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B. J. Holzer et al.
6.4 Concept of Luminosity
Werner Herr · Bernhard Holzer · Bruno Muratori
6.4.1 Introduction
In particle physics experiments the energy available for the production of new
effects is the most important parameter. Besides the energy the number of useful
interactions (events), is important. The quantity that measures the ability of a
particle accelerator to produce the required number of interactions is called the
luminosity (see Chap. 2) and is the proportionality factor between the number of
events per second dR/dt and the cross section σ p :
dR
dt
= L · σ p
(6.22)
The unit of the luminosity is therefore cm −2 s −1 .
Here we will derive a general expression for the luminosity and give formulae for
basic cases. Additional complications such as crossing angle and offset collisions
are added to the calculation. Other effects such as the hourglass effect are estimated
from the generalized expression.
In the final section we will discuss the measurement and calibration of the
luminosity for both e + e − as well as hadron colliders.
6.4.2 Computation of Luminosity
In the case of two colliding bunches, both serve as “target” as well as “incoming”
beam at the same time. A schematic picture is shown in Fig. 6.18. The overlap
integral which is proportional to the luminosity L can be written as [37]:
L ∝ KN 1 N 2 ·
+∞
−∞
ρ 1 (x, y, s, −s 0 ) ρ 2 (x, y, s, s 0 ) dxdydsds 0
(6.23)
Here ρ 1 (x, y, s, s 0 ) and ρ 2 (x, y, s, s 0 ) are the time dependent beam density distribution functions and N 1 and N 2 the number of particles per bunch. We assume, that
the two bunches meet at s 0 = 0 and s 0 = c · t is used as the “time” variable. Because
the beams are moving against each other, we have to introduce the kinematic factor
[38]:
K =
− → ν 1 − − → ν 2
2 −
− → ν 1 × − → ν 2
2 /c 2
(6.24)
B. J. Holzer et al.
6.4 Concept of Luminosity
Werner Herr · Bernhard Holzer · Bruno Muratori
6.4.1 Introduction
In particle physics experiments the energy available for the production of new
effects is the most important parameter. Besides the energy the number of useful
interactions (events), is important. The quantity that measures the ability of a
particle accelerator to produce the required number of interactions is called the
luminosity (see Chap. 2) and is the proportionality factor between the number of
events per second dR/dt and the cross section σ p :
dR
dt
= L · σ p
(6.22)
The unit of the luminosity is therefore cm −2 s −1 .
Here we will derive a general expression for the luminosity and give formulae for
basic cases. Additional complications such as crossing angle and offset collisions
are added to the calculation. Other effects such as the hourglass effect are estimated
from the generalized expression.
In the final section we will discuss the measurement and calibration of the
luminosity for both e + e − as well as hadron colliders.
6.4.2 Computation of Luminosity
In the case of two colliding bunches, both serve as “target” as well as “incoming”
beam at the same time. A schematic picture is shown in Fig. 6.18. The overlap
integral which is proportional to the luminosity L can be written as [37]:
L ∝ KN 1 N 2 ·
+∞
−∞
ρ 1 (x, y, s, −s 0 ) ρ 2 (x, y, s, s 0 ) dxdydsds 0
(6.23)
Here ρ 1 (x, y, s, s 0 ) and ρ 2 (x, y, s, s 0 ) are the time dependent beam density distribution functions and N 1 and N 2 the number of particles per bunch. We assume, that
the two bunches meet at s 0 = 0 and s 0 = c · t is used as the “time” variable. Because
the beams are moving against each other, we have to introduce the kinematic factor
[38]:
K =
− → ν 1 − − → ν 2
2 −
− → ν 1 × − → ν 2
2 /c 2
(6.24)