Collider Measurements: Theory → Variable → Result

A collider experiment is a pipeline that turns a QFT prediction into a number with an uncertainty. This doc traces that pipeline end-to-end for the most important measurements, showing for each one:

1. What theory predicts (the QFT computation, formula, free parameter(s))
2. Which detector-level variable carries the prediction's signal
3. How experimentalists reconstruct it from raw detector data
4. The published result + dominant systematic uncertainty

The companion observables docs (electroweak.md, standard-model.md, cross-sections.md, decay-rates.md) catalogue what is measured; this doc explains how the measurement is actually done.

1. The Generic Collider-Measurement Pipeline

Every collider measurement follows the same chain. The vocabulary differs by measurement type but the structure does not:

$$\boxed{\underset{\text{theory}}{\underbrace{{\mathcal{L}}_{\text{SM}},\text{parameters}}}\stackrel{\text{Feynman + PDFs}}{\to }\underset{\text{differential prediction}}{\underbrace{d\sigma /dX}}\stackrel{\text{event generation}}{\to }\underset{\text{parton/particle level}}{\underbrace{\text{truth-level }X}}\stackrel{\text{detector simulation}}{\to }\underset{\text{detector level}}{\underbrace{X_{\text{reco}}}}\stackrel{\text{compare}}{⟷}\underset{\text{measured}}{\underbrace{X_{\text{data}}}}\stackrel{\text{unfold / fit}}{\to }\underset{\text{measured parameter}}{\underbrace{\hat{\theta }}}.}$$

The right end ($\hat{\theta }$ — a mass, coupling, cross section, asymmetry) is what gets published; everything to its left is what makes the comparison possible. The five chain steps map to standard software stacks:

Two general design principles:

• Calibrate against a known reference. Every measurement uses another measurement as a calibration anchor. $m_W$ is calibrated against $m_Z$; cross sections are normalized to a known luminosity (itself measured via van der Meer scans against forward elastic scattering); jet energies are calibrated using photon+jet balance. Nothing is purely ab initio.
• Make a histogram, fit a template. Almost every published result reduces to: build a histogram of some variable $X$; predict its shape from SM + nuisance parameters; vary the parameter(s) of interest + nuisance parameters until the predicted histogram matches data. The "fit" is a maximum-likelihood / profile-likelihood-ratio procedure with hundreds to thousands of nuisance parameters.

2. Anatomy of a Modern Collider Experiment

Modern $pp$ colliders (LHC) and earlier $e^{+}{e}^{-}$ colliders (LEP, SLC, BaBar, Belle) all share a common geometry:

• Beam pipe — vacuum, with bunches of accelerated particles crossing at the interaction point (IP) every $\sim 25\mathrm{ns}$ (LHC).
• Tracker — silicon pixel + strip layers in a solenoidal magnetic field ($\sim 2-4\mathrm{T}$). Measures charged-particle momenta via curvature; resolves primary (collision) and secondary (displaced, from $b/c$-decays) vertices.
• Electromagnetic calorimeter (ECAL) — measures energies of $e^{\pm },\gamma$ by total absorption.
• Hadronic calorimeter (HCAL) — measures energies of hadrons (charged + neutral) by absorption.
• Muon spectrometer — outermost, beyond the calorimeters. Muons are the only charged particles that punch through; identified by hits in muon chambers.
• Trigger — multi-level system reducing the $\sim 40\mathrm{MHz}$ bunch-crossing rate to the $\sim 1\mathrm{kHz}$ that can be written to disk, by online cuts on $p_T$, energy, isolation.

From these raw signals, physics objects are reconstructed:

The neutrinos, the dark matter, and any new weakly-interacting BSM particles leave no signal — they show up as $/E_T$, the vector sum of all visible-transverse momenta with a sign flip.

Every measurement that follows uses these objects as primitives.

