14.4. 3-jet events in e
+ e
− annihilation
99
quark of type ‘a’. Note the minus sign in (14.59): the antiquark coupling is
−g s . In (14.59), ∈
∗ν (λ) is the polarization vector of the outgoing gluon with
polarization λ; a c is the colour wavefunction of the gluon (c = 1 . . . . . . 8),
and λ c is the corresponding Gell-Mann matrix introduced in section 12.2; the
colour parts of the q and q ¯ wavefunctions are understood to be included in
the u and v factors; and ( p 1 + p 3 )/2p 1 · p 3 is the virtual quark propagator
(cf (L.6) in appendix L of volume 1) before gluon radiation, and similarly for
the antiquark. Since the colour parts separate from the Dirac trace parts, we
shall ignore them to begin with, and reinstate the result of the colour sum
(via problem (14.5)) in the final answer (14.73).
Averaging over e
± spins and summing over final state quark spins and
gluon polarization λ (using (8.171), and noting the discussion after (13.93)),
we obtain (problem 14.6)
1 ∑
e
4 e
2
|M
L
μν
q¯ qg |
2 =
a g
2
s
(k 1 , k 2 )H μν (p 1 , p 2 , p 3 )
(14.60)
4
Q 4
spins,λ
where the lepton tensor is, as usual (equation (8.119)),
L
μν
μ
(k 1 , k ) = 2(k
ν
ν μ
μν
2
1 k 2 + k 1 k 2 − k 1 · k 2 g )
(14.61)
and the hadron tensor is
1
H μν (p 1 , p 2 , p 3 ) =
[L μν (p 2 , p 3 ) − L μν (p 1 , p 1 ) + L μν (p 1 , p 2 )]
p 1 · p 3
1
+
[L μν (p 1 , p 3 ) L μν (p 2 , p 2 )
p 2 p 3
−
·
+ L μν (p 1 , p 2 )]
p 1
+
· p 2
[2L
·
μν (p 1 , p 2 ) + L μν (p 1 , p 3 )
(p 1 p 3 )(p 2 · p 3 )
+ L μν (p 2 , p 3 )]
(14.62)
Combining (14.61) and (14.62) allows complete expressions for the five-fold
differential cross section to be obtained (Ellis et al. 1976).
For the subsequent discussion it will be useful to integrate over the three
angles describing the orientation (relative to the beam axis) of the production plane containing the three partons. After this integration, the (doubly
differential) cross section is a function of two independent Lorentz invariant
variables, which are conveniently taken to be two of the three s ij defined by
s ij = (p i + p
2
j ) .
(14.63)
Since we are considering the massless case p
2
i = 0 throughout, we may also
write
s ij = 2p i · p j .
(14.64)
These variables are linearly related by
2(p 1 · p 2 + p
2
2 · p 3 + p 3 · p 1 ) = Q
(14.65)
/
/
+ e
− annihilation
99
quark of type ‘a’. Note the minus sign in (14.59): the antiquark coupling is
−g s . In (14.59), ∈
∗ν (λ) is the polarization vector of the outgoing gluon with
polarization λ; a c is the colour wavefunction of the gluon (c = 1 . . . . . . 8),
and λ c is the corresponding Gell-Mann matrix introduced in section 12.2; the
colour parts of the q and q ¯ wavefunctions are understood to be included in
the u and v factors; and ( p 1 + p 3 )/2p 1 · p 3 is the virtual quark propagator
(cf (L.6) in appendix L of volume 1) before gluon radiation, and similarly for
the antiquark. Since the colour parts separate from the Dirac trace parts, we
shall ignore them to begin with, and reinstate the result of the colour sum
(via problem (14.5)) in the final answer (14.73).
Averaging over e
± spins and summing over final state quark spins and
gluon polarization λ (using (8.171), and noting the discussion after (13.93)),
we obtain (problem 14.6)
1 ∑
e
4 e
2
|M
L
μν
q¯ qg |
2 =
a g
2
s
(k 1 , k 2 )H μν (p 1 , p 2 , p 3 )
(14.60)
4
Q 4
spins,λ
where the lepton tensor is, as usual (equation (8.119)),
L
μν
μ
(k 1 , k ) = 2(k
ν
ν μ
μν
2
1 k 2 + k 1 k 2 − k 1 · k 2 g )
(14.61)
and the hadron tensor is
1
H μν (p 1 , p 2 , p 3 ) =
[L μν (p 2 , p 3 ) − L μν (p 1 , p 1 ) + L μν (p 1 , p 2 )]
p 1 · p 3
1
+
[L μν (p 1 , p 3 ) L μν (p 2 , p 2 )
p 2 p 3
−
·
+ L μν (p 1 , p 2 )]
p 1
+
· p 2
[2L
·
μν (p 1 , p 2 ) + L μν (p 1 , p 3 )
(p 1 p 3 )(p 2 · p 3 )
+ L μν (p 2 , p 3 )]
(14.62)
Combining (14.61) and (14.62) allows complete expressions for the five-fold
differential cross section to be obtained (Ellis et al. 1976).
For the subsequent discussion it will be useful to integrate over the three
angles describing the orientation (relative to the beam axis) of the production plane containing the three partons. After this integration, the (doubly
differential) cross section is a function of two independent Lorentz invariant
variables, which are conveniently taken to be two of the three s ij defined by
s ij = (p i + p
2
j ) .
(14.63)
Since we are considering the massless case p
2
i = 0 throughout, we may also
write
s ij = 2p i · p j .
(14.64)
These variables are linearly related by
2(p 1 · p 2 + p
2
2 · p 3 + p 3 · p 1 ) = Q
(14.65)
/
/