--- title: Accelerators, Colliders, and Luminosity module: Accelerators and Detectors moduleNumber: 11 lessonNumber: 1 order: 1101 summary: > Fixed-target machines waste energy in the center-of-mass motion of the whole system, so the reachable $\sqrt s$ grows only as the square root of the beam energy, while colliders put the full beam energy into the collision. Circular electron machines are limited by synchrotron radiation scaling as $E^4/m^4R$; proton machines are limited by bending fields. Luminosity, set by beam current and focusing, converts a cross section into an event rate through $R=\mathcal L\,\sigma$, and integrated luminosity sets the total event count. topics: [Accelerators and Detectors] draft: false sources: - book: Perkins ref: "Ch. 9 (accelerators and beams; fixed-target versus colliding beams; luminosity)" - book: Thomson ref: "Ch. 1 §1.4 and Appendix (particle accelerators, the LHC)" - book: Griffiths ref: "Ch. 1 §1.11 (colliders and fixed-target machines)" - book: Halzen & Martin ref: "Ch. 4 (kinematics of the invariant mass and available energy)" --- An accelerator delivers the energy that a reaction needs and the intensity that a measurement needs. The energy sets which final states are kinematically open; the intensity sets how many events accumulate before statistics limit the answer. The first quantity is fixed by relativistic kinematics and the choice between a fixed target and colliding beams; the second is fixed by the luminosity. This lesson develops both, then applies them to the machines that define the energy frontier. Natural units and $\eta^{\mu\nu}=\operatorname{diag}(+,-,-,-)$ throughout, with the invariant mass of a system written $\sqrt s$ where $s=(\sum_i p_i)^2$. ## Available energy: fixed target versus collider The quantity that decides which particles a collision can create is the invariant mass of the initial state, $\sqrt s$, because $s$ is frame-independent and equals the total energy in the center-of-momentum frame. A final state of total rest mass $M$ is accessible only when $\sqrt s \ge M$. For a beam of energy $E_{\text{beam}}$ and mass $m$ striking a stationary target of mass $M_t$, the two four-momenta are $p_1=(E_{\text{beam}},\vec p)$ and $p_2=(M_t,\vec 0)$, so $$ s = (p_1+p_2)^2 = m^2 + M_t^2 + 2E_{\text{beam}}M_t . $$ At high energy $E_{\text{beam}}\gg m,M_t$ the first two terms are negligible and $$ \sqrt s \;\simeq\; \sqrt{2 M_t E_{\text{beam}}} \qquad (\text{fixed target}). $$ The center-of-mass energy grows only as the **square root** of the beam energy. The reason is visible in the four-momentum: the produced system inherits the beam's momentum $\vec p$ and therefore carries a large kinetic energy of overall motion that cannot be spent on making new particles. Most of the beam energy ends up as the recoil of the center of mass. A collider removes that waste. Two beams of energy $E$ and opposite momenta meet head-on, so the total momentum is zero and the lab frame **is** the center-of-mass frame. For equal beams of the same mass, $$ s = (E+E)^2 - (\vec p - \vec p)^2 = 4E^2, \qquad \sqrt s = 2E \qquad (\text{collider, head-on, equal beams}). $$ Every unit of beam energy appears in the collision. The contrast is dramatic at high energy: to reach a given $\sqrt s$, a fixed-target machine needs a beam energy that scales as $s$, whereas a collider needs a beam energy that scales as $\sqrt s$. > **Worked example.** To produce a system of mass $M=100\,\text{GeV}$ on a > stationary proton target ($M_t\simeq0.94\,\text{GeV}$) requires > $$ > E_{\text{beam}} \simeq \frac{s}{2M_t} = \frac{M^2}{2M_t} > = \frac{(100)^2}{2(0.94)}\,\text{GeV} \approx 5.3\times10^{3}\,\text{GeV}. > $$ > A collider reaches the same $\sqrt s=100\,\text{GeV}$ with $E=50\,\text{GeV}$ per > beam. The fixed-target machine needs roughly a hundred times the beam energy. The penalty is the whole reason the energy frontier is a collider frontier. Fixed targets survive where their advantages matter: a stationary target can be dense and thick, giving high interaction rates, and it permits a secondary beam of unstable particles (pions, kaons, neutrinos, muons) to be produced and steered. Colliders own the high-$\sqrt s$ regime; fixed-target beams own rate and beam-species flexibility. $$ % caption: Available center-of-mass energy versus beam energy. The collider line is % linear, root-s equals twice the beam energy; the fixed-target curve rises only as % the square root of beam energy, so the same root-s costs vastly more beam energy on % a stationary target. