286
9. Deep Inelastic Electron–Nucleon Scattering and the Parton Model
There is, of course, another contribution for which the antiquark has fraction
x 1 and the quark x 2 :
q ¯ a (x 1 ) dx 1 q a (x 2 ) dx 2 .
(9.90)
Thus the Drell–Yan prediction is
d
2 σ(pp → μ
+ μ
− + X)
∑
(9.91)
4πα
2
2
=
e [q a (x 1 )¯ q a (x 2 ) + q ¯ a (x 1 )q a (x 2 )] dx 1 dx 2
a
9q 2
a
where we have included a factor
1 to account for the colour of the quarks:
3
in order to make a colour singlet photon, one needs to match the colours of
quark and antiquark. Equation (9.91) is the master formula. Its importance
lies in the fact that the same quark distribution functions are measured in
deep inelastic lepton scattering so one can make absolute predictions.
3 For
example, if the photon in figure 9.11 is replaced by a W(Z), one can predict
W(Z) production cross sections, as we shall see in volume 2.
We would expect some ‘scaling’ property to hold for this cross section, following from the point-like constituent cross section (9.88). One way to exhibit
2
this is to use the variables q and x F = x 1 − x 2 as discussed in problem 9.6.
There it is shown that the dimensionless quantity
d
2 σ
4
q
(9.92)
dq 2 dx F
should be a function of x F and the ratio τ = q
2 /s. The data bear out this
prediction well – see figure 9.12.
Furthermore, the assumption that the lepton pair is produced via quark–
antiquark annihilation to a virtual photon can be checked by observing the
angular distribution of either lepton in the dilepton rest frame, relative to the
incident proton beam direction. This distribution is expected to be the same
as in e
+ e
−
→ μ
+ μ
− , namely (cf (8.194))
dσ/dΩ ∝ (1 + cos
2 θ)
(9.93)
as is indeed observed (figure 9.13). Note that figure 9.13 provides evidence
that the quarks have spin1 : if they are assumed to have spin-0, the angular
2
distribution would be (see problem 9.7) proportional to (1 − cos
2 θ), and this
is clearly ruled out.
3 QCD corrections make the connection more complicated, but still perturbatively computable.
9. Deep Inelastic Electron–Nucleon Scattering and the Parton Model
There is, of course, another contribution for which the antiquark has fraction
x 1 and the quark x 2 :
q ¯ a (x 1 ) dx 1 q a (x 2 ) dx 2 .
(9.90)
Thus the Drell–Yan prediction is
d
2 σ(pp → μ
+ μ
− + X)
∑
(9.91)
4πα
2
2
=
e [q a (x 1 )¯ q a (x 2 ) + q ¯ a (x 1 )q a (x 2 )] dx 1 dx 2
a
9q 2
a
where we have included a factor
1 to account for the colour of the quarks:
3
in order to make a colour singlet photon, one needs to match the colours of
quark and antiquark. Equation (9.91) is the master formula. Its importance
lies in the fact that the same quark distribution functions are measured in
deep inelastic lepton scattering so one can make absolute predictions.
3 For
example, if the photon in figure 9.11 is replaced by a W(Z), one can predict
W(Z) production cross sections, as we shall see in volume 2.
We would expect some ‘scaling’ property to hold for this cross section, following from the point-like constituent cross section (9.88). One way to exhibit
2
this is to use the variables q and x F = x 1 − x 2 as discussed in problem 9.6.
There it is shown that the dimensionless quantity
d
2 σ
4
q
(9.92)
dq 2 dx F
should be a function of x F and the ratio τ = q
2 /s. The data bear out this
prediction well – see figure 9.12.
Furthermore, the assumption that the lepton pair is produced via quark–
antiquark annihilation to a virtual photon can be checked by observing the
angular distribution of either lepton in the dilepton rest frame, relative to the
incident proton beam direction. This distribution is expected to be the same
as in e
+ e
−
→ μ
+ μ
− , namely (cf (8.194))
dσ/dΩ ∝ (1 + cos
2 θ)
(9.93)
as is indeed observed (figure 9.13). Note that figure 9.13 provides evidence
that the quarks have spin1 : if they are assumed to have spin-0, the angular
2
distribution would be (see problem 9.7) proportional to (1 − cos
2 θ), and this
is clearly ruled out.
3 QCD corrections make the connection more complicated, but still perturbatively computable.