306
14 Amplitude Factorization and Feynman Diagrams
The vertices satisfy the following identity for g ≥ 0 and n ≥ 1 [3, pp. 41–42]:
0 =
g 1 ,g 2 ≥0
g 1 +g 2 =g
n 1 ,n 2 ≥0
n 1 +n 2 =n
n!
n 1 ! n 2 !
V g 1 ,n 1 +1
n 1 , , g 2 ,n 2 ((
n 2 )
+ (−1)
|φ s |
V g−1,n+2
φ s , b
−
0 φ
c
s , ,
n
.
(14.70)
The last term is absent for g = 0. It is a consequence of the definition of the vertices
as the missing region from gluing lower-order vertices.
14.3.2 Feynman Graph Interpretation
The vertices must satisfy a certain number of conditions to be interpreted as
Feynman diagrams. The first is that they must be symmetric under permutations
of the states. Not every choice of local coordinates satisfies this requirement: this
can be solved by defining the vertex over a generalized section. In this case, the
vertex is defined as the average of the integrals over N sections S
(a)
g,n of P g,n :
V g,n (V 1 , . . . , V n ) =
1
N
N
a=1
R
(a)
g,n
ω
g,n
M g,n
(V 1 , . . . , V n ).
(14.71)
Example 14.2: 3-Point Vertex
The cubic vertex must be symmetric under permutations
V 0,3 (V 1 , V 2 , V 3 ) = V 0,3 (V 3 , V 1 , V 2 ) + · · ·
(14.72)
Taking the vertex to be given by a section S 0,3 with local coordinates f i
V 0,3 (V 1 , V 2 , V 3 ) = ω
0,3
0 (V 1 , V 2 , V 3 )| S 0,3 = =f 1 ◦ V 1 (0)f 2 ◦ V 2 (0)f 3 ◦ V 3 (0),
(14.73)
one finds that a permutation looks different
V 0,3 (V 3 , V 1 , V 2 ) = =f 1 ◦ V 3 (0)f 2 ◦ V 1 (0)f 3 ◦ V 2 (0) = V 0,3 (V 1 , V 2 , V 3 ),
(14.74)
unless the local coordinates satisfy special properties (remember that the local
coordinates are specified by the vertex state V and not by the external states V i ,
so a permutation of them does not permute the local maps). Obviously, both
amplitudes agree on-shell since the dependence in the local coordinates gets
cancelled (equivalently one can rotate the punctures using SL(2, C)).
14 Amplitude Factorization and Feynman Diagrams
The vertices satisfy the following identity for g ≥ 0 and n ≥ 1 [3, pp. 41–42]:
0 =
g 1 ,g 2 ≥0
g 1 +g 2 =g
n 1 ,n 2 ≥0
n 1 +n 2 =n
n!
n 1 ! n 2 !
V g 1 ,n 1 +1
n 1 , , g 2 ,n 2 ((
n 2 )
+ (−1)
|φ s |
V g−1,n+2
φ s , b
−
0 φ
c
s , ,
n
.
(14.70)
The last term is absent for g = 0. It is a consequence of the definition of the vertices
as the missing region from gluing lower-order vertices.
14.3.2 Feynman Graph Interpretation
The vertices must satisfy a certain number of conditions to be interpreted as
Feynman diagrams. The first is that they must be symmetric under permutations
of the states. Not every choice of local coordinates satisfies this requirement: this
can be solved by defining the vertex over a generalized section. In this case, the
vertex is defined as the average of the integrals over N sections S
(a)
g,n of P g,n :
V g,n (V 1 , . . . , V n ) =
1
N
N
a=1
R
(a)
g,n
ω
g,n
M g,n
(V 1 , . . . , V n ).
(14.71)
Example 14.2: 3-Point Vertex
The cubic vertex must be symmetric under permutations
V 0,3 (V 1 , V 2 , V 3 ) = V 0,3 (V 3 , V 1 , V 2 ) + · · ·
(14.72)
Taking the vertex to be given by a section S 0,3 with local coordinates f i
V 0,3 (V 1 , V 2 , V 3 ) = ω
0,3
0 (V 1 , V 2 , V 3 )| S 0,3 = =f 1 ◦ V 1 (0)f 2 ◦ V 2 (0)f 3 ◦ V 3 (0),
(14.73)
one finds that a permutation looks different
V 0,3 (V 3 , V 1 , V 2 ) = =f 1 ◦ V 3 (0)f 2 ◦ V 1 (0)f 3 ◦ V 2 (0) = V 0,3 (V 1 , V 2 , V 3 ),
(14.74)
unless the local coordinates satisfy special properties (remember that the local
coordinates are specified by the vertex state V and not by the external states V i ,
so a permutation of them does not permute the local maps). Obviously, both
amplitudes agree on-shell since the dependence in the local coordinates gets
cancelled (equivalently one can rotate the punctures using SL(2, C)).