70
13. Local Non-Abelian (Gauge) Symmetries
namely (from (13.44) and (12.48))
μ
μ
ˆ
ˆ
D = ∂
μ δ ij + g∈ kij W k ,
(13.141)
ij
in this case. The result (13.140) is derived in standard books of quantum
field theory, for example Cheng and Li (1984), Peskin and Schroeder (1995)
or Ryder (1996). We should add the caution that the form of the ghost
Lagrangian depends on the choice of the gauge-fixing term; there are gauges
in which the ghosts are absent. Feynman rules for non-Abelian gauge field
theories are given in Cheng and Li (1984), for example. We give the rules for
tree diagrams, for which there are no problems with ghosts, in appendix Q.
Problems
13.1 Verify that (13.34) reduces to (13.26) in the infinitesimal case.
13.2 Verify equation (13.45).
13.3 Using the expression for D
μ in (13.47), verify (13.48).
13.4 Verify the transformation law (13.51) of F
μν under local SU(2) transformations.
F
μν
13.5 Verify that F μν ·
is invariant under local SU(2) transformations.
13.6 Verify that the (infinitesimal) transformation law (13.56) for the SU(3)
gauge field A
μ is consistent with (13.55).
a
13.7 By considering the commutator of two D
μ ’s of the form (13.53), verify
(13.61).
13.8 Verify that (13.84) reduces to (13.86) (omitting the (2π)
4 δ
4 factors).
13.9 Verify that the replacement of ∈ 1 by k 1 in (13.93) leads to (13.94).
13.10 Verify that when ∈ 1 is replaced by k 1 in (13.95), the resulting amplitude
cancels the contribution (13.94), provided that δ = 1.
13.11 Show that P
μν of (13.122), with the ∈’s specified by the conditions
(13.123) and (13.124), is given by (13.125).
13. Local Non-Abelian (Gauge) Symmetries
namely (from (13.44) and (12.48))
μ
μ
ˆ
ˆ
D = ∂
μ δ ij + g∈ kij W k ,
(13.141)
ij
in this case. The result (13.140) is derived in standard books of quantum
field theory, for example Cheng and Li (1984), Peskin and Schroeder (1995)
or Ryder (1996). We should add the caution that the form of the ghost
Lagrangian depends on the choice of the gauge-fixing term; there are gauges
in which the ghosts are absent. Feynman rules for non-Abelian gauge field
theories are given in Cheng and Li (1984), for example. We give the rules for
tree diagrams, for which there are no problems with ghosts, in appendix Q.
Problems
13.1 Verify that (13.34) reduces to (13.26) in the infinitesimal case.
13.2 Verify equation (13.45).
13.3 Using the expression for D
μ in (13.47), verify (13.48).
13.4 Verify the transformation law (13.51) of F
μν under local SU(2) transformations.
F
μν
13.5 Verify that F μν ·
is invariant under local SU(2) transformations.
13.6 Verify that the (infinitesimal) transformation law (13.56) for the SU(3)
gauge field A
μ is consistent with (13.55).
a
13.7 By considering the commutator of two D
μ ’s of the form (13.53), verify
(13.61).
13.8 Verify that (13.84) reduces to (13.86) (omitting the (2π)
4 δ
4 factors).
13.9 Verify that the replacement of ∈ 1 by k 1 in (13.93) leads to (13.94).
13.10 Verify that when ∈ 1 is replaced by k 1 in (13.95), the resulting amplitude
cancels the contribution (13.94), provided that δ = 1.
13.11 Show that P
μν of (13.122), with the ∈’s specified by the conditions
(13.123) and (13.124), is given by (13.125).