14.2 Feynman Diagrams and Feynman Rules
303
Hence, we have
V 0,2 (V 1 , V 2 ) := =V 1 | c
+
0 c
−
0 L
+
0 δ L
−
0 ,0 |V 2 .
(14.57)
Note that V 0,2 is not the 2-point scattering amplitude.
We denote the tree-level 1-point and 0-point vertices as V 0,1 (V 1 ) and V 0,0 . The
first can be interpreted as a classical source in the action, while the second is a
classical vacuum energy. They are set to zero in most applications and can be safely
ignored. However, they appear when formulating the theory on a background that
does not solve the equation of motion [4].
Finally, the 1-loop vacuum energy V 1,0 can also be defined as the partition
function of the worldsheet CFT integrated over the torus modulus.
This allows to define the vertices V g,n for all g, n ∈ N. We define the sum of all
loop contributions for a fixed n as
V n (V 1 , . . . , V n ) :=
g≥0
( ¯
hg
2
s )
g
V g,n (V 1 , . . . , V n ).
(14.58)
14.2.4 Stubs
In Sect. 12.3.4, we have indicated that the plumbing fixture can be modified by
adding stubs or, equivalently, by rescaling the local coordinates. This amounts to
introducing a cut-off (12.53) on the variable s such that
q = e
−s+iθ ,
s ∈ [s 0 , ∞),
θ ∈ [0, 2π).
(14.59)
instead of (14.37). In this case, the s-integral in the propagator (14.36) is modified
to
∞
s 0
ds e
−sL
+
0 =
e
−s 0 L
+
0
L
+
0
.
(14.60)
This leads to a new expression for the propagator:
(s 0 ) = b
+
0
e
−s 0 L
+
0
L
+
0
b
−
0 δ L
−
0 ,0 .
(14.61)
In momentum space, this reads
αβ (k) :=
e
−
α s 0
2 (k 2 +m 2
α )
k 2 + m 2
α
M αβ (k).
(14.62)