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13 Off-Shell Amplitudes
Example 13.1: On-Shell Amplitude A 0,4
The transition functions are given by (see Fig. 12.1):
C 1 : w 1 = z 1 − y 1 ,
C 3 : w 3 = z 2 − y 3 ,
C 5 : z 1 = z 2 ,
C 2 : w 2 = z 1 − y 2 ,
C 4 : w 4 = z 2 − y 4 .
(13.23)
Three of the parameters (y 1 , y 2 and y 3 ) are fixed, while the single complex
modulus of M 0,4 is taken to be y 4 . Since we are interested in the on-shell
amplitude, it is not necessary to introduce local coordinates and the associated
parameters.
A variation of the modulus
y 4 −→ y 4 + δy 4 ,
¯
y 4 −→ ¯
y 4 + δ ¯
y 4
(13.24)
is equivalent to a change in the transition function of C 4 . This translates in turn
into a transformation of z 2
z
2 = z 2 + δy 4 ,
¯
z
2 = ¯
z 2 + δ ¯
y 4 .
(13.25)
Then, the tangent vector V = ∂ y 4 is associated to the vector field
v = 1,
¯
v = 0,
(13.26)
with support on C 4 . For V = ∂ ¯
y 4 , one finds
v = 0,
¯
v = 1.
(13.27)
The Beltrami 1-forms for the unit vectors are
B(∂ y 4 ) =
C 4
dz 2 b(z 2 )(+1),
B(∂ ¯
y 4 ) =
C 4
d¯ z 2 ¯
b(¯ z 2 )(+1),
(13.28)
with both contours running anti-clockwise.
The components of the 2-form read
ω 2 (∂ y 4 , ∂ ¯
y 4 ) =
1
2π i
B(∂ y 4 )B(∂ ¯
y 4 )
4
i=1
V i
0,4
=
1
2π i
C 4
dz 2 b(z 2 )
C 4
d¯ z 2 ¯
b(¯ z 2 )
4
i=1
V i
0,4
.