3. Case Studies

3.1 Measuring the $Z$ boson mass

Theory. In $e^{+}{e}^{-}\to Z^0\to f\bar{f}$ near $\sqrt{s}\approx m_Z$, the cross section is a relativistic Breit–Wigner resonance:

$$\sigma (s)=\frac{12\pi }{m_Z^2}\frac{s{\Gamma }_{ee}{\Gamma }_{ff}}{(s-m_Z^2{)}^2+s^2{\Gamma }_Z^2/m_Z^2}[1+\text{QED + EW corrections}].$$

The peak position fixes $m_Z$; the FWHM fixes ${\Gamma }_Z$; the peak height (combined with ${\Gamma }_{ee}$, known from $e^{+}{e}^{-}\to e^{+}{e}^{-}$) fixes the absolute coupling normalization. Three independent quantities from one curve.

Variable. ${\sigma }_{\text{had}}(\sqrt{s})$ measured at $\sim 8$ energy points around the peak (a scan). Hadronic channel because ${\Gamma }_{\text{had}}/{\Gamma }_Z\approx 70\%$ — highest statistics.

Reconstruction. $e^{+}{e}^{-}\to \text{hadrons}$ events are trivially identified at LEP1 (huge $\sim 30\mathrm{GeV}$ visible energy, $\sim 20$ charged tracks; $e^{+}{e}^{-}\to \mu \mu$ has 2 tracks, $\to \tau \tau$ has 2–6 tracks with displaced vertices — all separable). Counted with $\sim 99\%$ efficiency.

The hard part is calibrating $\sqrt{s}$:

• LEP measured beam energies via resonant depolarization: transversely polarized $e^{\pm }$ beams have a spin-precession frequency ${\nu }_s=E_{\text{beam}}/(440.65\mathrm{MeV})$; sweep an RF kicker, find the depolarizing resonance, read $E_{\text{beam}}$ to $\Delta E_{\text{beam}}/E_{\text{beam}}\sim 10^{-5}$.
• Corrections were applied for: lunar tides distorting the LEP tunnel (1 cm radial deformation ↔ 1 MeV beam-energy shift), train passages on the Geneva–Bellegarde TGV line (RF coupling to the LEP RF system), seasonal water-table changes (Versoix river level), and even the time of day.

Result. $m_Z=91.1876\pm 0.0021\mathrm{GeV}$ — relative uncertainty $2.3\times 10^{-5}$. Beam-energy calibration is the dominant systematic; statistical uncertainty was negligible after $\sim 17$M $Z$s.

Pipeline summary.

$$\text{Breit-Wigner}\to \sigma (\sqrt{s})\to \text{count events at 8 scan points}\to \text{fit BW shape}\to m_Z,{\Gamma }_Z,{\sigma }_{\text{peak}}.$$

3.2 Measuring the $W$ boson mass

Theory. Unlike the $Z$, the $W^{\pm }$ cannot be produced via $e^{+}{e}^{-}\to W$ (charge conservation forbids it), and its decay $W\to \ell \nu$ contains a neutrino whose full 4-momentum is not reconstructible — only the transverse component ${\overrightarrow{/E}}_T$. So $m_W$ never appears as a clean pole in a directly-measured $\sigma (s)$; it shows up in three cross-section formulas, each carrying $m_W$ differently:

1. Pair-threshold formula ($e^{+}{e}^{-}\to W^{+}{W}^{-}$ at LEP2):

   $$\sigma (e^{+}{e}^{-}\to W^{+}{W}^{-})(s)=\frac{\pi {\alpha }^2}{s}{\beta }_W(s)F\left(\frac{s}{m_W^2},{\theta }_W\right),{\beta }_W\equiv \sqrt{1-\frac{4m_W^2}{s}}.$$

   Near $\sqrt{s}=2m_W$ the cross section rises from zero as $\sigma \propto {\beta }_W$; the rate of rise pins down $m_W$. LEP2 scanned $\sqrt{s}=161,170,183,\dots$ GeV across threshold and the on-shell region.