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[right, black] {beam energy}; \draw[->, black] (0,0) -- (0,4.0) node[above, black] {available energy}; % collider: linear, sqrt s = 2E -> slope chosen to fill frame \draw[very thick, domain=0:6.0, samples=2, variable=\x] plot ({\x}, {0.60*\x}); % fixed target: sqrt s ~ sqrt(2 Mt E), sublinear \draw[very thick, dashed, domain=0.02:6.0, samples=80, variable=\x] plot ({\x}, {1.45*sqrt(\x)}); \node[right] at (5.4,3.55) {collider}; \node[right] at (5.0,1.55) {stationary target}; \draw[black, dashed] (0,3.0) -- (6.0,3.0); \node[black, left] at (0,3.0) {target}; % markers where each reaches the target energy \fill (5.0,3.0) circle (1.6pt); \fill (4.28,3.0) circle (1.6pt); \end{tikzpicture} $$ ## Linear and circular machines A charged particle gains energy only from an electric field along its motion. All accelerators therefore reduce to radio-frequency (RF) cavities that present an oscillating longitudinal field, timed so that a particle crossing the gap always sees an accelerating phase. The two geometries differ in how often a bunch reuses the same cavities. - **Linear accelerator (linac).** Cavities are laid in a straight line; each bunch passes once. The final energy is the sum of the gap voltages, so reaching high energy means building a long machine. There is no synchrotron loss and no bending, and the beam is used once per pass, which suits it to a collider only if the spent beams are dumped rather than stored. - **Circular accelerator (synchrotron).** Bending magnets return the bunch to the same RF cavities on every turn, so a modest voltage compounds over millions of revolutions. The magnetic field is ramped in step with the rising momentum to hold the orbit radius fixed — the feature that names the machine. A stored beam can circulate for hours, and two counter-rotating beams in the same ring collide at fixed points every turn. The bend that makes a synchrotron efficient also forces the beam to radiate. ## Synchrotron radiation and the electron limit A relativistic charge on a circular orbit is transversely accelerated and radiates. For a particle of energy $E$, mass $m$, and orbit radius $R$, the energy lost to **synchrotron radiation** per revolution is $$ \Delta E \;=\; \frac{e^2}{3\varepsilon_0}\,\frac{\beta^3\gamma^4}{R} \;\xrightarrow{\;\beta\to1\;}\; \frac{e^2}{3\varepsilon_0 R}\left(\frac{E}{m}\right)^{\!4}. $$ The $\gamma^4=(E/m)^4$ dependence is decisive. Because the loss scales as the inverse fourth power of the mass, an electron ($m_e=0.511\,\text{MeV}$) radiates $(m_p/m_e)^4\approx1.1\times10^{13}$ times more than a proton of the same energy on the same orbit. For electrons a convenient practical form is $$ \Delta E\,[\text{keV}] \;\approx\; 88.5\,\frac{(E\,[\text{GeV}])^4}{R\,[\text{m}]} . $$ At the LEP collider ($R\approx3.1\,\text{km}$, $E=100\,\text{GeV}$ per beam) this is about $2.9\,\text{GeV}$ radiated **per turn** — a substantial fraction of the beam energy that the RF system had to replace every revolution. LEP was the practical ceiling for a circular electron collider: the radiated power scales as $E^4/R$, so pushing the energy higher would have demanded an impractically large ring or an impractically large RF plant. Higher-energy electron colliders are therefore linear by design, where the beam never bends and never radiates. Protons escape this limit by their mass. At the LHC ($E=7\,\text{TeV}$, $R\approx4.3\,\text{km}$) the synchrotron loss is only a few keV per turn. The proton machine is limited instead by the **bending field**: the momentum a ring can hold is $$ p\,[\text{GeV}] \;=\; 0.3\,B\,[\text{T}]\;R\,[\text{m}], $$ so a $7\,\text{TeV}$ beam in the $27\,\text{km}$ LHC tunnel requires superconducting dipoles of about $8.3\,\text{T}$. The electron frontier is set by radiation; the proton frontier is set by magnet technology and tunnel size. $$ % caption: Synchrotron energy loss per turn versus beam energy on a fixed orbit. The % loss climbs as the fourth power of energy, and for a fixed energy it is enormous for % the light electron and negligible for the heavy proton, the reason high-energy % electron colliders are linear and proton colliders are circular. