Observables of QFT

QFT predicts numbers measured by experiments. The list of types of such numbers is wider than scattering cross sections alone — particle physics, atomic physics, condensed-matter QFT, and cosmology each draw from different parts of the formalism. This page is the map of those observables: what each one is, what mathematical object inside the QFT machinery produces it, and where in this repository it lives (or should live).

Cross sections and decay rates are by far the most prominent and have their own dedicated pages (cross-sections.md, decay-rates.md); this doc places them inside the broader inventory. Per-theory observable inventories live in electroweak.md (EW-sector observables with overlap tags) and standard-model.md (cross-sector SM observables). The methodology layer — how a theory prediction becomes a published number — is in collider-measurements.md, and the direct-vs-inferred taxonomy (what the detector actually records vs. what is fit-extracted) is in direct-vs-inferred.md.

1. The Three Fundamental Sources

Almost every QFT-derived experimental number falls into one of three structural classes, each tied to a different feature of the underlying machinery:

Most observables are reducible to one of these, possibly through a long but mechanical chain.

2. The Inventory

2.1 Cross sections $\sigma$ (Class A)

The dominant observable in collider physics. For 2 → $n$ scattering processes:

$$d\sigma =\frac{1}{F}\overline{|\mathcal{M}{|}^2}d{\Pi }_n.$$

Subtypes — total, differential ($d\sigma /d\Omega$, $d\sigma /dt$, ...), inclusive (sum over final states), exclusive (specific final state). Full treatment in cross-sections.md.

Where measured. Colliders (LHC, LEP, B-factories), fixed-target experiments (Rutherford, electron–nucleon DIS), neutrino experiments.

2.2 Decay rates $\Gamma$ and lifetimes $\tau =\hslash /\Gamma$ (Class A)

For 1 → $n$ processes, with the analogous master formula

$$d\Gamma =\frac{1}{2M}\overline{|\mathcal{M}{|}^2}d{\Pi }_n.$$

Conceptually distinct from cross sections — measured for unstable particles in their rest frame. Shares the same $\mathcal{M}$ machinery; covered in decay-rates.md.

Where measured. Particle lifetimes ($B$-meson, $\tau$-lepton, muon, neutron), nuclear $\beta$-decay, cosmological evolution of unstable particles.

2.3 Branching ratios $\mathrm{BR}(X\to f)=\Gamma (X\to f)/{\Gamma }_{\text{total}}$ (Class A, derived)

Ratios of partial decay widths. Cancel many normalization uncertainties (overall coupling constants, wavefunction renormalizations) and are often the cleanest predictions from QFT for hadronic processes.

Examples. $\mathrm{BR}(B\to K^{\ast }{\mu }^{+}{\mu }^{-})$, $\mathrm{BR}(H\to \gamma \gamma )$, neutron $\beta$-decay branching to $p{e}^{-}\bar{\nu }$ versus radiative modes.

2.4 Asymmetries (Class A, derived)

Designed-to-cancel ratios of differential cross sections / decay rates. They expose subtle effects (interferences, parity violation, $CP$ violation) by removing dominant common factors.

2.5 Distribution shapes / kinematic spectra (Class A)

The shape of an observable distribution rather than its absolute rate:

• Invariant-mass spectra — peaks reveal new particles ($J/\psi$, ${\rm Y}$, $H\to \gamma \gamma$, $H\to ZZ\to 4\ell$).
• Transverse-momentum $p_T$ distributions.
• Angular distributions of decay products (e.g. $H\to Z{Z}^{\ast }\to 4\ell$ angles probe spin/parity of the Higgs).
• Rapidity / pseudorapidity distributions.

These are differential cross sections viewed as functions; the underlying machinery is identical.

2.6 Bound-state energy levels (Class B)

Spectroscopy of bound systems (atoms, hadrons). The bound-state masses appear as poles in the appropriate channel of multi-particle correlators, computed via:

• Bethe–Salpeter equation — relativistic two-body bound-state framework.
• NRQED / NRQCD — effective theories integrating out the relativistic scale.
• Lattice QCD — Euclidean two-point functions in the meson / baryon channels.

Examples. Hydrogen Lamb shift (see QED/hydrogen.md), positronium / muonium / hydrogenic-ion spectra, hyperfine splittings, the entire light hadron spectrum (lattice QCD), quarkonium ($J/\psi$, ${\rm Y}$ families) levels.

Conceptually distinct from cross sections. A bound state is not an asymptotic in/out state — it is a singularity of the off-shell propagator, not an entry in $S$.

2.7 Static electromagnetic / weak properties (Class C)

Properties of a particle as seen by an external static probe. Extracted from the on-shell vertex form factors $F_1(q^2),F_2(q^2)$ in the parameterization

$$⟨p^{\prime }|j^{\mu }|p⟩=\bar{u}(p^{\prime })\left[{\gamma }^{\mu }{F}_1(q^2)+i\frac{{\sigma }^{\mu \nu }{q}_{\nu }}{2m}{F}_2(q^2)\right]u(p),q=p^{\prime }-p.$$

These are all static (zero or near-zero momentum-transfer) limits of vertex functions, not cross sections.

