1. van der Meer Scans

2. Basic Formula

3. Beam Height Essentially the width of the beam (name is historical) i.e. if all the luminosity of the beam were delivered in a rectangular beam profile, for a given luminosity the width of the rectangle is h Formalism derived for CERN ISR (first proton colliding machine). Effective beam height given by

4. Formalism van der Meer proposed to measure effective height using displaced beams. Rate for a given separation is so rate for head-on collisions is just same expression with h=0

5. van der Meer scan “Trick” comes in realizing that the integrated normalized counts if the beams are scanned across each other in a given direction gives the numerator of effective height distribution

6. van der Meer procedure Get numerator by summing contribution from each step in displacement scan (in steps of h) Get denominator by measuring head-on rate Then Integrated counts from vdM scan Head-on rate

7. Procedure Do two scans, one in x direction and one in y direction, to obtain effective beam heights in two dimensions This gives a geometrical measurement of the beam, good for a given implementation of the beam (crossing angle, emittance, etc.) so doesn’t depend on choice of trigger.

8. Assumptions Does assume x and y profiles are independent and therefore factorizable Needs distance calibration (so the integration steps dh, which are implemented by changing beam currents, correspond to known steps in x.) This is done by moving both beams in the same direction together, which moves the centroid of the beam spot (unlike the scan itself, which moves them in opposite directions, thus keeping the beam-spot centroid constant.) The inner tracker reconstruction, built for heavy flavour decays, should be easily good enough to measure the beam spot centroid displacement for a given change in currents. Gives as by-product a “trigger cross-section” for the process used for calibration. The rate for the trigger cross-section can then be used to monitor the luminosity for as long as the beam settings persist. Cross-check: use several different triggers for the scan. The trigger cross- sections can differ significantly, but the effective heights (and therefore the luminosities) should agree irrespective of the calibration trigger chosen.