--- title: From Collisions to Discoveries module: Accelerators and Detectors moduleNumber: 11 lessonNumber: 3 order: 1103 summary: > A discovery is a peak that survives statistics. Events are reconstructed into invariant masses, a signal accumulates as a bump over a smooth background, and its significance is judged by a p-value; the field's threshold is five sigma. The expected yield is a product — luminosity times cross section times branching ratio times acceptance and efficiency — that must be balanced by a trigger and data-reduction chain against an overwhelming rate. Worked reconstructions of $Z\to\ell\ell$, the $J/\psi$, and the Higgs show the same peak-over-background logic at three scales. topics: [Accelerators and Detectors] draft: false sources: - book: Perkins ref: "Ch. 1 and Ch. 9 (event reconstruction, statistics, and discovery)" - book: Thomson ref: "Ch. 1 (invariant-mass reconstruction and resonances)" - book: Griffiths ref: "Ch. 3 §3.4 (invariant mass) and Ch. 1 (resonances and discovery)" - book: Halzen & Martin ref: "Ch. 2 (resonances and the Breit-Wigner line shape)" --- A collision is not a discovery. The detector records energy deposits; software reconstructs particles; the particles combine into candidate events; and only a statistical argument over many events turns a candidate excess into a claim. This lesson follows that chain: reconstructing an invariant mass, recognizing a signal as a peak over background, quantifying it with a significance, and accounting for the yield through the event budget and the trigger. The physics of every earlier module returns here as a signature to be dug out of data. ## Reconstructing the invariant mass The central tool is the invariant mass of a set of final-state particles. For $n$ measured four-momenta the combination $$ M^2 = \Big(\sum_{i=1}^{n} p_i\Big)^2 = \Big(\sum_i E_i\Big)^2 - \Big(\sum_i \vec p_i\Big)^2 $$ is Lorentz-invariant and, when the particles are the decay products of a single parent, equals the parent's rest mass. A parent produced with a definite mass appears as a **peak** in the distribution of $M$ over many events, at the parent mass. The width of the peak has two sources: the parent's intrinsic decay width $\Gamma$, and the detector's finite momentum and energy resolution, which smears the reconstructed $M$ even for a perfectly sharp state. For a long-lived narrow state the observed width is dominated by resolution; for a broad resonance the intrinsic $\Gamma$ dominates. A resonance of mass $M_R$ and width $\Gamma$ has the **Breit-Wigner** line shape $$ \sigma(E) \;\propto\; \frac{1}{(E-M_R)^2 + \Gamma^2/4}, $$ a peak of full width $\Gamma$ at half maximum centered at $M_R$. The width fixes the lifetime through $\tau=\hbar/\Gamma$: a narrow peak is a long-lived state, a broad peak a fleeting one. Reconstructing $M$ event by event and histogramming it is the single most productive operation in experimental particle physics, and every discovery below is a peak in such a histogram. $$ % caption: The Breit-Wigner resonance. The reconstructed invariant-mass distribution % peaks at the parent mass with a full width at half maximum equal to the decay width % Gamma; a narrow peak is a long-lived state, a broad peak a short-lived one, through % tau equals h-bar over Gamma. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[right, black] {invariant mass}; \draw[->, black] (0,0) -- (0,4.0) node[above, black] {events}; % narrow BW centered at x=3 \draw[very thick, domain=0.3:5.9, samples=140, variable=\x] plot ({\x}, {3.2*0.09/((\x-3.0)*(\x-3.0)+0.09)}); % half-max line \draw[black, dashed] (2.7,1.6) -- (3.3,1.6); \node[black, above] at (3.0,3.35) {$M_R$}; \node[black, right] at (3.32,1.6) {FWHM}; \end{tikzpicture} $$ ## Signal, background, and significance A peak sits on a background. Most reconstructed combinations do not come from the sought parent: they are random pairings of particles from unrelated processes, forming a smooth **combinatorial background** under the signal. A discovery is the claim that the number of events in the peak region exceeds what the background alone would produce, by more than statistical fluctuation can explain. Because event counts are Poisson-distributed, a background expectation of $B$ events fluctuates with standard deviation $\sqrt B$. A signal of $S$ events on top is therefore judged against that fluctuation, and in the large-count limit the **significance** is $$ Z \;=\; \frac{S}{\sqrt B}. $$ The significance is quoted in units of the standard deviation ("sigma"), and it maps to a **p-value** — the probability that background alone would fluctuate up to at least the observed excess. A one-sided Gaussian gives $$ Z=3\sigma \Rightarrow p\approx1.3\times10^{-3}, \qquad Z=5\sigma \Rightarrow p\approx2.9\times10^{-7}. $$ The field's conventions attach names to thresholds: $3\sigma$ is **evidence**, $5\sigma$ is **discovery**. The five-sigma bar — a one-in-three-million chance of a background fluctuation — is deliberately stringent because a large experiment tests many mass bins and many channels, so modest excesses appear somewhere by chance. This **look-elsewhere effect** means the local p-value at the peak must be corrected to a global one accounting for the whole search range; the corrected significance is what a discovery claim reports. $$ % caption: Significance as a signal over the square-root of the background. A given % number of signal events is more significant against a smaller background; the plotted % contours mark the three-sigma evidence and five-sigma discovery thresholds as the % signal grows relative to the background fluctuation. