the physical $\chi$-like excitation lies above the allowed two-$\phi$ threshold and its on-shell decay amplitude is nonzero. At order $g^2$ the self-energy then acquires an imaginary part above threshold, and a narrow-width resummation can display the continued complex-pole signal. Schwartz 2014, §§ 24.1.1 and 24.1.4, pp. 455–456, 461–463 supports precisely this perturbative statement and its narrow-width qualification.

To state the exact spectral contrast, add the hypothesis that the full theory has no stable embedded $\chi$-like eigenstate in that channel. Under that extra hypothesis the physical spectral measure has continuum support but no delta atom for an unstable $\chi$ particle; in a resonance regime, a suitable amplitude or correlator has a pole only after continuation through the two-particle cut. If the physical mass lies below threshold instead, the decay is kinematically closed and an isolated stable pole can remain.

For the charged QED pattern under the Gauss-law hypotheses above, the issue is the opposite. The electron is not being diagnosed as an unstable decay resonance. Rather, charged states are inseparable from arbitrarily soft photons, so continuous spectrum reaches the nominal electron mass and the isolated atom is absent. No continuation to a complex pole is required for that conclusion.

Question	Stable scalar sector	Scalar resonance above an open channel	Charged infraparticle sector
Where is the decisive datum?	Physical mass spectrum	Analytic continuation across a named channel cut	Physical charged-sector mass spectrum
Is there an isolated atom at the quoted mass?	Yes	No unstable-particle atom	No, under the stated infrared hypotheses
What is the analytic marker?	Real physical-sheet pole	Complex pole on a named nonphysical sheet	Real lower spectral edge with no isolated pole; detailed singularity is model-dependent
Is it a normalizable one-particle eigenstate?	An invariant one-particle sector exists	No unstable eigenstate	No sharp-mass charged eigenstate under the theorem’s hypotheses
What physical language is appropriate?	Stable particle	Unstable resonance in a channel	Stable charge with an inseparable soft cloud
May it be used as an ordinary LSZ external leg?	Only after the additional scattering hypotheses are checked	No	Not without a modified infrared framework

The resonance loses a stable state because decay products form an open channel. The infraparticle loses an isolated mass shell because infinitely soft radiation removes the gap without requiring decay. Conflating the two replaces two distinct mechanisms with the vague phrase “a broadened particle.”

Regulator and finite-resolution cautions

Both patterns can be hidden by limits that have not yet been taken.

Finite spatial volume. At fixed volume the spectrum is discrete and the correlator is meromorphic. A resonance is encoded indirectly in the volume dependence of levels and matrix elements; it is not a complex finite-volume energy eigenvalue. An infrared threshold likewise becomes a set of discrete soft levels.

Photon-mass or soft-energy regulator. Giving the photon a small mass, imposing an infrared cutoff, or restricting the box can create an artificial gap and an apparently isolated charged pole. The pole proves an infraparticle has been avoided only if its isolation and nonzero weight survive the controlled massless, infinite-volume limit. The order of limits is part of the claim.

Finite detector resolution. Convolution with a resolution function can broaden a stable line, smooth a cusp, or make a threshold enhancement look resonance-like. Resolution changes an observed line shape, not the underlying sheet or spectral projection.

Truncated perturbation theory. A finite-order propagator, a partial resummation, or a fitted Breit–Wigner form may be useful, but none alone proves that the exact analytic continuation contains the claimed pole. Likewise, infrared logarithms in a gauge-fixed two-point function do not alone establish the theorem-level charged-sector spectrum.

