106
14. QCD I: Introduction, Tree Graph Predictions, and Jets
tinguishable mass-degenerate states. This is the reason for the finiteness of
the Sterman-Weinberg 2-jet cross section, in an analogous QCD case.
Returning to (14.76), it is important to note that the angular distribution of this well-defined two-jet process is given precisely by the lowest-order
expression (9.102), just as was hoped in the simple parton model of section
9.5. Of course, the cross section depends on the jet parameters δ and ∈. The
formula (14.76) can be used, for example, to estimate the angular radius of
the jets, as a function of E.
Although the Sterman-Weinberg jet definition was historically the first,
it is not the only possible one. Another, in some ways simpler, definition
(Kramer and Lampe 1987) is directly phrased in terms of the offending de−1
−1
nominators s and s in (14.67). Let us introduce the dimensionless jet
13
23
mass variables
y ij = s ij /Q
2 = 2E i E j (1 − cos θ ij )/Q
2
(14.77)
for any two partons i and j; s 12 will be included, though no singularity is
involved. Here E i and E j are the (massless) parton energies, and θ ij is the
angle between their 3-momenta, in the overall CMS. Then i and j are defined
to be in one jet if y ij is less than some given number y. Note that for small
θ ij , s ij ≈ E i E j θ
2 /Q
2 , so the single parameter y provides effectively both an
ij
energy and an angle cut. Clearly this definition is equivalent to a formulation
in terms of strips 1 ≤ x k < 1 − y on figure 14.7, as discussed earlier. Including
contributions, as before, from figures 14.22, 14.21, and 14.16, the resulting
2-jet cross section is found to be (Kramer and Lampe 1987)
2 α s (2 ln
2 y + 3 ln y − 4y ln y + 1 − π
2 /3)].
(14.78)
σ 2−jet = σ pt [1 − 3 π
Terms of order y were calculated numerically. These include the contribution
from the (non-singular) region y 12 < y, where the two quarks are in one jet
and the other jet is a pure gluon jet. Plainly the IRC singularities have been
eliminated from (14.78), at the cost of the jet mass resolution parameter y.
Kramer and Lampe also calculated the order α
2 corrections to (14.78).
s
These two ways of regulating the IRC divergences in the 2-jet partonic
cross section have each been extensively developed into jet algorithms, as we
shall briefly discuss in section 14.6.2.
14.6 Further developments
14.6.1 Test of non-Abelian nature of QCD in e + e − → 4 jets
We have seen in section 14.3.1 how the colour factors associated with different
QCD vertices (problem 14.5) play an important part in determining the relative weights of different parton-level processes. The quark-gluon colour factor
14. QCD I: Introduction, Tree Graph Predictions, and Jets
tinguishable mass-degenerate states. This is the reason for the finiteness of
the Sterman-Weinberg 2-jet cross section, in an analogous QCD case.
Returning to (14.76), it is important to note that the angular distribution of this well-defined two-jet process is given precisely by the lowest-order
expression (9.102), just as was hoped in the simple parton model of section
9.5. Of course, the cross section depends on the jet parameters δ and ∈. The
formula (14.76) can be used, for example, to estimate the angular radius of
the jets, as a function of E.
Although the Sterman-Weinberg jet definition was historically the first,
it is not the only possible one. Another, in some ways simpler, definition
(Kramer and Lampe 1987) is directly phrased in terms of the offending de−1
−1
nominators s and s in (14.67). Let us introduce the dimensionless jet
13
23
mass variables
y ij = s ij /Q
2 = 2E i E j (1 − cos θ ij )/Q
2
(14.77)
for any two partons i and j; s 12 will be included, though no singularity is
involved. Here E i and E j are the (massless) parton energies, and θ ij is the
angle between their 3-momenta, in the overall CMS. Then i and j are defined
to be in one jet if y ij is less than some given number y. Note that for small
θ ij , s ij ≈ E i E j θ
2 /Q
2 , so the single parameter y provides effectively both an
ij
energy and an angle cut. Clearly this definition is equivalent to a formulation
in terms of strips 1 ≤ x k < 1 − y on figure 14.7, as discussed earlier. Including
contributions, as before, from figures 14.22, 14.21, and 14.16, the resulting
2-jet cross section is found to be (Kramer and Lampe 1987)
2 α s (2 ln
2 y + 3 ln y − 4y ln y + 1 − π
2 /3)].
(14.78)
σ 2−jet = σ pt [1 − 3 π
Terms of order y were calculated numerically. These include the contribution
from the (non-singular) region y 12 < y, where the two quarks are in one jet
and the other jet is a pure gluon jet. Plainly the IRC singularities have been
eliminated from (14.78), at the cost of the jet mass resolution parameter y.
Kramer and Lampe also calculated the order α
2 corrections to (14.78).
s
These two ways of regulating the IRC divergences in the 2-jet partonic
cross section have each been extensively developed into jet algorithms, as we
shall briefly discuss in section 14.6.2.
14.6 Further developments
14.6.1 Test of non-Abelian nature of QCD in e + e − → 4 jets
We have seen in section 14.3.1 how the colour factors associated with different
QCD vertices (problem 14.5) play an important part in determining the relative weights of different parton-level processes. The quark-gluon colour factor