returns in output the variables we have to integrate over to compute the value of this diagram. In this case we will have a six-dimensional integral to perform.
In[4]:=
MomVars
@integrand2031
Out[4]=
[1]
ρ
,
[2]
θ
,
[2]
ρ
,
[3]
θ
,
[3]
ρ
,
[3]
ϕ
Let's look at the case with the tadpole-like subdiagram substituted
In[5]:=
VisualizeDiagram
[2,0,3,1,"Substitutions""Analytics"],
IntegrandDiagram
[2,0,3,1,"Substitutions""Analytics"]
Out[5]=
,
12
In[6]:=
WriteExplicit
@
IntegrandDiagram
[2,0,3,1,"Substitutions""Analytics"]
Out[6]=
-
Log
4
3
1536
3
π
This is directly the value of this Feynman diagram in
d=3
.
The substitution of this tadpole-like subdiagram with sunset reduces the number of loop of the diagram by three. Let's look at an example: the
st
1
diagram for
(0)
Γ
for the
4
ϕ
theory with vertices
v4
=6 quartic vertices, without and with the subdiagrams substitutions.
In[7]:=
VisualizeDiagram
[0,0,6,1],
VisualizeDiagram
[0,0,6,1,"Substitutions""Analytics"]
Out[7]=
,
In[8]:=
IntegrandDiagram
[0,0,6,1]
IntegrandDiagram
[0,0,6,1,"Substitutions""Analytics"]
Out[8]=
1
576
[[1]][[2]]
2
[[3]]
[-[1]-[2]+[3]]
2
[[4]]
2
[[5]]
[[6]][-[5]-[6]-[7]][[7]]
Out[8]=
1
576
ℬ[0]
2
The 3 substitutions reduced the number of loops from 7 to 0, as we can see printed using the function