2. Single-$W$ Breit–Wigner in transverse mass ($pp/p\bar{p}\to W\to \ell \nu$, used by Tevatron + LHC):

   $$\frac{d\sigma }{d{m}_T^2}\propto \frac{1}{(m_T^2-m_W^2{)}^2+m_W^2{\Gamma }_W^2}\times \mathcal{J}(m_T;\text{PDFs, recoil, FSR}),$$

   where the transverse mass

   $$m_T\equiv \sqrt{2p_T^{\ell }/E_T(1-\cos \Delta {\varphi }_{\ell \nu })}$$

   plays for the $W$ the role $s$ plays for the $Z$ — the variable in which the cross section has a Breit–Wigner pole. For a $W$ at rest $m_T\to m_W$; for a recoiling $W$ the distribution has a Jacobian peak at $m_T=m_W$. The lepton-$p_T$ spectrum has the analogous Jacobian peak at $p_T^{\ell }\approx m_W/2$. The kinematic Jacobian $\mathcal{J}$ contains all the hadron-collider complexity (PDFs, $W$ recoil distribution, QED FSR off the lepton) and is the source of most systematic uncertainty.

3. Propagator factor in $W$-mediated processes (Drell–Yan, deep-inelastic $\nu N$, $b\to c\ell \nu$, etc.):

   $$\mathcal{M}\propto \frac{1}{q^2-m_W^2},\frac{d\sigma }{d{Q}^2}\propto \frac{G_F^2}{(1+Q^2/m_W^2{)}^2}F(x,Q^2).$$

   At $Q^2\ll m_W^2$ the propagator collapses to $-1/m_W^2$ and gives Fermi's $G_F\propto g^2/m_W^2$ — the source of the historical "weak force is weak". At $Q^2\sim m_W^2$ (deep-inelastic neutrino scattering, $W^{\prime }$-search regions) the full propagator resolves and the cross section is directly sensitive to $m_W$ as an independent parameter. Historically used to constrain $m_W$ from $\nu N$ DIS before its 1983 direct discovery.

Modern precision measurements use formula 2 — the transverse-mass Breit–Wigner — fit as a template.

Variable. Three nearly equivalent variables, often combined:

• $m_T$ distribution (most robust, less sensitive to $W$ recoil)
• Lepton $p_T^{\ell }$ distribution (most sensitive to $m_W$ at the Jacobian peak)
• $/E_T$ distribution (least used because of pileup contamination)

Reconstruction.

• Charged lepton ${\vec{p}}_T^{\ell }$ — directly from tracker + muon-system (muons) or tracker + ECAL (electrons). Lepton momentum scale is the leading systematic; calibrated against $Z\to {\ell }^{+}{\ell }^{-}$ events whose $m_Z$ is known to $0.002\%$ from LEP1.
• ${\overrightarrow{/E}}_T$ — $-\sum {\vec{p}}_T^{\text{visible}}$ over the entire detector. Requires modeling the "hadronic recoil": every particle in the event that is not the signal lepton.
• Hadronic recoil response — calibrated using $Z\to {\ell }^{+}{\ell }^{-}$ events, where the recoil is fully measurable (no missing $\nu$). The same calibration is then transferred to the $W$ analysis.
• PDF inputs — proton PDFs (NNPDF, MSHT, CT) determine the $W$ rapidity / $p_T$ template; PDF uncertainties contribute $\sim 5\mathrm{MeV}$ to the systematic budget.

The measurement is a template fit: simulate $d\sigma /d{m}_T$ for many values of $m_W$ (with all nuisance parameters profiled — calibration scales, PDFs, QED FSR), find the value of $m_W$ that best matches the data histogram.

Result. Per-experiment values (transverse-mass + $p_T^{\ell }$ combination):

The CDF result is $\sim 7\sigma$ above the others. Subsequent ATLAS, LHCb, and CMS measurements all agreed with the non-CDF world average; the CDF anomaly remains unresolved as of 2026 and is the single most active open puzzle in EW precision physics.

Pipeline summary.

$$\text{QCD+EW templates of }d\sigma /d{m}_T(m_W)\to \text{histogram in data}\to \text{template fit profiling 50+ nuisances}\to m_W\pm \delta m_W^{\text{cal}}\pm \delta m_W^{\text{PDF}}\pm \delta m_W^{\text{stat}}.$$

3.3 Measuring the Higgs boson mass

Theory. The Higgs is produced (Section 3.5) and decays. Two clean channels give an invariant-mass peak over a smooth background:

• $h\to \gamma \gamma$: SM BR $=0.23\%$, but two photons are cleanly reconstructed with $\sim 1-2\%$ energy resolution.
• $h\to Z{Z}^{\ast }\to 4\ell$ ($\ell =e,\mu$): SM BR $\sim 1.2\times 10^{-4}$, but signal-to-background is enormous and resolution on $m_{4\ell }$ is $\sim 1\mathrm{GeV}$.

