Deadline 23rd April 2013
This is the baisic outline of the problem, but I will upload the actual question sheet.
1) Solve the interaction Lagrangian to show the interactions between the Z boson with quarks.2) Derive expressions (in terms of the Weinberg angle) for the decay rate of Z to each type of quark and lepton.Evaluate these decay rates (in units of GeV; remember these formulae are quoted inh = c = 1 units). Add them up to show that the total decay rate of the Z (i.e. the widthof the Z resonance) is 2.5 GeV.
Specific requirements:Must have step by step workings and adhere to the form of the Lagrangian in the question. No index notation to be used.Second question requires that:CL for neutrinos = 1/2CL for electrons = -1/2 + Sin^2(Theta_w)CR for electrons = Sin^2(Theta_w)
PH 335 PARTICLE PHYSICS II Problem Sheet
2
Return by Wednesday, 24 April (assessed)
1. In the SU(2)L × U(1)Y electroweak theory, the couplings of the first generation of
quarks to the gauge bosons are described by the interaction Lagrangian:
Lint = (ūL d̄L)
(
gTaWa +
1
2
g′Y B
)(uL
dL
)
+
1
2
g′B ūRY uR +
1
2
g′B d̄RY dR
where the Ta matrices ar
e
T1 =
1
2
(
0 1
1 0
)
, T2 =
1
2
(
0 −i
i 0
)
. T3 =
1
2
(
1 0
0 −1
)
and the U(1)Y charges are: Y = 1/3 for uL,dL, Y = 4/3 for uR and Y = −2/3 for dR.
(Cabibbo mixing is neglected in this question.)
By considering the mixing between the SU(2)L and U(1)Y gauge bosons W
3 and B
to give the physical gauge bosons Z and γ, together with the fundamental interactions of
W3 and B with the quarks, show that the interaction of the Z boson with quarks is of the
form:
e
sin θW cos θW
Z
[
cL q̄LqL + cR q̄RqR
]
where
cL =
1
2
−
2
3
sin2 θW , for q = u,c,t
cL = −
1
2
+
1
3
sin2 θW , for q = d,s,b
cR = −
2
3
sin2 θW , for q = u,c,t
cR =
1
3
sin2 θW , for q = d,s,b
[15 marks]
2. The decay rate for the Z into a fermion-antifermion pair is
Γ[Z → f̄f] =
1
6
mZ
α
sin2 θW cos2 θW
(
(c
f
L)
2 + (c
f
R)
2
)
Using the results given in the lectures for the Z couplings cL, cR to leptons, together with
the results of question 1, derive expressions (in terms of the Weinberg angle) for the decay
rate of Z to each type of quark and lepton.
Evaluate these decay rates (in units of GeV; remember these formulae are quoted in
h̄ = c = 1 units). Add them up to show that the total decay rate of the Z (i.e. the width
of the Z resonance) is 2.5 GeV. (Use α = 1/128, which is the value of the fine structure
constant evaluated at the Z mass, and sin2 θW = 0.23.)
Notice that the precision measurement of the Z width therefore gives a limit on the
number of light generations of quarks and leptons.
[10 marks]