2.8 IR-safe QCD observables (Class A, specialised)

In QCD, infrared and collinear divergences make naive $|\mathcal{M}{|}^2$ for individual quark/gluon final states ill-defined. IR-safe observables are constructed to be insensitive to these divergences:

• Jet cross sections — defined via jet algorithms (anti-$k_T$, Cambridge–Aachen) on inclusive final states.
• Event shapes — thrust $T$, sphericity, $C$-parameter; characterize the "shape" of a multi-jet event.
• Inclusive structure functions $F_1(x,Q^2)$, $F_2(x,Q^2)$, $F_3$ in deep inelastic scattering.
• Total hadronic cross section $R=\sigma (e^{+}{e}^{-}\to \text{hadrons})/\sigma (e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-})$.

All are class A in the sense that they reduce to $|\mathcal{M}{|}^2$, but with sums-over-final-states designed for IR safety.

2.9 Particle masses and coupling constants (Class B + RG)

These are parameters of any specific QFT, but they are also observables — fixed by experiment to specify a renormalization scheme.

• Pole masses — $m_{\text{pole}}$ from the location of the propagator pole. Universal but suffers from renormalon ambiguities at higher loops.
• $\overline{\text{MS}}$ masses — running masses defined by minimal-subtraction renormalization. Scheme-dependent but theoretically clean.
• Running coupling constants ${\alpha }_{\text{em}}(Q^2)$, ${\alpha }_s(Q^2)$, ${\sin }^2{\theta }_W(Q^2)$ — extracted from fits to many observables across an energy range, satisfying RG equations.
• Mixing matrices — CKM (quark) and PMNS (neutrino) matrix elements. Extracted from a global fit to a large set of decay rates and asymmetries.

2.10 Vacuum and topological observables (misc)

Less common in particle-physics tables, but real:

• Vacuum stability bounds — does the SM electroweak vacuum decay? Computed from the effective potential at large field values.
• Anomaly coefficients — the chiral anomaly ${\partial }_{\mu }{j}_5^{\mu }=(e^2/16{\pi }^2)F\tilde{F}$ has measurable consequences (e.g. ${\pi }^0\to \gamma \gamma$ rate).
• Theta-vacuum / instanton effects — ${\theta }_{\text{QCD}}$ measured via the neutron EDM bound; baryogenesis via electroweak sphalerons.
• CMB / cosmological observables — primordial scalar and tensor power spectra from inflationary QFT, BBN abundances from electroweak-era equilibrium.

2.11 Information-theoretic / quantum-correlation observables

A growing area in modern QFT, motivated by quantum information and condensed matter:

• Entanglement entropy of a spatial region (computed from the reduced density matrix of the QFT vacuum).
• Bell-type inequalities in particle physics (recent measurements in top-quark pair production at LHC).
• Out-of-time-order correlators (OTOCs) — diagnostic of quantum chaos / scrambling, e.g. $⟨[W(t),V(0){]}^2⟩$.

These are not "cross sections" in any classical sense but are real QFT observables increasingly being measured.

3. Relating Each Class to the QFT Machinery

4. Where each lives in this repository

5. The Logical Flow Across Observables

The chain that connects QFT to any observable:

$$\underset{\text{theory data}}{\underbrace{\text{Lagrangian / S-matrix}}}\to \underset{⟨0|T\varphi \cdots \varphi |0⟩}{\underbrace{\text{Correlators}}}\to \{\begin{array}{ll}\overline{|\mathcal{M}{|}^2}& \stackrel{\text{LSZ}}{\to }\text{Class A: cross sections, decays, asymmetries}\\ \text{poles, residues}& \stackrel{\text{spectral analysis}}{\to }\text{Class B: masses, bound-state energies}\\ \text{matrix elements}& \stackrel{\text{on-shell projection}}{\to }\text{Class C: form factors, }g-2\end{array}$$

So every observable in QFT goes through correlators of local operators. The differences between cross sections, bound-state energies, and form factors are differences in what part of the correlator structure you extract — not in the underlying machinery.

6. Pointers

• cross-sections.md — the master $d\sigma$ formula, flux factor, Lorentz-invariant phase space, Mandelstam variables, optical theorem, units (barns).
• decay-rates.md — the master $d\Gamma$ formula, lifetimes, branching ratios, the muon-lifetime worked example.
• direct-vs-inferred.md — the taxonomy of what is directly measured by detectors (track hits, calorimeter deposits, magnetic-field curvature, timing, polarization) vs. what is fit-extracted (essentially everything else in particle physics); per-observable inference-distance table; the "which mass?" renormalization-scheme ambiguity.
• collider-measurements.md — the theory → variable → reconstruction → result pipeline, with 8 worked case studies ($m_Z$ scan, $m_W$ transverse mass, $m_h$ peak, Higgs discovery, cross section, branching ratio, asymmetry, coupling modifier).
• electroweak.md — per-class inventory of EW observables (gauge-boson masses, $Z$-pole asymmetries, $G_F$, CKM elements, Higgs sector, neutrinos), tagged by overlap with QED/QCD/SM.
• standard-model.md — cross-sector SM observables (CKM unitarity triangle, global EW fit, GIM, lepton universality, cosmological constraints).
• LSZ Reduction Formula — extracts S-matrix elements from correlators.
• QED/compton.md — worked tree-level cross-section calculation.
• QED/hydrogen.md — bound-state observables.
• QED § Successes and Tested Predictions — historical highlights of QED-derived observables.