Standard Model Pseudo-Observables and Unstable Particles

An unstable Standard Model particle is characterized by singularity data of amplitudes between stable external states, not by assigning it an LSZ state. The gauge-invariant core is the complex pole and the coefficients of the full amplitude’s pole expansion; masses, widths, effective couplings, and production–decay factorizations are useful pseudo-observables only after their pole, radiation, background, and acceptance conventions are fixed.

Required background. Unstable-particle observables and resonance approximations supplies resonance kinematics and distributional limits. Electroweak renormalization and input schemes supplies the renormalized parameters whose scheme must remain fixed in a pole extraction.

Helpful background. Unstable-particle effective theory organizes systematic expansions when the width and other small scales must be power counted together.

Complex poles belong to amplitudes with stable external states

Let $s$ be the invariant mass flowing through a resonant channel of an amplitude $\mathcal A(s,\Omega)$; $\Omega$ denotes all remaining kinematics and quantum numbers. Analytic continuation to the appropriate unphysical sheet gives an isolated simple pole $s_\star$. In a neighborhood containing no other singularity,

$$\mathcal A(s,\Omega) =\frac{R(\Omega)}{s-s_\star}+N(s,\Omega),$$

where $N$ is regular at $s_\star$. Analyticity makes this a Laurent expansion, not a diagrammatic choice. For a physical amplitude between stable states, the pole position, its residue, and the regular remainder defined by the expansion are separately gauge independent; selecting only “resonant diagrams” generally does not inherit that property Stuart 1991, pp. 114–116.

This page uses

$$s_\star=\left(M_\star-\frac{i}{2}\Gamma_\star\right)^2 =M_\star^2-iM_\star\Gamma_\star-\frac{\Gamma_\star^2}{4}.$$

Another common convention writes $s_\star=\mu^2-i\mu\gamma$. The pole $s_\star$ is the invariant object; $M_\star,\Gamma_\star$ and $\mu,\gamma$ are different real parameterizations of it beyond leading order in $\Gamma/M$. A quoted “mass” or “width” is therefore incomplete without its convention.

For a scalar propagator written locally as

$$\Delta(s)=\frac{i}{D(s)}, \qquad D(s)=s-m_0^2-\Sigma(s),$$

the pole solves $D(s_\star)=0$. Taylor expansion gives

$$\Delta(s) =\frac{iZ_\star}{s-s_\star}+O(1), \qquad Z_\star=\frac{1}{1-\Sigma'(s_\star)}.$$

The sign in $Z_\star$ follows the displayed definition of $D$; reversing the sign convention for $\Sigma$ reverses the derivative sign. In a gauge theory, neither a resummed propagator nor $Z_\star$ taken in isolation is automatically an observable. Vertices, mixing, and nonresonant terms must be combined into the residue of the full stable-state amplitude Stuart 1995, Eqs. (10)–(13), pp. 4–6, PDF.

No asymptotic ket $|V\rangle$ is introduced for an unstable vector boson. Consequently, a “partial width” is a residue-derived pseudo-observable with a specified treatment of radiation and final-state definitions, not the modulus squared of an exact S-matrix element with $V$ as an external particle.

From the pole expansion to production and decay

Near one isolated narrow resonance, the squared propagator factor has the distributional limit

$$\frac{1}{(s-M^2)^2+M^2\Gamma^2} \xrightarrow[\Gamma/M\to0]{\text{distribution}} \frac{\pi}{M\Gamma}\,\delta(s-M^2).$$

To see the normalization, set $x=(s-M^2)/(M\Gamma)$ in an integral against a smooth test function $f$:

$$\begin{aligned} \int ds\,\frac{f(s)}{(s-M^2)^2+M^2\Gamma^2} &=\frac{1}{M\Gamma}\int dx\,\frac{f(M^2+M\Gamma x)}{1+x^2}\\ &\longrightarrow \frac{\pi}{M\Gamma}f(M^2). \end{aligned}$$

If the residue factorizes into production and decay factors and the measurement is sufficiently inclusive and smooth across the resonant region, this produces the familiar schematic relation

$$d\sigma(i\to V\to f) \simeq d\sigma(i\to V)\,\frac{d\Gamma(V\to f)}{\Gamma}.$$

The approximation requires more than $\Gamma/M\ll1$: the pole must be isolated; thresholds and phase-space boundaries must not vary on the width scale; cuts must not select only a distorted tail; and interference with regular amplitudes must not be parametrically enhanced. Its error is therefore not universally “of order $\Gamma/M$.” A calculation must test the actual observable Stuart 1995, pp. 1–3, PDF.

For two resonances with virtualities $s_1,s_2$, a double-pole expansion has the form

$$\mathcal A =\frac{R_{22}}{(s_1-s_{1\star})(s_2-s_{2\star})} +\frac{R_{12}}{s_1-s_{1\star}} +\frac{R_{21}}{s_2-s_{2\star}} +N.$$

The double-pole term may dominate in a doubly resonant region, but the single-pole and regular terms are part of the amplitude. Dropping them is an approximation with a phase-space-dependent error, not a gauge-invariant definition of “signal.”

