Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
GSberveglieri`Phi4tools`
TriangleSubdiagram  | |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
Needs["GSberveglieri`Phi4tools`"]
Let's look at the diagram for for the theory with =2 cubic vertices and =2 quartic vertices, let's visualize it and print its integrand:
th
9
(4)
Γ
4
ϕ
v3
v4
In[2]:=
[4,2,2,9],
[4,2,2,9]
 | VisualizeDiagram |
| IntegrandDiagram |
Out[2]=
,2[[1]][[1]-[2]][-[1]+[2]]
2
[[2]]
The function prints the integrand in in terms of the three-dimensional components in spherical coordinates ready to be integrated.
d=3
In[3]:=
integrand4229=
@
[4,2,2,9]
 | WriteExplicit |
| IntegrandDiagram |
Out[3]=
2
1+
2
[1]
ρ
2
1+
2
[2]
ρ
2
1+-2Cos[]+
2
[1]
ρ
[2]
θ
[1]
ρ
[2]
ρ
2
[2]
ρ
The function returns in output the variables we have to integrate over to compute the value of this diagram. In this case we will have a three-dimensional integral to perform.
In[4]:=
 | MomVars |
Out[4]=
{,,}
[1]
ρ
[2]
θ
[2]
ρ
Let's look at the case with the triangle subdiagram substituted
In[5]:=
[4,2,2,9,"Substitutions""Analytics"],
[4,2,2,9,"Substitutions""Analytics"]
| VisualizeDiagram |
| IntegrandDiagram |
Out[5]=
,2[-[1],[1],0]
2
[[1]]
In[6]:=
integrand4229triangle=
@
[4,2,2,9,"Substitutions""Analytics"]
| WriteExplicit |
| IntegrandDiagram |
Out[6]=
1
4π4+
2
1+
2
[1]
ρ
2
[1]
ρ
The integral to perform is just one-dimensional. Notice how the wrote as its expression in .
[-[1],[1],0]
d=3
Every triangle subdiagram substitution reduces the number of loop of the diagram by one. Let's look at another example: the diagram for for the theory with vertices =6 quartic vertices, without and with the triangle subdiagram substitutions.
th
19
(0)
Γ
4
ϕ
v4
In[7]:=
[0,0,6,19],
[0,0,6,19,"Substitutions""Analytics"]
| VisualizeDiagram |
| VisualizeDiagram |
Out[7]=
,
In[8]:=
| IntegrandDiagram |
| IntegrandDiagram |
Out[8]=
1
48
Out[8]=
1
48
The 4 substitutions reduced the number of loops from 7 to 3, as we can see printed using the function on the respective integrands
In[9]:=
| CountLoops |
 | IntegrandDiagram |
| IntegrandDiagram |
Out[9]=
{7,3}
 |
|
 |
|
""