LHC

Prerequisites: Collider Physics. Full path (34 pages)

Best explanations

Intuitive Introduction to the LHC
Concrete Crash Course in the LHC by Nima Arkani-Hamed

Intuitive
Concrete
Abstract
Why is it interesting?
FAQ

Intuitive

The Large Hadron Collider (LHC) at CERN, near Geneva, is a ring about 27 km around, built in the underground tunnel that previously held the LEP electron-positron collider. Two beams of protons travel around the ring in opposite directions, in two separate beam pipes, and are brought together at four points. At each of these points sits a large detector: ATLAS, CMS, LHCb and ALICE. For part of the running time the LHC collides lead nuclei instead of protons.

The reason for colliding particles at high energy is that we want to produce heavy particles that do not exist in ordinary matter, like the Higgs boson, the top quark or particles that are not yet known. Mass and energy are interchangeable, so the more energy is available in a collision, the heavier the particles that can be created. Short wavelengths also mean that we can resolve structure at shorter distances.

Protons are not elementary. Each proton is a bound state of quarks and gluons, and in a collision usually one quark or gluon from each proton does the hard scattering. Each of them carries only a fraction of the proton's energy, and that fraction is different in every collision. This makes the LHC a discovery machine that scans a wide range of collision energies at once, at the price of messier events than in electron-positron collisions.

Most collisions produce only well-known physics. Interesting processes are rare, so the LHC collides bunches of protons tens of millions of times per second, and the experiments have to select in real time the tiny fraction of events worth storing. In 2012 the ATLAS and CMS collaborations announced the discovery of the Higgs boson using LHC data.

Introduction to the LHC by Matt Strassler
A Zeptospace Odyssey: A Journey Into the Physics of the LHC by Gian Francesco Giudice

Concrete

Beam energy and magnets. A particle of charge $e$ and momentum $p$ moving perpendicular to a magnetic field $B$ follows a circle of radius $\rho$ with $p = eB\rho$ . In practical units,

$p\,[\text{GeV}/c] \approx 0.3\, B\,[\text{T}]\; \rho\,[\text{m}].$

The LHC dipoles are superconducting magnets operated at about $1.9\,\text{K}$ and designed for about $8.3\,\text{T}$ . The dipoles fill only part of the ring, and the bending radius inside them is about $2.8\,\text{km}$ , so the design momentum is $p \approx 0.3 \times 8.3 \times 2800\,\text{GeV}/c \approx 7\,\text{TeV}/c$ . The energy of a ring collider is limited by the magnetic field it can reach and the size of the tunnel.

Centre-of-mass energy. For two beams of energy $E$ colliding head on, with $E \gg m_p c^2$ , the invariant $s = (p_1 + p_2)^2$ (with $c = 1$ ) gives $\sqrt{s} = 2E$ . The design value is $\sqrt{s} = 14\,\text{TeV}$ . The main proton-proton runs were at $\sqrt{s} = 7$ , $8$ , $13$ and $13.6\,\text{TeV}$ .

Partons. A hard collision happens between a parton (quark or gluon) carrying momentum fraction $x_1$ of one proton and a parton carrying $x_2$ of the other. The partonic system has invariant mass and rapidity

$\hat s = M^2 = x_1 x_2 s, \qquad y = \frac{1}{2}\ln\frac{x_1}{x_2},$

so that $x_{1,2} = (M/\sqrt{s})\, e^{\pm y}$ . Producing a state of mass $M = 125\,\text{GeV}$ at $\sqrt{s} = 13\,\text{TeV}$ with $y = 0$ needs $x_1 = x_2 \approx 0.01$ . Since $x_1$ and $x_2$ are unknown event by event, the partonic system is boosted along the beam axis by an unknown amount. For this reason analyses use quantities that are invariant under boosts along the beam (transverse momentum $p_T$ , differences of rapidity) and the pseudorapidity $\eta = -\ln\tan(\theta/2)$ , which equals $y$ for massless particles.