Pseudo-observables and fiducial observables answer different questions

Object	Definition	What must accompany it	Typical limitation
Complex pole	$s_\star$ of the analytically continued full amplitude	Sheet and real pole parameterization	Does not alone specify channel-dependent residues
Pole residue or effective coupling	Coefficient of $(s-s_\star)^{-1}$, projected onto stated structures	Tensor basis, normalization, input and radiation scheme	May not map uniquely to a Lagrangian coupling
Pole partial width	Channel-specific residue combination under a radiation convention	Final-state definition, QED/QCD radiator, inclusiveness	Not an unstable-particle LSZ matrix element
Line-shape pseudo-observable	Compact parameters obtained after removing specified radiation/background effects	Extraction map, nonresonant model, acceptance and covariance	Model dependence enters through the map
Fiducial cross section	Stable-particle cross section with an explicit measurement function	Object definitions, cuts, binning, dressing and units	Detector correction and extrapolation still require a response model

For example, the conventional pole-level quantity

$$\sigma^0_{\rm had} =\frac{12\pi}{M_Z^2}\frac{\Gamma_e\Gamma_{\rm had}}{\Gamma_Z^2}$$

summarizes pole residues after specified photon-radiation, interference, and line-shape treatments; it is not the cross section obtained by applying detector cuts to events. The LEP line-shape construction explicitly separates measured cross sections and asymmetries from the pseudo-observable fit and records their correlations LEP and SLD Electroweak Working Groups 2006, §§1.5 and 2.3–2.6, pp. 33–50.

A defensible map therefore records both endpoints:

Field	Required content
Pseudo-observable definition	Pole convention, projected residue or width definition, radiation and background subtraction
Fiducial definition	Stable-particle objects, recombination/dressing, cuts, bin edges, normalization and units
Map	Acceptance or response, perturbative order, generator or analytic method, nonresonant and interference treatment
Inference object	Data ordering, covariance or likelihood, nuisance meanings and shared correlations
Identity	Dataset and table identifiers, exact version, DOI or stable URL, checksum when supplied
Lifecycle	Corrections, superseded objects, and the version used by a result

This record is part of the scientific definition. A newer corrected table must not silently replace the table used in a previously reported fit.

Independent checks

Dimensions. Since a four-dimensional invariant amplitude has convention-dependent overall dimension, check the displayed decomposition consistently: $s-s_\star$ has mass dimension two, so its residue has two more powers of mass than $\mathcal A$. The factor $\pi/(M\Gamma)\,\delta(s-M^2)$ has dimension $M^{-4}$, matching the Breit–Wigner denominator.

Gauge parameter. Vary a gauge-fixing parameter in an explicit calculation. The extracted full-amplitude pole and residue must be unchanged, even though self-energy, vertex, and box subsets can vary.

Stable and narrow limits. As the couplings producing the width are turned off, the pole approaches the real axis. Integrating a normalized Breit–Wigner over a smooth region must reproduce the delta-function limit.

Recombination. Expanding the pole term and regular remainder to the fixed perturbative order must reproduce the fixed-order amplitude in their common domain. A selective resummation that fails this check has changed terms inconsistently.

Threshold and cut sensitivity. Move phase-space boundaries relative to the resonance and compare the pole approximation with the full calculation. Large changes signal that production–decay factorization is not uniform for that observable.

Common pitfalls

Putting an unstable particle in the asymptotic Hilbert space. LSZ reduction applies to stable one-particle poles on the physical spectrum. Define resonance information through stable-state amplitudes and their analytic continuation.

Resumming only a convenient self-energy. A selectively resummed propagator mixed with fixed-order vertices can violate gauge cancellations and perturbative counting. Use a systematically matched pole expansion or another gauge-consistent scheme.

Calling every width a physical observable. A pole width is conventionally parameterized from $s_\star$; a channel width additionally requires residue, radiation, and final-state conventions. State which one is meant.

Using a narrow-width approximation in a sculpted tail. Small inclusive width does not control a cut that removes the pole region or magnifies interference. Validate the approximation for the measurement function actually used.

Informal self-check

Show that the two real pole parameterizations agree at leading relative order in the width, and find their first difference.

Solution

Equating

$$\mu^2-i\mu\gamma =M_\star^2-iM_\star\Gamma_\star-\frac{\Gamma_\star^2}{4}$$

gives $\mu^2=M_\star^2-\Gamma_\star^2/4$ and $\mu\gamma=M_\star\Gamma_\star$. Hence $\mu=M_\star+O(\Gamma_\star^2/M_\star)$ and $\gamma=\Gamma_\star+O(\Gamma_\star^3/M_\star^2)$. They agree through leading order but differ once terms quadratic in $\Gamma/M$ are retained. The pole itself has not changed.

Handoffs

• Use electroweak precision observables when pole quantities enter a correlated electroweak prediction.
• Use Higgs precision and coupling inference when production, decay, and total-width assumptions must be separated.
• Use collider measurements and likelihood provenance when mapping a pole or fiducial prediction to a released result.
• Use unstable-particle effective theory when a systematic expansion in the width is required.

References

• LEP Collaborations, ALEPH, DELPHI, L3, OPAL, SLD Collaborations, LEP Electroweak Working Group, SLD Electroweak and Heavy Flavour Groups. “Precision Electroweak Measurements on the $Z$ Resonance.” Physics Reports 427 (2006) 257–454. DOI · Open PDF
• Stuart, Robin G. “Gauge Invariance, Analyticity and Physical Observables at the $Z^0$ Resonance.” Physics Letters B 262 (1991) 113–119. DOI
• Stuart, Robin G. “Unstable Particles.” In Electroweak Physics and the Early Universe, NATO ASI Series B 338 (1995) 461–470. Open PDF