389
l
22.3. Simple (tree-level) predictions
FIGURE 22.12
Neutrino-electron graphs involving Z
0 exchange.
Quark pairs couple as in (22.71), the GIM mechanism ensuring that all flavourchanging terms cancel. The total width to u¯ u, dd ¯ , c¯ c, s¯ s and b ¯
b channels
(allowing 3 for colour and neglecting masses) is then 1538 MeV, producing an estimated total width of approximately 2.22 GeV. (QCD corrections
will increase these estimates by a factor of order 1.1). The branching ratio
to charged leptons is approximately 3.4%, to the three (invisible) neutrino
channels 20.5%, and to hadrons (via hadronization of the q¯ q channels) about
69.3%. In section 22.4.3 we shall see how a precise measurement of the total
Z
0 width at LEP determined the number of light neutrinos to be 3.
Cross sections for neutrino–lepton scattering proceeding via Z
0 exchange
can be calculated (for k
2
≪ M
2 ) using the currents (22.59) and (22.60), and
Z
the method of section 20.5. Examples are
−
ν μ e
−
→ ν μ e
(22.84)
and
−
ν ¯ μ e
−
→ ν ¯ μ e
(22.85)
as shown in figure 22.12. Since the neutral current for the electron is not pure
l
L |
2
expect to see terms involving both |c
|
2 , and possibly an interference term. The cross section for (22.84) is
found to be (’t Hooft 1971c)
V–A, as was the charged current, we
l
and |c R
1
∗
∗ l
R |
2 (1 − y)
2
−
R )ym e
(22.86)
where E is the energy of the incident neutrino in the ‘laboratory’ system, and
y = (E − E
′ )/E as before, where E
′ is the energy of the outgoing neutrino in
the ‘laboratory’ system
2 . Equation (22.86) may be compared with the ν μ e
−
→
μ
− ν e (charged current) cross section of (20.84) by noting that t = −2m e Ey:
dσ/dy = (2G
2
F Em e
l
/π)[|c L |
2
l
+ |c
l
l
c
c c
R L + L
l
(c
/E],
2
the |c L |
2 term agrees with the pure V–A result (20.84), while the |c R |
2 term
2 In the kinematics, lepton masses have been neglected wherever possible.
l
l
22.3. Simple (tree-level) predictions
FIGURE 22.12
Neutrino-electron graphs involving Z
0 exchange.
Quark pairs couple as in (22.71), the GIM mechanism ensuring that all flavourchanging terms cancel. The total width to u¯ u, dd ¯ , c¯ c, s¯ s and b ¯
b channels
(allowing 3 for colour and neglecting masses) is then 1538 MeV, producing an estimated total width of approximately 2.22 GeV. (QCD corrections
will increase these estimates by a factor of order 1.1). The branching ratio
to charged leptons is approximately 3.4%, to the three (invisible) neutrino
channels 20.5%, and to hadrons (via hadronization of the q¯ q channels) about
69.3%. In section 22.4.3 we shall see how a precise measurement of the total
Z
0 width at LEP determined the number of light neutrinos to be 3.
Cross sections for neutrino–lepton scattering proceeding via Z
0 exchange
can be calculated (for k
2
≪ M
2 ) using the currents (22.59) and (22.60), and
Z
the method of section 20.5. Examples are
−
ν μ e
−
→ ν μ e
(22.84)
and
−
ν ¯ μ e
−
→ ν ¯ μ e
(22.85)
as shown in figure 22.12. Since the neutral current for the electron is not pure
l
L |
2
expect to see terms involving both |c
|
2 , and possibly an interference term. The cross section for (22.84) is
found to be (’t Hooft 1971c)
V–A, as was the charged current, we
l
and |c R
1
∗
∗ l
R |
2 (1 − y)
2
−
R )ym e
(22.86)
where E is the energy of the incident neutrino in the ‘laboratory’ system, and
y = (E − E
′ )/E as before, where E
′ is the energy of the outgoing neutrino in
the ‘laboratory’ system
2 . Equation (22.86) may be compared with the ν μ e
−
→
μ
− ν e (charged current) cross section of (20.84) by noting that t = −2m e Ey:
dσ/dy = (2G
2
F Em e
l
/π)[|c L |
2
l
+ |c
l
l
c
c c
R L + L
l
(c
/E],
2
the |c L |
2 term agrees with the pure V–A result (20.84), while the |c R |
2 term
2 In the kinematics, lepton masses have been neglected wherever possible.
l