14.1 Amplitude Factorization
291
A g 1 ,n 1 and A g 2 ,n 2 contain a delta function for the momenta:
A g 1 ,n 1 ∼ δ
(D)
k
(1)
1 + · · · + k
(1)
n 1 −1 + k
, A g 2 ,n 2 ∼ δ
(D)
k
(2)
1 + · · · + k
(2)
n 2 −1 + k
.
(14.20)
As a consequence, the second momentum integral can be performed and yields a
delta function:
F g,n ∼ δ
(D)
k
(1)
1 + · · · + k
(1)
n 1 −1 + k
(2)
1 + · · · + k
(2)
n 2 −1
.
(14.21)
Hence, the momentum flowing in the internal line is fixed, and this ensures the
overall momentum conservation as expected.
• The ghost numbers of the states φ α and φ β are also fixed (in terms of the external
states). Indeed, because of the ghost number anomaly, the amplitudes on M g 1 ,n 1
and M g 2 ,n 2 are non-vanishing only if the ghost numbers of these states satisfy
N gh (φ α ) = 2n 1 −
n 1 −1
i=1
N gh
V
(1)
i
,
N gh (φ β ) = 2n 2 −
n 2 −1
j =1
N gh
V
(2)
j
.
(14.22)
The non-vanishing of F g,n also gives another relation:
N gh (φ α ) + N gh (φ β ) = 4.
(14.23)
In particular, if the external states are on-shell with N gh = 2, we find
N gh (φ α ) = N gh (φ β ) = 2.
(14.24)
As indicated in Chap. 10, such states are appropriate at the classical level since
they do not contain spacetime ghosts.
• The sum over α and β is over an infinite number of states and could diverge. In
fact, the sum can be made convergent by tuning the stub parameter (Sect. 14.2.4).
Properties of Feynman graphs and amplitudes in the momentum space will be
discussed further in Chap. 18.
14.1.2 Non-separating Case
Next, we consider the non-separating plumbing fixture (Sect. 12.3.2). The computations are almost identical to the separating case; thus, we outline only the general
steps.
Part of the moduli space M g,n is covered by #M g 1 ,n 1 , with g = g 1 + 1 and
n = n 1 − 2. The local coordinates are denoted as w i for i = 1, . . . , n 1 and the
291
A g 1 ,n 1 and A g 2 ,n 2 contain a delta function for the momenta:
A g 1 ,n 1 ∼ δ
(D)
k
(1)
1 + · · · + k
(1)
n 1 −1 + k
, A g 2 ,n 2 ∼ δ
(D)
k
(2)
1 + · · · + k
(2)
n 2 −1 + k
.
(14.20)
As a consequence, the second momentum integral can be performed and yields a
delta function:
F g,n ∼ δ
(D)
k
(1)
1 + · · · + k
(1)
n 1 −1 + k
(2)
1 + · · · + k
(2)
n 2 −1
.
(14.21)
Hence, the momentum flowing in the internal line is fixed, and this ensures the
overall momentum conservation as expected.
• The ghost numbers of the states φ α and φ β are also fixed (in terms of the external
states). Indeed, because of the ghost number anomaly, the amplitudes on M g 1 ,n 1
and M g 2 ,n 2 are non-vanishing only if the ghost numbers of these states satisfy
N gh (φ α ) = 2n 1 −
n 1 −1
i=1
N gh
V
(1)
i
,
N gh (φ β ) = 2n 2 −
n 2 −1
j =1
N gh
V
(2)
j
.
(14.22)
The non-vanishing of F g,n also gives another relation:
N gh (φ α ) + N gh (φ β ) = 4.
(14.23)
In particular, if the external states are on-shell with N gh = 2, we find
N gh (φ α ) = N gh (φ β ) = 2.
(14.24)
As indicated in Chap. 10, such states are appropriate at the classical level since
they do not contain spacetime ghosts.
• The sum over α and β is over an infinite number of states and could diverge. In
fact, the sum can be made convergent by tuning the stub parameter (Sect. 14.2.4).
Properties of Feynman graphs and amplitudes in the momentum space will be
discussed further in Chap. 18.
14.1.2 Non-separating Case
Next, we consider the non-separating plumbing fixture (Sect. 12.3.2). The computations are almost identical to the separating case; thus, we outline only the general
steps.
Part of the moduli space M g,n is covered by #M g 1 ,n 1 , with g = g 1 + 1 and
n = n 1 − 2. The local coordinates are denoted as w i for i = 1, . . . , n 1 and the