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[right, black] {signal events}; \draw[->, black] (0,0) -- (0,4.0) node[above, black] {$Z$}; % Z = S/sqrt(B): with fixed B, linear in S \draw[very thick, domain=0:6.0, samples=2, variable=\x] plot ({\x}, {0.58*\x}); % thresholds \draw[black, dashed] (0,1.7) -- (6.0,1.7); \draw[black, dashed] (0,2.85) -- (6.0,2.85); \node[black, left] at (0,1.7) {$3$}; \node[black, left] at (0,2.85) {$5$}; \node[black, right] at (0.2,1.9) {evidence}; \node[black, right] at (0.2,3.05) {discovery}; \end{tikzpicture} $$ ## The event budget Whether a peak is even reachable is decided before any data arrive, by the expected number of signal events. That number is a product of five factors, $$ N \;=\; L_{\text{int}}\;\times\;\sigma\;\times\;\mathcal B\;\times\;A\;\times\;\epsilon, $$ each of which the experiment either controls or must accept: - **$L_{\text{int}}$** — the integrated luminosity delivered by the machine. - **$\sigma$** — the production cross section for the process, fixed by physics. - **$\mathcal B$** — the branching ratio into the observed final state; a rare decay mode shrinks the yield. - **$A$** — the geometric and kinematic **acceptance**, the fraction of events falling within the detector's angular coverage and momentum thresholds. - **$\epsilon$** — the reconstruction and selection **efficiency**, the fraction of accepted events actually reconstructed and passing the analysis cuts. The product form makes the strategy plain: a small cross section or a tiny branching ratio can be compensated only by large integrated luminosity, and every inefficiency multiplies straight into the yield. It also sets the background: the same product with the background cross section and its acceptance gives $B$, and the analysis cuts are tuned to maximize $S/\sqrt B$ rather than $S$ alone, since a cut that removes some signal but more background improves the significance. > **Worked example.** A process with $\sigma=1\,\text{pb}$ and a final-state > branching ratio $\mathcal B=10^{-2}$, observed with acceptance times efficiency > $A\epsilon=0.2$, collected against $L_{\text{int}}=100\,\text{fb}^{-1}$, yields > $$ > N = (100\,\text{fb}^{-1})(1\,\text{pb})(10^{-2})(0.2). > $$ > With $1\,\text{pb}=10^{3}\,\text{fb}$, this is > $N = 100\times10^{3}\times10^{-2}\times0.2 = 2\times10^{2}$ signal events — enough > for a peak if the background under it is not much larger than a few hundred. ## The trigger and data-reduction chain The event budget assumes the interesting collisions were kept. They nearly were not. At the LHC, bunches cross every $25\,\text{ns}$, a rate of $4\times10^{7}$ crossings per second, each producing on average tens of overlapping proton-proton interactions. The raw data rate is far beyond what can be written to storage, so a **trigger** decides in real time which events to keep and discards the rest permanently. The reduction is staged: - a fast hardware **first-level trigger** examines coarse, low-latency information (calorimeter energy sums, muon track stubs) and reduces the rate from tens of MHz to tens or hundreds of kHz within microseconds; - a software **high-level trigger** runs partial reconstruction on the surviving events and reduces the rate further to the low-kHz range written to disk. Every stage is a bottleneck that throws away most events, so the trigger menu must be designed around the signatures the physics program needs — high-momentum leptons, large missing transverse momentum, energetic jets — because an event not selected by the trigger is lost before any analysis can see it. The trigger is thus a physics choice as much as an engineering one: it defines the acceptance $A$ at its very first factor. $$ % caption: The data-reduction funnel. The bunch-crossing rate of tens of megahertz is % cut by a fast hardware trigger to hundreds of kilohertz, then by a software % high-level trigger to the few kilohertz written to storage; each stage discards most % events permanently. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % funnel as stacked trapezoids narrowing downward \draw[thick] (-3.4,2.4) -- (3.4,2.4) -- (2.6,1.4) -- (-2.6,1.4) -- cycle; \node[align=center] at (0,1.9) {crossings\\40 MHz}; \draw[thick] (-2.6,1.4) -- (2.6,1.4) -- (1.8,0.3) -- (-1.8,0.3) -- cycle; \node[align=center] at (0,0.85) {hardware\\100 kHz}; \draw[thick] (-1.8,0.3) -- (1.8,0.3) -- (1.1,-0.8) -- (-1.1,-0.8) -- cycle; \node[align=center] at (0,-0.25) {software\\1 kHz}; \draw[acc, very thick, ->] (0,-0.8) -- (0,-1.6); \node[acc, below] at (0,-1.6) {stored}; \end{tikzpicture} $$ ## Worked reconstructions at three scales The same peak-over-background logic recurs across the energy range, differing only in the parent mass and the background level. - **$Z\to\ell^+\ell^-$.