In either case, theory predicts a Breit–Wigner of width ${\Gamma }_h=4.07\mathrm{MeV}$ (intrinsic) broadened by detector resolution to $\sim 1-2\mathrm{GeV}$.

Variable.

• $m_{\gamma \gamma }$ — diphoton invariant mass.
• $m_{4\ell }$ — four-lepton invariant mass.

Reconstruction.

• Diphoton. Two ECAL clusters above $p_T>25\mathrm{GeV}$ (or so), each isolated from tracks (to reject ${\pi }^0\to \gamma \gamma$ jets). Photon energy scale calibrated using $Z\to ee$ (electrons treated as photons after dropping the track requirement). $m_{\gamma \gamma }=\sqrt{2E_1{E}_2(1-\cos {\theta }_{12})}$.
• Four-lepton. Four well-identified, isolated leptons; reconstruct $m_{Z_1}$ closest to $m_Z$, $m_{Z_2}$ the other pair, then $m_{4\ell }$. Lepton momentum scale again calibrated on $Z\to \ell \ell$.

Result. $m_h=125.20\pm 0.11\mathrm{GeV}$ (ATLAS + CMS Run-2 combination). Photon-energy / lepton-momentum scale is the dominant systematic; statistics enter at the same level after Run 2.

Pipeline summary.

$$\text{Higgs Lagrangian}\to BR(h\to \gamma \gamma ,4\ell )\to m_{\gamma \gamma },m_{4\ell }\text{ histograms}\to \text{Gaussian-signal + falling-background fit}\to m_h.$$

3.4 Discovering a new particle: the Higgs in 2012

Theory. Compute expected signal yield $N_s=\mathcal{L}\cdot {\sigma }_{pp\to h}\cdot \text{BR}(h\to Z{Z}^{\ast })\cdot \text{efficiency}$ for a Higgs of mass $m_h$ in a chosen channel, vs. expected background yield $N_b$ from QCD continuum (estimated from data sidebands or simulation).

Variable. The signal channel's invariant-mass distribution ($m_{\gamma \gamma }$, $m_{4\ell }$, $m_{WW}$).

Reconstruction + statistical extraction.

1. Reconstruct events passing the analysis selection; histogram them in $m$.
2. Define a likelihood ratio: $$\Lambda (m_h)=\frac{\mathcal{L}\left(\text{data}\right|\mu \cdot N_s(m_h)+N_b)}{\mathcal{L}\left(\text{data}\right|\mu =0)},$$ where $\mu$ is the signal-strength modifier (best-fit signal divided by SM expectation) and the likelihood is built from a Poisson product over mass bins.
3. The test statistic $q=-2\ln \Lambda$ has a ${\chi }^2$-like distribution under the no-signal null; convert observed $q$ to a $p$-value, then to a number of Gaussian-equivalent $\sigma$.

Result. Both ATLAS and CMS reported $\ge 5\sigma$ excesses at $m_h\approx 125\mathrm{GeV}$ on July 4, 2012, combining $h\to \gamma \gamma$ and $h\to Z{Z}^{\ast }\to 4\ell$ (and $h\to W{W}^{\ast }\to 2\ell 2\nu$ as supporting evidence). The "5$\sigma$ discovery threshold" corresponds to $p<2.87\times 10^{-7}$ for a single channel — chosen conservatively to account for the look-elsewhere effect (the fact that "a bump anywhere in the allowed mass range" is more likely than "a bump at this specific mass").