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[right, black] {beam energy}; \draw[->, black] (0,0) -- (0,4.0) node[above, black] {loss per turn}; % electron: steep quartic \draw[very thick, domain=0:3.35, samples=60, variable=\x] plot ({\x}, {0.08*(\x)^4}); % proton: essentially flat on this scale \draw[very thick, dashed, domain=0:6.0, samples=40, variable=\x] plot ({\x}, {0.004*(\x)^4}); \node[right] at (3.05,3.55) {electron}; \node[right] at (4.6,0.95) {proton}; \end{tikzpicture} $$ ## Luminosity and event rate Energy decides what a collision can make; luminosity decides how often it does. For a process of cross section $\sigma$, the event rate is $$ R \;=\; \mathcal L\,\sigma, $$ which defines the **instantaneous luminosity** $\mathcal L$, a property of the machine alone with units of inverse area per unit time (conventionally $\text{cm}^{-2}\,\text{s}^{-1}$). The luminosity is set by how much beam is packed into how small a spot, how often. For two Gaussian bunches of $N_1$ and $N_2$ particles colliding head-on at frequency $f$, transverse sizes $\sigma_x$ and $\sigma_y$, $$ \mathcal L \;=\; f\,\frac{N_1 N_2}{4\pi\,\sigma_x\,\sigma_y}. $$ High luminosity wants many particles per bunch, a high crossing frequency, and tight focusing at the interaction point (small $\sigma_x\sigma_y$). The final focusing is the job of strong quadrupole magnets that squeeze the beams just before they meet. The LHC reaches $\mathcal L\sim2\times10^{34}\,\text{cm}^{-2}\,\text{s}^{-1}$, packing $\sim10^{11}$ protons into each of thousands of bunches and focusing them to a transverse size of order tens of micrometers. The quantity that governs a data set is the time-integrated luminosity, $$ L_{\text{int}} \;=\; \int \mathcal L\,\d t, $$ with the same units as an inverse cross section. It is quoted in inverse barns: $1\,\text{b}=10^{-24}\,\text{cm}^2$, so $1\,\text{fb}^{-1}=10^{39}\,\text{cm}^{-2}$. The **expected number of events** of a process is $$ N \;=\; \sigma\,L_{\text{int}}. $$ A cross section of $1\,\text{pb}$ collected against $L_{\text{int}}=100\,\text{fb}^{-1}$ yields $N=(10^{-12}\,\text{b})(10^{5}\,\text{b}^{-1})\times10^{24}\ldots$ — kept in consistent units, $\sigma=1\,\text{pb}=10^{-3}\,\text{fb}$ times $100\,\text{fb}^{-1}$ gives $N=10^{5}$ events. Integrated luminosity is the currency in which discovery reach is priced: a rare process with a small cross section becomes observable only once enough inverse femtobarns have been accumulated. $$ % caption: The luminosity chain. Two focused bunches cross at frequency f; the % overlap of N1 and N2 particles in a transverse area of order sigma-x times sigma-y % sets the instantaneous luminosity, which multiplied by a cross section gives the % event rate and, integrated over time, the total event count. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % two bunches approaching an interaction point \draw[very thick, ->] (-6.4,0.0) -- (-4.7,0.0); \draw[very thick, ->] (-1.4,0.0) -- (-3.1,0.0); \draw[black] (-4.7,0.0) ellipse (0.55 and 0.28); \draw[black] (-3.1,0.0) ellipse (0.55 and 0.28); \node[above] at (-4.7,0.28) {bunch 1}; \node[above] at (-3.1,0.28) {bunch 2}; \node[below] at (-3.9,-0.35) {interaction point}; \node[black, below] at (-3.9,-0.95) {crossing rate $f$}; % flow boxes \draw[acc] (0.4,-0.5) rectangle (2.4,0.5); \node[acc, align=center] at (1.4,0.0) {luminosity\\$\mathcal L$}; \draw[->, black] (2.4,0.0) -- (3.2,0.0); \draw[black] (3.2,-0.5) rectangle (5.2,0.5); \node[align=center] at (4.2,0.0) {rate\\$R$}; \draw[->, black] (4.2,-0.5) -- (4.2,-1.2); \draw[black] (3.2,-2.2) rectangle (5.2,-1.2); \node[align=center] at (4.2,-1.7) {events\\$N$}; \end{tikzpicture} $$ ## Machine choices: the electron and proton trade-off The two collider families answer different questions, and the difference traces back to the two limits above. - **Electron-positron colliders** ($e^+e^-$). The colliding particles are pointlike, so the full $\sqrt s$ is available to a single elementary reaction and the initial state is known exactly. Events are clean: little hadronic debris, calculable QED backgrounds, and a fixed, tunable collision energy that can be parked on a resonance. LEP scanning the $Z$ pole and measuring its line shape is the archetype. The cost is synchrotron radiation, which caps the energy of a circular machine and forces a linear design above the LEP scale. - **Proton-proton (or proton-antiproton) colliders** ($pp$, $p\bar p$). Protons are composite, so the elementary collision is between partons that each carry an unknown fraction of the proton momentum; the parton-level $\sqrt{\hat s}$ is a distribution below the machine $\sqrt s$, and the events are messy with beam remnants and QCD radiation. In exchange, protons do not radiate, so a circular machine reaches far higher energy, and the range of parton energies means one machine probes many scales at once. The LHC is the archetype: a discovery machine that trades a clean initial