108
14. QCD I: Introduction, Tree Graph Predictions, and Jets
The JADE algorithm (Bartel et al. 1986, Bethke et al. 1988) is a promi−
nent early example of sequential recombination algorithms applied in e
+ e
annihilation reactions. Particles are clustered in a jet iteratively as long as
the quantity y ij of (14.77) is less than some prescribed value y c . If for some
pair (i, j), y ij < y c , particles i and j are combined into a compound object
(with the resultant 4-momentum, typically), and the process continues by
pairing the compound with a new particle k. The procedure stops when all
y ij distances are greater than y c , and the compounds that remain at this stage
are the jets, by definition.
One drawback with this scheme is that in higher orders of perturbation
theory one meets terms of the form α
2 ln
2n y (generalizations of the α s ln
2 y
s
term in (14.78)). Such terms can be large enough to invalidate a perturbative
approach. Also, it is possible for two soft particles moving in opposite directions to get combined in the early stages of clustering, which runs counter
to the intuitive notion of a jet being restricted in angular radius. The k t
algorithm (Catani et al. 1991) avoids these problems by replacing the y ij of
(14.77) by
y ij = 2min.[E i
2 , E j
2 ](1 − cos θ ij )/Q
2 .
(14.81)
This amounts to defining ‘distance’ by the minimum transverse momentum
k t of the particles in nearby collinear pairs. The use of the minimum energy
ensures that the distance between two soft, back-to-back particles is larger
than that between a soft particle and a hard one that is close to it in angle.
The k t algorithm was widely used at LEP.
The basic idea of the k t algorithm was extended to hadron colliders (Ellis
and Soper 1993, Catani et al. 1993), where the total energy of the hard
scattering particles is not well defined experimentally. The distance measure
y ij is replaced by
2p 2p
d ij = min.[p ti , p tj ][(y i − y j )
2 + (φ i − φ j )
2 ]/R
2
(14.82)
where, for particle i, p ti is the transverse momentum with respect to the
1
(beam) z-axis, y i is the rapidity along the beam axis (defined by y i = ln[(E i +
2
p zi )/(E i −p zi )]), φ i is the azimuthal angle in the plane transverse to the beam,
and R is a jet parameter. The variables y i , φ i have the property that they are
invariant under boosts along the beam direction. In addition, recombination
with the beam jets is controlled by the quantity d ij = k
2p , which is included
ti
along with the d ij ’s when recombining all the particles into (i) jets with nonzero transverse momentum, and (ii) beam jets. The power parameter p takes
the value 1 in the (extended) k t algorithm, and -1 in the ‘anti-k t ’ algorithm
(Cacciari et al. 2008). Whereas the former (and p = 0) leads to irregularly
shaped jet boundaries, the latter leads to cone-like boundaries. The choice
p = −1 was made in early LHC analyses.
14. QCD I: Introduction, Tree Graph Predictions, and Jets
The JADE algorithm (Bartel et al. 1986, Bethke et al. 1988) is a promi−
nent early example of sequential recombination algorithms applied in e
+ e
annihilation reactions. Particles are clustered in a jet iteratively as long as
the quantity y ij of (14.77) is less than some prescribed value y c . If for some
pair (i, j), y ij < y c , particles i and j are combined into a compound object
(with the resultant 4-momentum, typically), and the process continues by
pairing the compound with a new particle k. The procedure stops when all
y ij distances are greater than y c , and the compounds that remain at this stage
are the jets, by definition.
One drawback with this scheme is that in higher orders of perturbation
theory one meets terms of the form α
2 ln
2n y (generalizations of the α s ln
2 y
s
term in (14.78)). Such terms can be large enough to invalidate a perturbative
approach. Also, it is possible for two soft particles moving in opposite directions to get combined in the early stages of clustering, which runs counter
to the intuitive notion of a jet being restricted in angular radius. The k t
algorithm (Catani et al. 1991) avoids these problems by replacing the y ij of
(14.77) by
y ij = 2min.[E i
2 , E j
2 ](1 − cos θ ij )/Q
2 .
(14.81)
This amounts to defining ‘distance’ by the minimum transverse momentum
k t of the particles in nearby collinear pairs. The use of the minimum energy
ensures that the distance between two soft, back-to-back particles is larger
than that between a soft particle and a hard one that is close to it in angle.
The k t algorithm was widely used at LEP.
The basic idea of the k t algorithm was extended to hadron colliders (Ellis
and Soper 1993, Catani et al. 1993), where the total energy of the hard
scattering particles is not well defined experimentally. The distance measure
y ij is replaced by
2p 2p
d ij = min.[p ti , p tj ][(y i − y j )
2 + (φ i − φ j )
2 ]/R
2
(14.82)
where, for particle i, p ti is the transverse momentum with respect to the
1
(beam) z-axis, y i is the rapidity along the beam axis (defined by y i = ln[(E i +
2
p zi )/(E i −p zi )]), φ i is the azimuthal angle in the plane transverse to the beam,
and R is a jet parameter. The variables y i , φ i have the property that they are
invariant under boosts along the beam direction. In addition, recombination
with the beam jets is controlled by the quantity d ij = k
2p , which is included
ti
along with the d ij ’s when recombining all the particles into (i) jets with nonzero transverse momentum, and (ii) beam jets. The power parameter p takes
the value 1 in the (extended) k t algorithm, and -1 in the ‘anti-k t ’ algorithm
(Cacciari et al. 2008). Whereas the former (and p = 0) leads to irregularly
shaped jet boundaries, the latter leads to cone-like boundaries. The choice
p = −1 was made in early LHC analyses.