Event rates. The number of events per second for a process with cross section $\sigma$ is

$\frac{dN}{dt} = \mathcal{L}\,\sigma,$

where $\mathcal{L}$ is the instantaneous luminosity. The design value is $\mathcal{L} = 10^{34}\,\text{cm}^{-2}\text{s}^{-1}$ . The total number of events is $N = \sigma \int \mathcal{L}\, dt$ , and the integrated luminosity is quoted in inverse femtobarns, $1\,\text{fb}^{-1} = 10^{39}\,\text{cm}^{-2}$ . Two examples at $\sqrt{s} = 13\,\text{TeV}$ :

The inelastic proton-proton cross section is of order $80\,\text{mb} = 8\times 10^{-26}\,\text{cm}^2$ . At design luminosity this gives about $10^9$ inelastic collisions per second, which with bunches crossing every $25\,\text{ns}$ means a few tens of collisions per crossing ("pile-up").
The total Higgs production cross section is about $50\,\text{pb}$ , so each $\text{fb}^{-1}$ contains about $5\times 10^4$ Higgs bosons, roughly one per $2\times 10^9$ inelastic collisions.
CRASH COURSE IN THE LHC by Nima Arkani-Hamed

Abstract

The theoretical description of hadron collisions at the LHC rests on QCD collinear factorization. For a hard process producing a final state $X$ at a large scale $Q \gg \Lambda_{\text{QCD}}$ ,

$\sigma(pp \to X + \text{anything}) = \sum_{i,j} \int_0^1 dx_1 \int_0^1 dx_2\; f_{i/p}(x_1, \mu_F^2)\, f_{j/p}(x_2, \mu_F^2)\; \hat\sigma_{ij \to X}\!\left(x_1 x_2 s, \mu_F^2, \mu_R^2\right) + \text{power corrections},$

where the sum runs over quarks, antiquarks and the gluon, $f_{i/p}$ are the parton distribution functions (PDFs), $\hat\sigma_{ij\to X}$ is the partonic cross section computed in perturbation theory in $\alpha_s(\mu_R^2)$ , and $\mu_F$ , $\mu_R$ are the factorization and renormalization scales. The power corrections are suppressed by powers of $\Lambda_{\text{QCD}}/Q$ . The PDFs are universal (the same for all processes) and non-perturbative. They are extracted from data, while their dependence on $\mu_F$ follows from the DGLAP equations,

$\mu^2 \frac{\partial f_{i/p}(x, \mu^2)}{\partial \mu^2} = \frac{\alpha_s(\mu^2)}{2\pi} \sum_j \int_x^1 \frac{dz}{z}\, P_{ij}(z)\, f_{j/p}\!\left(\frac{x}{z}, \mu^2\right)$

at leading order, with the splitting functions $P_{ij}$ . The dependence of a truncated perturbative result on $\mu_F$ and $\mu_R$ is of higher order in $\alpha_s$ , and its size is used as an estimate of the missing higher orders.

With $\tau = x_1 x_2 = \hat s/s$ the cross section can be written in terms of parton luminosities,

$\sigma = \sum_{i,j} \int_{\tau_0}^1 d\tau\; \frac{d\mathcal{L}_{ij}}{d\tau}\, \hat\sigma_{ij}(\tau s), \qquad \frac{d\mathcal{L}_{ij}}{d\tau} = \int_\tau^1 \frac{dx}{x}\, f_{i/p}(x)\, f_{j/p}(\tau/x),$

where $\tau_0$ is the production threshold. This makes the trade-off between beam energy and reach explicit: raising $\sqrt{s}$ moves a fixed mass $M$ to smaller $\tau = M^2/s$ , where the parton luminosities, in particular the gluon-gluon luminosity, grow quickly.

The kinematics is organised by the Lorentz group restricted to boosts along the beam axis and rotations around it. Rapidity $y = \frac{1}{2}\ln\frac{E + p_z}{E - p_z}$ is additive under longitudinal boosts, so the shape of a distribution in $y$ and the transverse quantities ( $p_T$ , azimuthal angle $\phi$ , transverse mass) do not depend on the unknown longitudinal boost of the partonic system. Distances between objects are measured in the boost invariant $\Delta R = \sqrt{(\Delta y)^2 + (\Delta\phi)^2}$ , which is also the distance measure used by jet algorithms such as anti- $k_t$ .

Functional specification LHC modes

Why is it interesting?

The LHC is currently the most powerful particle collider and discovered the Higgs particle.

“The L.H.C. is considered the best—some would say the only—hope for testing the theories of “new physics” against material reality. Once the collider begins operating at full power—in early 2008, if all goes well— nearly half the particle physicists in the world will be involved in analyzing its four-million-megabyte-per-hour stream of data. Few events in the history of science have had a bigger buildup. It’s been suggested that the L.H.C. will unlock the secrets of the universe or, barring that, prove this ambition to be hopeless.” http://www.newyorker.com/magazine/2007/05/14/crash-course

FAQ

see https://www.symmetrymagazine.org/article/inside-the-large-hadron-collider

LHC

Intuitive

Concrete

Abstract

Why is it interesting?

FAQ

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