Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
GSberveglieri`Phi4tools`
SquareSubdiagram  | |
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Details and Options
Examples
(2)
Basic Examples
(2)
In[1]:=
Needs["GSberveglieri`Phi4tools`"]
Let's look at the diagram for for the theory with =4 quartic vertices, let's visualize it and print its integrand:
th
6
(4)
Γ
4
ϕ
v4
In[2]:=
[4,0,4,6],
[4,0,4,6]
 | VisualizeDiagram |
| IntegrandDiagram |
Out[2]=
,[[1]][[2]][[1]+[2]-[3]][-[1]-[2]+[3]]
1
4
2
[[3]]
The function prints the integrand in in terms of the three-dimensional components in spherical coordinates ready to be integrated.
d=3
In[3]:=
integrand4046=
@
[4,0,4,6]
 | WriteExplicit |
| IntegrandDiagram |
Out[3]=
141+1+
2
[1]
ρ
2
[2]
ρ
2
1+
2
[3]
ρ
2
1+++
2
(+Cos[]-Cos[])
[1]
ρ
[2]
θ
[2]
ρ
[3]
θ
[3]
ρ
2
Sin[]-CosSin[]
[2]
ρ
[2]
θ
[3]
ϕ
[3]
ρ
[3]
θ
2
[3]
ρ
2
Sin[]
[3]
θ
2
Sin
[3]
ϕ
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 six-dimensional integral to perform.
In[4]:=
 | MomVars |
Out[4]=
,,,,,
[1]
ρ
[2]
θ
[2]
ρ
[3]
θ
[3]
ρ
[3]
ϕ
Let's look at the case with the subdiagrams substituted
In[5]:=
[4,0,4,6,"Substitutions""Analytics"],
[4,0,4,6,"Substitutions""Analytics"]
| VisualizeDiagram |
| IntegrandDiagram |
Out[5]=
,ℬ[[1]][0,-[1],0,[1]]
1
4
In[6]:=
integrand4046bubblesquare=
@
[4,0,4,6,"Substitutions""Analytics"]
| WriteExplicit |
| IntegrandDiagram |
Out[6]=
ArcTan
[1]
ρ
2
64
2
π
[1]
ρ
2
4+
2
[1]
ρ
The integral to perform is just one-dimensional. Notice how the wrote as its expression in .
[0,-[1],0,[1]]
d=3
The function for the square subdiagram with all momenta different from zero is quite long and cumbersome in .
d=3
In[1]:=
Simplify@
[[[1],[2],[3],[4]]]
| WriteExplicit |
Out[1]=
Every square 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 subdiagrams substitutions.
th
8
(0)
Γ
4
ϕ
v4
In[2]:=
[0,0,6,8],
[0,0,6,8,"Substitutions""Analytics"]
| VisualizeDiagram |
| VisualizeDiagram |
Out[2]=
,
In[3]:=
| IntegrandDiagram |
| IntegrandDiagram |
Out[3]=
1
64
2
[[3]]
Out[3]=
1
64
2
ℬ[[1]]
2
ℬ[[2]]
The 5 substitutions, 4 bubble subdiagrams and 1 square one, reduced the number of loops from 7 to 2, as we can see printed using the function on the respective integrands
In[4]:=
| CountLoops |
 | IntegrandDiagram |
| IntegrandDiagram |
Out[4]=
{7,2}
 |
|
 |
|
""