30
G. Altarelli and S. Forte
value. If we leave A μ as it is and simply replace the linearized expression for φ,
we obtain the following quadratic terms (those important for propagators):
L quad = −
1
4
A
F
A
μν F
Aμν
+
1
2
M
2 A μ A
μ
+
+
1
2
(∂ μ ζ )
2
+ MA μ ∂
μ ζ +
1
2
(∂ μ h)
2
− h
2 μ
2
(2.64)
The mixing term between A μ and ∂ μ ζ does not allow to directly write diagonal mass
matrices. But this mixing term can be eliminated by an appropriate modification of
the covariant gauge fixing term given in Eq. (2.35) for the unbroken theory. We now
take:
GF = −
1
2ξ
(∂
μ A μ − ξMζ )
2 .
(2.65)
By adding GF to the quadratic terms in Eq. (2.64) the mixing term cancels (apart
from a total derivative that can be omitted) and we have:
L quad = −
1
4
A
F
A
μν F
Aμν
+
1
2
M
2 A μ A
μ
−
1
2ξ
(∂
μ A μ )
2
+
+
1
2
(∂ μ ζ )
2
−
ξ
2
M
2 ζ
2
+
1
2
(∂ μ h)
2
− h
2 μ
2
(2.66)
We see that the ζ field appears with a mass
√
ξ M and its propagator is:
iD ζ =
i
k 2 − ξM 2 + ii
.
(2.67)
The propagators of the Higgs field h and of gauge field A μ are:
iD h =
i
k 2 − 2μ 2 + ii
,
(2.68)
iD μν (k) =
−i
k 2 − M 2 + ii
(g μν − (1 − ξ)
k μ k ν
k 2 − ξM 2 ) .
(2.69)
As anticipated, all propagators have a good behaviour at large k 2 . This class of
gauges are called “R ξ gauges” [20]. Note that for ξ = 1 we have a sort of
generalization of the Feynman gauge with a Goldstone of mass M and a gauge
propagator:
iD μν (k) =
−ig μν
k 2 − M 2 + ii
.
(2.70)