state for reach. A common division of labor follows. A hadron collider is a **discovery** machine — broad reach, high energy, the tool that first sees a new state. An electron machine is a **precision** machine — clean events at a known energy, the tool that measures the new state's properties. The discovery of the $W$ and $Z$ at the $Sp\bar pS$ hadron collider followed by their precision study at LEP is the template, and the same pairing motivates proposals for an $e^+e^-$ "Higgs factory" to follow the LHC's discovery of the Higgs. $$ % caption: The energy frontier over time. Hadron colliders (solid) climb fastest in % reach; electron machines (dashed) sit lower because synchrotron radiation caps a % circular design, but deliver clean precision at a known energy. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[right, black] {year}; \draw[->, black] (0,0) -- (0,4.0) node[above, black] {reach (log)}; % hadron colliders rising \draw[very thick] (0.3,0.6) -- (1.6,1.3) -- (2.8,1.9) -- (4.2,3.0) -- (5.8,3.6); \fill (0.3,0.6) circle (1.6pt); \fill (1.6,1.3) circle (1.6pt); \fill (2.8,1.9) circle (1.6pt); \fill (4.2,3.0) circle (1.6pt); \fill (5.8,3.6) circle (1.6pt); % electron machines lower \draw[very thick, dashed] (0.6,0.4) -- (2.0,0.9) -- (3.4,1.4) -- (4.6,1.7); \node[right] at (4.9,3.55) {hadron}; \node[right] at (4.5,1.55) {electron}; \end{tikzpicture} $$ ## The LHC as a case study The Large Hadron Collider gathers every element of this lesson into one machine. It is a $pp$ synchrotron of circumference $26.7\,\text{km}$, chosen because protons do not radiate and a large radius is affordable in reused tunnel. Its energy is set by the bending field: superconducting dipoles of about $8.3\,\text{T}$ hold each beam at $6.8\,\text{TeV}$, giving $\sqrt s=13.6\,\text{TeV}$ in Run 3 (design $7\,\text{TeV}$ per beam, $\sqrt s=14\,\text{TeV}$).[^lhc] Because the colliding objects are partons, the actual parton-level energies span a wide range below $\sqrt s$, so a single machine probes electroweak-scale physics and multi-TeV physics simultaneously. Its luminosity is set by tight focusing of dense bunches: several thousand bunches of $\sim10^{11}$ protons each, crossing every $25\,\text{ns}$, squeezed to a transverse size of order $10\,\mu\text{m}$ at the interaction points, reaching $\mathcal L\sim2\times10^{34}\,\text{cm}^{-2}\,\text{s}^{-1}$ and delivering more than $100\,\text{fb}^{-1}$ per year. That integrated luminosity is what made the Higgs observable: with a production cross section of tens of picobarns and a $\gamma\gamma$ branching ratio near $2\times10^{-3}$, only a machine delivering tens of inverse femtobarns produces enough events to lift the signal over the background, the subject of the [discovery lesson](/particle-physics/electroweak-higgs/higgs-boson-discovery). ## Summary The invariant mass $\sqrt s$ sets the accessible final states, and the choice of machine sets how much beam energy $\sqrt s$ costs. A fixed target wastes energy in the recoil of the center of mass, so $\sqrt s\simeq\sqrt{2M_t E_{\text{beam}}}$ grows as the square root of the beam energy; a collider puts the lab frame at rest, $\sqrt s=2E$, and spends every unit of beam energy. Linacs accelerate once and never bend; synchrotrons reuse cavities every turn but pay synchrotron radiation $\Delta E\propto (E/m)^4/R$, which caps circular electron machines at the LEP scale and leaves the proton frontier limited by bending field, $p=0.3BR$. Luminosity converts a cross section into a rate through $R=\mathcal L\sigma$, and its time integral fixes the event count $N=\sigma L_{\text{int}}$. Electron machines buy clean precision at a known energy; proton machines buy reach with a composite, messy initial state. The LHC realizes the proton strategy — a large low-radiation ring, strong dipoles, and enough integrated luminosity to make rare processes visible. What those collisions register in is the subject of the [next lesson](/particle-physics/experiment/detectors-subsystems). [^lhc]: LHC parameters — circumference $26.7\,\text{km}$, dipole field $\approx8.3\,\text{T}$, Run 3 $\sqrt s=13.6\,\text{TeV}$, peak luminosity $\sim2\times10^{34}\,\text{cm}^{-2}\,\text{s}^{-1}$ — and the fixed-target versus collider kinematics follow Thomson, Ch. 1 §1.4 and Appendix, and Perkins, Ch. 9. The synchrotron-radiation loss and the momentum-field relation $p=0.3BR$ are standard accelerator relations given in Perkins, Ch. 9, and Griffiths, §1.11. Machine and beam values are tabulated by the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).