The LHC design luminosity is 10^{34} cm^{-2} s^{-1}. The total inelastic proton-proton cross section at 13 TeV is approximately 80 mb = 80 x 10^{-27} cm^2. How many inelastic collisions occur per second, and what does this imply for the detector?
AAbout 80 collisions per second, which is easily manageable
BAbout 8 x 10^8 (800 million) collisions per second — with 2808 bunches crossing at 40 MHz, this corresponds to approximately 20-50 simultaneous collisions per bunch crossing (pileup), which the detectors must be designed to handle
CAbout 10^{34} collisions per second
DAbout 10^{10} collisions per second, but most are filtered by the trigger
dN/dt = L * sigma = 10^{34} * 80 x 10^{-27} = 8 x 10^8 per second. Divided among ~3000 filled bunches crossing at 40 MHz, this gives ~25 interactions per bunch crossing at design luminosity (pileup <mu> ~ 25). At Run 2/3, <mu> reached 30-60. Each pileup interaction produces ~30 charged particles, so the detector must reconstruct the physics objects from ~1000 tracks per bunch crossing while rejecting the pileup contribution. The HL-LHC will operate at <mu> ~ 200, requiring major detector upgrades.
Question 2 Short Answer
Luminosity at the LHC is calibrated using van der Meer (vdM) scans. How does this calibration work?
Think about your answer, then reveal below.
Model answer: In a van der Meer scan, the two beams are deliberately displaced transversely (in x and y) while monitoring the collision rate. The rate as a function of beam separation traces out the beam overlap profile. The peak rate and the effective beam widths (Sigma_x, Sigma_y) determine the luminosity through L = f_rev * n_b * N_1 * N_2 / (2*pi * Sigma_x * Sigma_y), where f_rev is the revolution frequency, n_b is the number of colliding bunch pairs, and N_1, N_2 are the bunch populations. This method directly measures the beam overlap without assumptions about the beam shape. The calibration is transferred to online luminosity monitors (luminosity detectors counting collision products) that then track the luminosity during physics running. The dominant uncertainties are from beam-beam effects, bunch population measurements, and non-Gaussian beam tails, achieving ~1-2% total uncertainty.
The vdM technique was invented by Simon van der Meer (who also invented stochastic cooling, enabling the SppS and the W/Z discovery). It remains the primary absolute luminosity calibration method at all hadron colliders.
Question 3 Multiple Choice
The HL-LHC (High-Luminosity LHC) aims to deliver 3000 fb^{-1} of integrated luminosity, compared to ~300 fb^{-1} from the LHC Runs 1-3 combined. Why does a factor of 10 more data significantly extend the physics reach?
ABecause all measurements improve by a factor of 10
BBecause statistical sensitivity scales as sqrt(L) for discovery (so 10x more data gives ~3x better significance for rare signals) and as 1/sqrt(L) for statistical uncertainties on measured quantities — additionally, more data enables measurements of extremely rare processes (Higgs self-coupling, rare Higgs decays) that require thousands of signal events to be observable above background
CBecause the beam energy also increases by a factor of 10
DBecause systematic uncertainties decrease proportionally to the luminosity
For a signal of S events on a background of B events, the significance goes as S/sqrt(B) proportional to sqrt(L). So 10x more luminosity gives ~3x better significance. This is crucial for rare processes: di-Higgs production (sigma ~ 31 fb at 14 TeV) yields ~100 events in 3000 fb^{-1}, barely enough for observation. Precision measurements of Higgs couplings improve as 1/sqrt(L) (for statistically limited channels) or plateau (for systematically limited ones). The HL-LHC is designed to exploit the full statistical potential of the LHC energy frontier.