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

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 are

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]