6Non-abelian gauge theory
III Advanced Quantum Field Theory
6.6 Feynman rules for Yang–Mills
In general, we cannot get rid of the ghost fields. However, we can use the
previous corollary to get rid of the h field. We add the BRST exact term
−i
ξ
2
Q(¯c
a
h
a
) =
ξ
2
h
a
h
a
d
d
x
to the Lagrangian, where
ξ
is some arbitrary constant. Then we can complete
the square, and obtain
ih
a
f
a
[A] +
ξ
2
h
a
h
a
=
ξ
2
h
a
+
i
ξ
f
a
[A]
2
+
1
2ξ
f
a
[A]f
a
[A].
Moreover, in the path integral, for each fixed A, we should have
Z
Dh exp
−
ξ
2
h
a
+
i
ξ
f
a
[A]
2
!
=
Z
Dh exp
−
ξ
2
(h
a
)
2
,
since we are just shifting all
h
by a constant. Thus, if we are not interested in
correlation functions involving
h
, then we can simply factor out and integrate
out
h
, and it no longer exists. Then the complete gauge-fixed Yang–Mills action
in Lorenz gauge with coupling to fermions ψ is then
S[A, ¯c, c,
¯
ψ, ψ]
=
Z
d
d
x
1
4
F
a
µν
F
µν,a
+
1
2ξ
(∂
µ
A
a
µ
)(∂
ν
A
a
ν
) − ¯c
a
∂
µ
∇
µ
c
a
+
¯
ψ(
/
∇ + m)ψ
,
where we absorbed the factor of g
Y M
into A.
Using this new action, we can now write down our Feynman rules. We have
three propagators
p
= D
ab
µν
(p) = −
δ
ab
p
2
δ
µν
− (1 − ξ)
p
µ
p
ν
p
2
p
= S(p) =
1
i
/
p + m
¯c
c
p
= C
ab
(p) =
δ
ab
p
2
We also have interaction vertices given by
A
a
µ
A
b
ν
A
c
λ
p
q
k
= g
Y M
f
abc
((k − p)
λ
δ
µν
+ (p − q)
µ
δ
νλ
+ (q − k)δ
µλ
)
A
a
µ
A
b
ν
A
c
λ
A
d
σ
=
−g
2
Y M
f
abe
f
cde
(δ
µλ
δ
νσ
− δ
µσ
δ
νλ
)
−g
2
Y M
f
ace
f
bde
(δ
µν
δ
σλ
− δ
µσ
δ
νλ
)
−g
2
Y M
f
ade
f
bce
(δ
µν
δ
σλ
− δ
µλ
δ
νσ
)
¯c
b
c
c
A
a
µ
(p)
p
= g
Y M
f
abc
p
µ
¯
ψ ψ
A
a
µ
(p)
= −g
Y M
γ
µ
t
a
f
.
What is the point of writing all this out? The point is not to use them.
The point is to realize these are horrible! It is a complete pain to work with
these Feynman rules. For example, a
gg → ggg
scattering process involves
∼
10000 terms when we expand it in terms of Feynman diagrams, at tree level!
Perturbation theory is not going to right way to think about Yang–Mills. And
GR is only worse.
Perhaps this is a sign that our theory is wrong. Surely a “correct” theory
must look nice. But when we try to do actual computations, despite these
horrific expressions, the end results tends to be very nice. So it’s not really
Yang–Mills’ fault. It’s just perturbation theory that is bad.
Indeed, there is no reason to expect perturbation theory to look good.
We formulated Yang–Mills in terms of a very nice geometric picture, about
principal
G
-bundles. From this perspective, everything is very natural and
simple. However, to do perturbation theory, we needed to pick a trivialization,
and then worked with this object
A
. The curvature
F
µν
was a natural geometric
object to study, but breaking it up into d
A
+
A ∧A
is not. The individual terms
that give rise to the interaction vertices have no geometric meaning — only
∇
and
F
do. The brutal butchering of the connection and curvature into these
non-gauge-invariant terms is bound to make our theory look messy.
Then what is a good way to do Yang–Mills? We don’t know. One rather
successful approach is to use lattice regularization, and use a computer to
actually compute partition functions and correlations directly. But this is very
computationally intensive, and it’s difficult to do complicated computations with
this. Some other people try to understand Yang–Mills in terms of string theory,
or twistor theory. But we don’t really know.
The only thing we can do now is to work with this mess.