Pipeline summary.

$${\sigma }_{pp\to h}\cdot \text{BR}\to \text{expected event counts}\to \text{observe excess in }m\text{-histogram}\to q=-2\ln \Lambda \to p\text{-value}\to \sigma \to \text{discovery}.$$

3.5 Measuring a cross section: Higgs production at the LHC

Theory. Factorization theorem (see QCD § Factorization):

$${\sigma }_{pp\to h+X}(s)=\sum _{i,j}\int d{x}_{1}d{x}_2{f}_i(x_1,{\mu }_F)f_j(x_2,{\mu }_F){\hat{\sigma }}_{ij\to h+X}(x_1{x}_{2}s,{\mu }_F,{\mu }_R).$$

For $gg\to h$ (the dominant channel, $\sim 87\%$), ${\hat{\sigma }}_{gg\to h}$ is computed via a top-quark loop. NNLO QCD K-factors are $\sim 2$ relative to LO; full N3LO (NNNLO) results are now standard. Predicted total ${\sigma }_{pp\to h}$ at $\sqrt{s}=13\mathrm{TeV}$ is $\approx 48.6\mathrm{pb}$ for $m_h=125\mathrm{GeV}$.

Variable. Number of observed $h\to X$ events:

$$N_{\text{signal}}=\mathcal{L}\cdot {\sigma }_{pp\to h}\cdot \text{BR}(h\to X)\cdot \mathcal{A}\cdot ϵ,$$

with $\mathcal{A}$ the kinematic acceptance (fraction of events passing fiducial cuts) and $ϵ$ the reconstruction efficiency.

Reconstruction.

• Luminosity $\mathcal{L}$. Measured by van der Meer scans: vertically and horizontally separate the colliding bunches, measure the rate vs. separation, fit a Gaussian, extract the bunch overlap integral. Cross-checked against forward elastic scattering (TOTEM, LHCf at the LHC). Per-experiment uncertainty: $\sim 2\%$.
• Signal count $N_s$. Bump-fit (Section 3.4) or counting in a signal region after sideband background subtraction.
• Acceptance $\mathcal{A}$ and efficiency $ϵ$. From signal Monte Carlo, validated against data control regions.

Result. Typical published format:

$${\sigma }_{pp\to h}(\sqrt{s}=13\mathrm{TeV})=48.6\pm 2.5\mathrm{pb}(\text{SM: }48.61\pm 2.4),$$

equivalently as a signal-strength modifier $\mu ={\sigma }^{\text{obs}}/{\sigma }^{\text{SM}}\approx 1.00\pm 0.05$. All five production modes (ggH, VBF, VH, $t\bar{t}H$, $b\bar{b}H$) are now individually measured.

Pipeline summary.

$$\text{PDFs}\otimes {\hat{\sigma }}_{ij\to h}\to {\sigma }_{\text{theory}}\to \text{predicted yield }{N}_s\to \text{counted events}\to \mu =N_s^{\text{obs}}/N_s^{\text{SM}}.$$

3.6 Measuring a branching ratio: $\mathrm{BR}(B_s\to {\mu }^{+}{\mu }^{-})$

Theory. A FCNC process forbidden at tree level (GIM), proceeding through a $W$-box + $Z$-penguin loop. The SM rate is

$$\mathrm{BR}(B_s\to {\mu }^{+}{\mu }^{-}{)}^{\text{SM}}=(3.66\pm 0.14)\times 10^{-9},$$

computed from CKM elements ($|V_{tb}{V}_{ts}{|}^2$), top-quark loop function, and the $B_s$ decay constant $f_{B_s}$ from lattice QCD.

Variable. Number of reconstructed $B_s\to {\mu }^{+}{\mu }^{-}$ candidates relative to a normalization channel (typically $B^{+}\to J/\psi K^{+}$):

$$\mathrm{BR}(B_s\to \mu \mu )=\mathrm{BR}(B^{+}\to J/\psi K^{+})\cdot \frac{f_u}{f_s}\cdot \frac{N_{\mu \mu }}{N_{J/\psi K}}\cdot \frac{{ϵ}_{J/\psi K}}{{ϵ}_{\mu \mu }}.$$

The normalization channel cancels luminosity, PDFs, and many systematics; $f_u/f_s$ is the fragmentation-fraction ratio (separately measured).

Reconstruction.

