Workshop on luminosity

Main method: from measurement of the total cross section

Total cross section in pp scattering at LHC using the optical theorem

Optical theorem: demonstration
The optical theorem says that Im f(0) = q*σ_tot/4π [Eq.1], where q is the CM momentum.
We combine this with the trivial formula dσ_el(0)/dΩ=|f(0)|² [Eq.2]
In fact we rewrite f(0) = Re f(0) + Im f(0) = Im f(0) * (1+ρ)
So, |f(0)|²=(Im f(0))²*(1+ρ)²
We take |f(0)|² from [Eq.2] and Im f(0) from [Eq.1], obtaining:
dσ_el(0)/dΩ=q²*σ_tot²*(1+ρ)²/16π²
where dσ_el(0)/dΩ is measured with forward detectors(*), and ρ is in principle calculable from theory(**).

(*) In reality you cannot measure it directly at 0 angle, so you extrapolate by taking the slope of dσ_el(t)/d(-t) at t=0.
(**) The QED contribution is very well known, but at very low angle the nuclear term is strong. There is a t value around which the Coulomb repulsion starts to dominate over the nuclear attraction: this is the Coulomb peak. This can be easily calculated. Somewhere in between the EM and nuclear domination domains, the EM and nuclear forces are comparable, producing an area of interference. This proves useful to measure ρ using dispersion relations (Δp/p). At the LHC energies, we expect ρ=0.10+-0.02.

Now that we know σ_tot, luminosity can be extracted as L=N/(σ_tot*Δt) (for N events taken in a period Δt).

NOTE: this cannot be done at design luminosity at LHC, so dedicated run will be used at smaller luminosity, at regular time intervals.
For online luminosity monitoring, better W's and Z's (at least to interpolate in time).
Extracting σ(W or Z) from N(W or Z) introduces an irreducible uncertainty from PDF's. But dN(W or Z)/dη constrains the PDF's themselves, so the PDF uncertainty can be kept at ~1%.

Luminosity: Van der Meer method (based on accelerator parameters)
L=B*N1*N2*f/(4π*σ_x*σ_y)
So you need to measure:

• N (total current in a bunch)
• σ_x,y

Luminosity in pp and AA colliders

Other methods to measure luminosity on line:

The pixel detector can count the number of primary vertexes in the event. Not very precise, but good for online.
In general, counting the number of interactions in a bunch crossing is very difficult and unreliable.

Solution: Zero counting method: (P-TDR I)
(This method uses the forward calorimeters.)
The number of interactions in a given bunch crossing is given by Poisson statistics:
P(n,μ)=e^-μμⁿ/n!
Reverting the logic, we count the number of bunch crossings without any interaction:
μ=-log[P(0)]
where P(0) is extracted as N(0)/N(tot).

Not very precise at high luminosity ("zero starvation"). Apart from the lack of zeroes, the problem is systematics (background from beam-halo and beam-gas interactions; also noise in the calorimeter?).

Solution: use the single highest-signal tower in the calorimeter, and count how many times it is above/below a certain threshold, such that μ~1 and the zero counting is again viable.
Each tower is a quasi-independent measurement.