Particle Physics, Interaction Lagrangian Su(2) X U(1), Electroweak theory.

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]