• Two opposite-sign muons forming a vertex displaced from the primary $pp$ vertex by $\sim 1\mathrm{cm}$.
• Vertex isolation cuts to reject background from $b\bar{b}\to \mu \mu X$.
• Multivariate classifier (BDT or neural net) trained on simulation, then transferred to data.

Result. LHCb 2022 + ATLAS + CMS combined:

$$\mathrm{BR}(B_s\to {\mu }^{+}{\mu }^{-})=(3.45_{-0.27}^{+0.29})\times 10^{-9}.$$

Consistent with SM at $\sim 1\sigma$. Constrains BSM models with new scalars or modified $Z$-penguins.

Pipeline summary.

$$\text{CKM + lattice }{f}_{B_s}\to \text{predicted BR}\to \text{tagged events vs. normalization}\to N_{\text{ratio}}\to \text{BR}.$$

3.7 Measuring an asymmetry: $A_{FB}$ at the $Z$-pole

Theory. In $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ near $\sqrt{s}=m_Z$, the differential cross section in $\cos {\theta }^{\ast }$ (the ${\mu }^{-}$ direction relative to the $e^{-}$ beam in the c.m.) is

$$\frac{d\sigma }{d\cos {\theta }^{\ast }}\propto 1+{\cos }^2{\theta }^{\ast }+\frac{8}{3}{A}_{FB}^{\mu }\cos {\theta }^{\ast }.$$

The asymmetry $A_{FB}^{\mu }=g_V^e{g}_A^e{g}_V^{\mu }{g}_A^{\mu }/(\text{V/A})$ depends on ${\sin }^2{\theta }_W$ through the $Z$ vector and axial couplings. SM predicts $A_{FB}^{\mu }\approx 0.0169$ at the peak — a small but very clean number.

Variable.

$$A_{FB}^{\mu }=\frac{N_F-N_B}{N_F+N_B},$$

with $N_F$ = events with $\cos {\theta }^{\ast }>0$, $N_B$ = events with $\cos {\theta }^{\ast }<0$.

Reconstruction. Identify $Z\to {\mu }^{+}{\mu }^{-}$ events (two opposite-sign isolated muons with $m_{\mu \mu }$ near $m_Z$). Compute $\cos {\theta }^{\ast }$ from the muon directions. Count.

Result. Combined from LEP1 with millions of events per flavor:

$$A_{FB}^{\mu }=0.0169\pm \mathrm{0.0013.}$$

The ratio cancels luminosity, efficiency-symmetric reconstruction effects, and most acceptance corrections — that's the point of asymmetries. Combined across all $f=\mu ,\tau ,b,c$ asymmetries, the $Z$-pole data pins down ${\sin }^2{\theta }_W^{\text{eff,lep}}=0.23153\pm 0.00016$.

Pipeline summary.

$${\sin }^2{\theta }_W\to \text{predicted }{A}_{FB}\to \text{count F vs. B}\to A_{FB}^{\text{obs}}.$$

3.8 Measuring a coupling: Higgs ${\kappa }_{\tau }$ from $h\to \tau \tau$

Theory. Higgs couples to fermions via Yukawa $y_f=m_f\sqrt{2}/v$. Predicted SM branching $\mathrm{BR}(h\to \tau \tau )=6.27\%$ for $m_h=125\mathrm{GeV}$. The coupling modifier

$${\kappa }_{\tau }\equiv \frac{y_{\tau }^{\text{obs}}}{y_{\tau }^{\text{SM}}}$$

is $1$ in the SM by definition.

Variable. Signal-strength modifier ${\mu }_{h\to \tau \tau }=\sigma \cdot {\mathrm{BR}}^{\text{obs}}/(\sigma \cdot \mathrm{BR}{)}^{\text{SM}}$, then convert: $\mu \propto {\kappa }_{\tau }^2\cdot {\kappa }_X^2/{\kappa }_h^2$ where $X$ depends on the production mode and ${\kappa }_h$ is the total-width modifier.

Reconstruction.

