296
14 Amplitude Factorization and Feynman Diagrams
will be exemplified in Chap. 15. Before describing (and generalizing) the vertices,
we describe first the properties of the propagator.
14.2.2 Propagator
The propagator has been defined in (14.18):
=
1
2π i
dq
q
∧
d ¯
q
¯
q
b 0 ¯
b 0 q
L 0 ¯
q
¯
L 0 .
(14.36)
The plumbing modulus q is parametrized by (12.31)
q = e
−s+iθ ,
s ∈ R + ,
θ ∈ [0, 2π),
(14.37)
such that the integration measure becomes
dq
q
∧
d ¯
q
¯
q
= −2i ds ∧ dθ.
(14.38)
Using the variables L
±
0 = L 0 ± ¯
L 0 and b
±
0 = b 0 ± ¯
b 0 , the propagator can be
recast as
=
1
2π
b
+
0 b
−
0
∞
0
ds e
−sL
+
0
2π
0
dθ e
iθL
−
0 .
(14.39)
The form of the first integral is recognized as the Schwinger parametrization of
the propagator, while the second is the Fourier transformation of the discrete delta
function:
∞
0
ds e
−sL
+
0 =
1
L
+
0
,
2π
0
dθ e
iθL
−
0 = 2π δ L
−
0 ,0 .
(14.40)
In fact, the first integral converges only if L
+
0 > 0. As argued in the introduction,
divergences for L
+
0 ≤ 0 are either non-physical or IR divergences that can be cured
by renormalization. For this reason, we take the RHS as a definition of the integral,
which would be the correct result if one starts with a field theory action instead of a
first-quantized formalism.
In this case, the propagator becomes
=
b
+
0
L
+
0
b
−
0 δ L
−
0 ,0 .
(14.41)