Definition 5.2 Constructions B, ℓ, 2
i
and E. i Construction B. Given any diagram line
η, say
τx − v, and given y 6= x , we define Construction B
η spat
y to be the operation in which τx − v is replaced by
τy − v λǫD
⋆
τx − y =
x
v y
, 5.8
and define Construction B
η temp
y to be the operation in which τx − v is replaced by
X
b= · ,y
τb − v λǫDb Pb, b
+
−→ x =
v y
x
, 5.9
where {b −→ x } = {b is occupied} ∩ {b −→ x } and v
+
= v, t
v
+
ǫ for v = v, t
v
. Con- struction B
η
y applied to τx − v is the sum of τx − vδ
x ,y
and the results of Construc- tion B
η spat
y and Construction B
η temp
y applied to τx − v. Construction B
η
s is the opera- tion in which Construction B
η
y, s is performed and then followed by summation over y ∈ Z
d
. Constructions B
η spat
s and B
η temp
s are defined similarly. We omit the superscript η and write,
e.g., Construction By when we perform Construction B
η
y followed by a sum over all pos-
sible lines η. We denote the result of applying Construction By to a diagram function F x
by F x ; By, and define F x ; B
spat
y and F x ; B
temp
y similarly. For example, we denote
the result of applying Construction B
spat
y to the line ϕx by
ϕx ; B
spat
y ≡ p
⋆
τx ; B
spat
y = δ
o ,y
λǫD
⋆
τx + ϕy λǫD
⋆
τx − y,
5.10 where
δ
o ,y
λǫD
⋆
τx is the contribution in which p of ϕ is replaced by λǫD.
ii Construction ℓ. Given any diagram line η, Construction ℓ
η
y is the operation in which a line to y is inserted into the line
η. This means, for example, that the 2-point function τu − v
corresponding to the line η is replaced by
X
z
τu − v; B
η
z τy − z.
5.11 We omit the superscript
η and write Construction ℓy when we perform Construction ℓ
η
y
followed by a sum over all possible lines
η. We write F v, y; ℓz for the diagram where
Construction ℓz is performed on the diagram F v, y. Similarly, for ~y = y
1
, . . . , y
j
, Con- struction
ℓ~y is the repeated application of Construction ℓy
i
for i = 1, . . . , j. We note that the order of application of the different Construction
ℓy
i
is irrelevant.
842
v y
−→
2
1
y
z
y
z
v
+ 5 other possibilities −→
2
z
w
z y
w
v
+ 53 other possibilities
Figure 10: Construction E
y
w in 5.14 applied to F v , y = τy − v − δ
v ,y
. The 6 = 4 + 2 possibilities of the result of applying Construction 2
1
y
z are due to the fact that Ly, u; z for some u
consists of 2 terms, and that the result of Construction B
η
u consists of 3 = 2 + 1 terms, one of which is the trivial contribution: F v , y
δ
y ,u
. The number of admissible lines in the resulting diagram is 2 for this trivial contribution, otherwise 1. Therefore, the number of resulting terms at
the end is 54, which is the sum of 6 due to the identity in 5.13, 24 = 4 × 6, due to the non-
trivial contribution in the first stage followed by Construction 2
z
w and 24 = 2 × 2 × 6, due to
the trivial contribution having 2 admissible lines followed by Construction 2
z
w .
iii Constructions 2
i
and E. For a diagram F v , u with two vertices carrying labels v and u and
with a certain set of admissible lines, Constructions 2
1
u
w and 2
u
w produce the diagrams F v ,
〈u〉; 2
1
〈u〉
w ≡
X
u
F v , u; 2
1
u
w =
X
η
X
u ,z
F v , u; B
η
z Lu, z; w , 5.12
F v , 〈u〉; 2
〈u〉
w = F v , w + F v , 〈u〉; 2
1
〈u〉
w , 5.13
where 〈u〉 is a dummy variable for u that is summed over Λ therefore, e.g., Fv, 〈u〉; 2
〈u〉
w is independent of u and
P
η
is the sum over the set of admissible lines for F v , u. We call the L-admissible lines of the added factor Lu, z; w in 5.12 the 2
1
-admissible lines for
F v , 〈u〉; 2
1
〈u〉
w . Construction E
y
w is the successive applications of Constructions 2
1
y
z
and 2
z
w followed by the summation over z ∈ Λ; see Figure 10: F v ,
〈y〉; E
〈y〉
w = F v , 〈y〉; 2
1
〈y〉
〈u〉, 2
〈u〉
w ≡ F v, 〈y〉; 2
1
〈y〉
w +
X
η
X
u ,z
F v , 〈y〉; 2
1
〈y〉
u, B
η
z Lu, z; w , 5.14
where P
η
is the sum over the 2
1
-admissible lines for F v , 〈y〉; 2
1
〈y〉
u. We further define the