• Identify $\tau$-leptons in their decay modes: $\tau \to \mu \nu \nu$ (one muon), $\tau \to e\nu \nu$ (one electron), ${\tau }_{\text{had}}\to {\pi }^{\pm }+n{\pi }^0+\nu$ (a narrow 1-prong or 3-prong $\tau$-jet). Six final-state combinations ($e\mu ,e{\tau }_h,\mu {\tau }_h,{\tau }_h{\tau }_h$, ...).
• Reconstruct $m_{\tau \tau }$ via the SVfit / collinear-approximation algorithm (the neutrinos add to $/E_T$, which is decomposed back along the $\tau$ directions).
• Multivariate analysis to separate signal from $Z\to \tau \tau$ background and from QCD multijet fakes.

Result. ATLAS + CMS combination: ${\kappa }_{\tau }=0.93\pm 0.07$ — consistent with SM (${\kappa }_{\tau }=1$) at $\sim 1\sigma$.

Similar fits for ${\kappa }_b,{\kappa }_t,{\kappa }_{\mu },{\kappa }_W,{\kappa }_Z,{\kappa }_g,{\kappa }_{\gamma }$ — all consistent with 1. Predicted couplings to first-generation fermions ($e,u,d$) are below current LHC sensitivity; HL-LHC and future colliders are needed.

Pipeline summary.

$$y_{\tau }\to \mathrm{BR}(h\to \tau \tau )\to \sigma \cdot \mathrm{BR}\to \text{events in }\tau \tau \text{ channels}\to {\mu }_{\tau \tau }\to {\kappa }_{\tau }.$$

4. Common Threads

The eight case studies above span very different physical observables — masses, cross sections, branching ratios, asymmetries, discovery significance, coupling modifiers — but the measurement structure is essentially the same:

A cross-cutting consequence: no LHC measurement is purely "data-driven". All of them depend on (a) Monte-Carlo simulation of signal + background, (b) NLO/NNLO QCD predictions, (c) PDF parametrizations, (d) parton-shower modeling, and (e) detector calibration anchors. The error budget always has a non-trivial "theory" component, and progress on the theory side (e.g. better PDFs, higher-order QCD calculations) reduces published uncertainties even without new data.

4.1 Direct vs. inferred observables

A second cross-cutting point: almost nothing in published collider results is "directly observed". The detector records a small universal set of primitives (track hits, calorimeter energy deposits, magnetic-field curvature, timing, polarization), and everything else — cross sections, masses, branching ratios, couplings, ${\sin }^2{\theta }_W$, CKM elements — is a fit-extracted parameter obtained by inferring the value that makes a theoretical model match a histogram of those primitives.

The full taxonomy — direct primitives, the per-class list of inferred observables (cross sections, masses, couplings, vertex form factors, oscillation parameters, composite global fits), the inference-distance spectrum (short for $m_Z$ and $m_h$, long for $m_W$ and $m_t$), and the renormalization-scheme wrinkle (the "which mass?" question) — lives in direct-vs-inferred.md. The short version of the chain:

$$\underset{\text{actually recorded}}{\underbrace{\text{detector primitives}}}\to \underset{\text{reconstruction}}{\underbrace{\text{physics objects (tracks, jets, leptons, }/E_T)}}\to \underset{\text{measured distribution}}{\underbrace{X_{\text{reco}}\text{ histogram}}}\to \underset{\text{inference}}{\underbrace{\text{template fit}}}\to \underset{\text{parameter in a scheme}}{\underbrace{\hat{\theta }(\text{scheme})}}\to \underset{\text{after scheme translation}}{\underbrace{\theta (\text{Lagrangian})}}.$$

The only left-end node is data; everything to its right is model + inference.

5. Where Each Measurement Lives in This Repo

6. See Also

• Cross Sections — master $d\sigma$ formula, flux factor, phase space.
• Decay Rates — master $d\Gamma$ formula, worked muon-lifetime example.
• Electroweak Observables — full inventory of what is measured at EW experiments.
• Standard Model Observables — cross-sector observables (UT triangle, global EW fit, GIM, lepton universality).
• Observables — General Map — the structural classification (Class A/B/C) all of the above instantiate.
• QCD § Factorization — the PDF × partonic-cross-section structure used by all hadron-collider measurements.
• QED/compton.md — the cleanest worked theory calculation in this repo, for comparison with the experimental extraction side covered here.