What this warning map does not establish

Observation	What may be concluded	What remains unproved here
A channel function has a pole after a declared continuation	A resonance candidate with a specified pole and sheet	Its extraction uncertainty, coupled-channel stability, residue interpretation, or observable line shape
A real-axis bump fits a Breit–Wigner curve	The parametrization describes those data in that range	Existence or location of a continued-sheet pole
A continuum begins at $s_0$	There is a threshold in that channel	A resonance, infraparticle, or particle at $s_0$
A charged sector satisfies the Buchholz hypotheses	Sharp mass eigenstates are excluded there	A universal threshold exponent or a theorem about every formulation of QED
A regulated charged correlator has a pole	The regulated theory has that analytic feature	Survival of a physical isolated pole after removing the infrared regulator
One operator’s correlator has no pole at a known stable mass	That operator has no isolated overlap there	Absence of the stable particle from the theory
No ordinary isolated pole exists	The standard stable-particle prerequisite fails	Absence of meaningful inclusive, dressed, detector-based, or algebraic particle descriptions

The page also does not decide confinement, finite-temperature quasiparticles, open-system resonances, or unstable particles in external backgrounds. Those settings require different state spaces and observables.

Check your understanding

Test the eigenstate claim. A normalized vector satisfies $H|\Psi\rangle=(M-i\Gamma/2)|\Psi\rangle$ with $\Gamma>0$. Can this hold for a self-adjoint $H$?

Check

No. Every eigenvalue of a self-adjoint operator is real. A complex resonance energy can label a pole of an analytically continued function or a generalized vector outside the Hilbert space, but not a normalized eigenvector of $H$.

Diagnose a bump. A cross section has a pronounced peak just above a threshold. Is a resonance established?

Check

No. A threshold cusp or interference can produce a peak. One must specify the relevant channel function, continue it through the correct cut, and locate a robust pole on a declared nonphysical sheet. Conversely, a pole need not make an obvious bump in every observable.

Read the mass spectral edge. Suppose $E_{P^2}(\{m_*^2\})=0$ but every interval $(m_*^2,m_*^2+\epsilon)$ has nonzero spectral projection. Is this a narrow resonance?

Check

Not from these data. They describe a real continuous lower edge without an atom. Under suitable long-range charged-sector hypotheses this is the infraparticle pattern; a resonance would require a pole obtained by analytic continuation to a specified nonphysical sheet.

Remove an infrared regulator. At photon mass $\lambda>0$, a calculation shows an isolated charged pole. What must be checked before claiming an ordinary particle at $\lambda=0$?

Check

Track the pole’s gap and weight while taking the massless and infinite-volume limits in a declared order. A pole present at every nonzero regulator can merge into the continuum or lose its weight in the limit. Finite resolution must be removed or kept explicitly as part of the observable.

Continue by diagnosis

For a resonance, continue to Resonance Poles, Riemann Sheets, and Unstable States, which develops channel continuation, pole extraction, residues, and controlled line-shape approximations. For theorem-level charged sectors, velocity superselection, and alternative particle concepts, continue to Infraparticles and Velocity Superselection. Modified infrared scattering is developed in Dressed States and Infrared-Finite Scattering, while Soft Photons and Infrared-Finite QED treats the QED application. Dated open questions and competing infrared frameworks belong to Infrared-Complete Scattering Observables.

The stable route does not require this page. If the physical spectrum instead contains an isolated positive atom, continue to From One-Particle Poles to the Scattering Handoff and test the additional asymptotic-state, gap, overlap, and infrared assumptions there.

References

• Buchholz, Detlev. “Gauss’ Law and the Infraparticle Problem.” Physics Letters B 174 (1986): 331–334. DOI. Open PDF.

• Duch, Paweł, and Wojciech Dybalski. “Infrared Problem in Quantum Electrodynamics.” In Encyclopedia of Mathematical Physics, 2nd ed., vol. 5, 304–316. Elsevier, 2025. DOI. Lawful-access manuscript: arXiv:2307.06114 (2023), Open manuscript PDF.

• Particle Data Group (D. M. Asner, C. Hanhart, and M. Mikhasenko, review authors). “Resonances.” In S. Navas et al., “Review of Particle Physics,” Physical Review D 110 (2024): 030001; 2025 review update. Official PDF.

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. First ed. Cambridge: Cambridge University Press, 2014. DOI.