292
14 Amplitude Factorization and Feynman Diagrams
plumbing fixture reads
w n 1 −1 w n 1 = q.
(14.25)
The g-loop n-point amplitude with external states {V
(1)
1 , . . . , V
(1)
n 1 −2 } is denoted as
A g,n =
S g,n
ω
g,n
M g,n
V
(1)
1 , . . . , V
(1)
n 1 −2
.
(14.26)
When the n 1 − 2 punctures and g 1 = g − 1 holes move lose to each other, the
form can be written as
ω M g,n
V
(1)
1 , . . . , V
(1)
M g 1 ,n 1
, ∂ q , ∂ ¯
q
= (2π i)
−M c
g,n
M g 1 ,n 1
λ=1
B
V
(1)
λ
B(∂ q )B(∂ ¯
q )
n 1 −2
i=1
V
(1)
i
g,n
.
(14.27)
To proceed, one needs to introduce the surface state n 1 −1,n 1 :
n 1 −1,n 1 |V
(1)
n 1 −1 ⊗ V
(1)
n 1
:= ω M g 1 ,n 1 (V
(1)
1 , . . . , V
(1)
n 1
).
(14.28)
Following the same step as in the previous section leads to
F g,n
V
(1)
i |
=
d D k
(2π) D
d D k
(2π) D A g 1 ,n 1
V
(1)
1 , . . . , V
(1)
n 1 −2 , φ α (k), φ β (k
)
αβ (k, k
),
(14.29)
where the propagator is given in (14.18). This is equivalent to an amplitude
for which two external legs are glued together with a propagator, giving a loop
(Fig. 14.3).
Since both states φ α and φ β are inserted on the same surface, their ghost numbers
are not fixed, even if the external states are physical. The non-vanishing of F g,n only
leads to the constraint:
N gh (φ α ) + N gh (φ β ) = 2n 1 −
n 1 −2
i=1
N gh
V
(1)
i
= 4.
(14.30)
As a consequence, loop diagrams force to introduce states of every ghost number.
Internal states with N gh = 2 correspond to spacetime ghosts.