9.6
Problems
295
(c) The general massless spinor u can be written
u = (P L + P R )u ≡ u L + u R
where u L , u R have the indicated helicities. Show that
uγ ¯
μ u = ¯
u L γ
μ u L + ¯
u R γ
μ u R
†
†
where ¯
u L = u γ
0 , ¯
u R = u γ
0 ; and deduce that in electromagnetic
L
R
interactions of massless fermions helicity is conserved.
(d) In weak interactions an axial vector current uγ
μ γ 5 u also enters. Is
¯
helicity still conserved?
¯
¯
(e) Show that the ‘Dirac’ mass term mψ ˆ ψ ˆ may be written as m(ψ ˆ
L ψ ˆ R +
¯ ˆ ˆ
ψ R ψ L ).
9.5 In the HERA colliding beam machine, positrons of total energy 27.5 GeV
collide head on with protons of total energy 820 GeV. Neglecting both the
positron and the proton rest masses, calculate the centre-of-mass energy in
such a collision process.
Some theories have predicted the existence of ‘leptoquarks’, which could
be produced at HERA as a resonance state formed from the incident positron
and the struck quark. How would a distribution of such events look, if plotted
versus the variable x?
(a) By the expedient of inserting a δ-function, the differential cross
√
section for Drell–Yan production of a lepton pair of mass q 2 may
be written as
∫
dσ
d
2 σ
= dx 1 dx 2
δ(q
2
− sx 1 x 2 ).
dq 2
dx 1 dx 2
Show that this is equivalent to the form
dσ
4πα
2 ∫
dq 2 = 9q 4
dx 1 dx 2 x 1 x 2 δ(x 1 x 2 − τ )
∑
×
e
2
a [q a (x 1 )¯ q a (x 2 ) + ¯
q a (x 1 )q a (x 2 )]
a
2
which, since q = sτ , exhibits a scaling law of the form
2
s
2 dσ/dq = F (τ ).
Problems
295
(c) The general massless spinor u can be written
u = (P L + P R )u ≡ u L + u R
where u L , u R have the indicated helicities. Show that
uγ ¯
μ u = ¯
u L γ
μ u L + ¯
u R γ
μ u R
†
†
where ¯
u L = u γ
0 , ¯
u R = u γ
0 ; and deduce that in electromagnetic
L
R
interactions of massless fermions helicity is conserved.
(d) In weak interactions an axial vector current uγ
μ γ 5 u also enters. Is
¯
helicity still conserved?
¯
¯
(e) Show that the ‘Dirac’ mass term mψ ˆ ψ ˆ may be written as m(ψ ˆ
L ψ ˆ R +
¯ ˆ ˆ
ψ R ψ L ).
9.5 In the HERA colliding beam machine, positrons of total energy 27.5 GeV
collide head on with protons of total energy 820 GeV. Neglecting both the
positron and the proton rest masses, calculate the centre-of-mass energy in
such a collision process.
Some theories have predicted the existence of ‘leptoquarks’, which could
be produced at HERA as a resonance state formed from the incident positron
and the struck quark. How would a distribution of such events look, if plotted
versus the variable x?
(a) By the expedient of inserting a δ-function, the differential cross
√
section for Drell–Yan production of a lepton pair of mass q 2 may
be written as
∫
dσ
d
2 σ
= dx 1 dx 2
δ(q
2
− sx 1 x 2 ).
dq 2
dx 1 dx 2
Show that this is equivalent to the form
dσ
4πα
2 ∫
dq 2 = 9q 4
dx 1 dx 2 x 1 x 2 δ(x 1 x 2 − τ )
∑
×
e
2
a [q a (x 1 )¯ q a (x 2 ) + ¯
q a (x 1 )q a (x 2 )]
a
2
which, since q = sτ , exhibits a scaling law of the form
2
s
2 dσ/dq = F (τ ).