Measurement of the Z boson cross-section in - Harvard University ...

Chapter 3: Luminosity Measurement at the LHC and in ATLAS 84
can be used to determine relative integrated luminosity offline. In this section, we
will briefly describe the subdetectors used for online luminosity monitoring and some
of the processes potentially useful for offline luminosity estimation.
3.3.1 ALFA
The ALFA (Absolute Luminosity F or Atlas) detector measures the elastic scat-
tering rate at very small angles, and uses the optical theorem to relate this rate to the
total cross-section and the luminosity. The total inelastic and elastic collision rates
measured in an experiment are related to the luminosity and the total cross-section:
˙Ninel + ˙ Nel = Lσtot
(3.16)
For small values of momentum transfer t 6 , the total cross-section is related to the
elastic cross-section by the optical theorem:
lim dσel
dt
= 1
L
d ˙
Nel
dt |t=0
(3.17)
But the elastic scattering amplitude can be expressed as a superposition of the
Coulomb scattering amplitude fc and the strong scattering amplitude fs. So we can
write:
lim dσel
dt
= 1
L
d ˙ Nel
dt |t=0 = π|fc + fs| 2 ≈ π| 2αem
−t
σtot
t
+ (ρ + i)eB 2 |≈
4π 4πσ2 em
t2 |
||t| →0
(3.18)
where ρ is the interference parameter between the Coulomb and strong interference
terms [58] and B is the nuclear slope parameter. If dσel
dt is measured accurately,
6 t is a Mandelstam variable, defined as t =(p1−p3) 2 =(p2−p4) 2 ,wherep1,p2 are the 4-momenta
of the particles incoming to a collision, and p3,p4 are the 4-momenta of the particles outgoing from
a collision.