2 Gauge Theories and the Standard Model
23
timelike one has a Coulomb or temporal gauge). The gauge fixing term is of the
form:
L GF = −
1
2λ
A
|n
μ V
A
μ |
2
(2.40)
With a procedure that can be found in QED textbooks [14] the corresponding
propagator, in Fourier space, is found to be:
D
AB
μν (q) =
i
q 2 + ii
[−g μν +
n μ q + n ν q μ
(nq)
−
n 2 q μ q ν
(nq) 2 ] δ
AB
(2.41)
In this case there are no ghost interactions because n μ V
A
μ , obtained by a gauge
transformation from n μ V A
μ , contains no gauge fields, once the gauge condition
n μ V A
μ = 0 has been taken into account. Thus the ghosts are decoupled and can
be ignored.
The introduction of a suitable regularization method that preserves gauge
invariance is essential for the definition and the calculation of loop diagrams and for
the renormalization programme of the theory. The method that is by now currently
adopted is dimensional regularization [15] which consists in the formulation of the
theory in n dimensions. All loop integrals have an analytic expression that is actually
valid also for non integer values of n. Writing the results for n = 4 − the loops
are ultraviolet finite for > 0 and the divergences reappear in the form of poles at
= 0.
2.7 Spontaneous Symmetry Breaking in Gauge Theories
The gauge symmetry of the SM was difficult to discover because it is well hidden
in nature. The only observed gauge boson that is massless is the photon. The gluons
are presumed massless but cannot be directly observed because of confinement, and
the W and Z weak bosons carry a heavy mass. Indeed a major difficulty in unifying
the weak and electromagnetic interactions was the fact that e.m. interactions have
infinite range (m γ = 0), whilst the weak forces have a very short range, owing to
m W,Z = 0.
The solution of this problem is in the concept of spontaneous symmetry breaking,
which was borrowed from statistical mechanics.
Consider a ferromagnet at zero magnetic field in the Landau–Ginzburg approximation. The free energy in terms of the temperature T and the magnetization M can
be written as
F (M, T ) F 0 (T ) + 1/2 μ
2 (T )M
2
+ 1/4 λ(T )(M
2 )
2
+ . . . .
(2.42)
23
timelike one has a Coulomb or temporal gauge). The gauge fixing term is of the
form:
L GF = −
1
2λ
A
|n
μ V
A
μ |
2
(2.40)
With a procedure that can be found in QED textbooks [14] the corresponding
propagator, in Fourier space, is found to be:
D
AB
μν (q) =
i
q 2 + ii
[−g μν +
n μ q + n ν q μ
(nq)
−
n 2 q μ q ν
(nq) 2 ] δ
AB
(2.41)
In this case there are no ghost interactions because n μ V
A
μ , obtained by a gauge
transformation from n μ V A
μ , contains no gauge fields, once the gauge condition
n μ V A
μ = 0 has been taken into account. Thus the ghosts are decoupled and can
be ignored.
The introduction of a suitable regularization method that preserves gauge
invariance is essential for the definition and the calculation of loop diagrams and for
the renormalization programme of the theory. The method that is by now currently
adopted is dimensional regularization [15] which consists in the formulation of the
theory in n dimensions. All loop integrals have an analytic expression that is actually
valid also for non integer values of n. Writing the results for n = 4 − the loops
are ultraviolet finite for > 0 and the divergences reappear in the form of poles at
= 0.
2.7 Spontaneous Symmetry Breaking in Gauge Theories
The gauge symmetry of the SM was difficult to discover because it is well hidden
in nature. The only observed gauge boson that is massless is the photon. The gluons
are presumed massless but cannot be directly observed because of confinement, and
the W and Z weak bosons carry a heavy mass. Indeed a major difficulty in unifying
the weak and electromagnetic interactions was the fact that e.m. interactions have
infinite range (m γ = 0), whilst the weak forces have a very short range, owing to
m W,Z = 0.
The solution of this problem is in the concept of spontaneous symmetry breaking,
which was borrowed from statistical mechanics.
Consider a ferromagnet at zero magnetic field in the Landau–Ginzburg approximation. The free energy in terms of the temperature T and the magnetization M can
be written as
F (M, T ) F 0 (T ) + 1/2 μ
2 (T )M
2
+ 1/4 λ(T )(M
2 )
2
+ . . . .
(2.42)