** The cleanest standard candle at a hadron collider. Two opposite-sign, high-momentum, isolated leptons of the same flavor are reconstructed, their invariant mass histogrammed, and a sharp peak appears at $M_Z\approx91.19\,\text{GeV}$.[^data] The two-lepton final state has little background, so the peak stands out with modest luminosity, and its position and width calibrate the lepton momentum scale and resolution for every other analysis. - **$J/\psi\to\ell^+\ell^-$.** The same dilepton reconstruction at lower mass yields a narrow peak at $M\approx3.097\,\text{GeV}$, the charmonium ground state whose sharp width (a long lifetime relative to typical hadrons) signalled a new conserved quantum number — charm — at its 1974 discovery. - **$H\to\gamma\gamma$ and $H\to ZZ^\ast\to4\ell$.** The Higgs at $M\approx125\,\text{GeV}$ is a small excess, because its production cross section is modest and the observed branching ratios are tiny. The two-photon channel is a narrow bump on a large, smoothly falling diphoton background; the four-lepton "golden channel" has almost no background but very few events. Combining the two — each on its own near the discovery threshold — produced the $5\sigma$ observation in 2012.[^higgs] Every feature of this lesson appears in that result: invariant-mass reconstruction, a peak over background, a significance corrected for the look-elsewhere effect, an event budget dominated by a tiny branching ratio, and a trigger tuned to keep photons and leptons. $$ % caption: A signal peak emerging over a smooth background. The falling curve is the % background expectation; the localized excess above it is the reconstructed signal % peak, and the discovery claim rests on that excess exceeding the background % fluctuation by five standard deviations. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[right, black] {invariant mass}; \draw[->, black] (0,0) -- (0,4.0) node[above, black] {events}; % smooth falling background \draw[black, thick, dashed, domain=0.3:6.0, samples=80, variable=\x] plot ({\x}, {3.2*exp(-0.42*\x)+0.3}); % background plus a localized peak near x=3.4 \draw[acc, very thick, domain=0.3:6.0, samples=160, variable=\x] plot ({\x}, {3.2*exp(-0.42*\x)+0.3 + 1.9*exp(-((\x-3.4)*(\x-3.4))/0.06)}); \node[acc, above] at (3.4,3.3) {signal}; \node[black, right] at (4.2,0.95) {background}; \end{tikzpicture} $$ ## Blind analysis A stringent significance threshold guards against statistical accidents; a **blind analysis** guards against the experimenter. Selection cuts, background models, and calibrations are fixed on simulation and on control regions of the data before the signal region is examined, so that the peak region is "unblinded" only after the analysis is frozen. The purpose is to prevent unconscious tuning of the cuts toward a fluctuation that looks like a signal — a bias that would inflate significance without any real physics. Blinding, together with the corrected p-value and the five-sigma threshold, is why a modern discovery claim is trusted. ## Summary A discovery is a reconstructed invariant-mass peak that survives statistics. The invariant mass $M^2=(\sum p_i)^2$ of a decay's products peaks at the parent mass, with a Breit-Wigner width set by the intrinsic $\Gamma$ and the detector resolution. The peak sits on a combinatorial background of $B$ events fluctuating by $\sqrt B$, so the significance is $Z=S/\sqrt B$, mapped to a p-value; $3\sigma$ is evidence and $5\sigma$ (a one-sided $p\approx3\times10^{-7}$) is discovery, corrected for the look-elsewhere effect. Whether the peak is reachable is set by the event budget $N=L_{\text{int}}\,\sigma\,\mathcal B\,A\,\epsilon$, and whether the events are even kept is set by a trigger that discards all but a few kHz of a 40 MHz crossing rate. The $Z$ dilepton peak, the narrow $J/\psi$, and the small Higgs excess are the same construction at three scales, and blind analysis keeps the claim honest. This closes the experimental module; the [next module](/particle-physics/beyond-standard-model/beyond-standard-model) turns to the questions the Standard Model leaves open. [^data]: Reference values — $M_Z\approx91.19\,\text{GeV}$, $M_{J/\psi}\approx3.097\,\text{GeV}$, $M_H\approx125\,\text{GeV}$ — and the statistics conventions (Poisson significance $S/\sqrt B$, the p-value scale, and the look-elsewhere effect) are from the Particle Data Group reviews, [pdg.lbl.gov](https://pdg.lbl.gov). Event reconstruction, invariant-mass resonances, and the discovery methodology follow Perkins, Ch. 1 and Ch. 9, Thomson, Ch. 1, and Griffiths, §3.4; the Breit-Wigner line shape is in Halzen & Martin, Ch. 2. [^higgs]: The $5\sigma$ observation of a Higgs boson near $125\,\text{GeV}$ in the $\gamma\gamma$ and $ZZ^\ast\to4\ell$ channels was reported by ATLAS, [arXiv:1207.7214](https://arxiv.org/abs/1207.7214), and CMS, [arXiv:1207.7235](https://arxiv.org/abs/1207.7235).