Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
GSberveglieri`Phi4tools`
BubbleSubdiagram |
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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 =1 quartic vertex, let's visualize it and print its integrand:
rd
3
(2)
Γ
4
ϕ
v3
v4
In[2]:=
[2,2,1,3],
[2,2,1,3]
 | VisualizeDiagram |
| IntegrandDiagram |
Out[2]=
,[[1]][-[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]:=
integrand2213=
@
[2,2,1,3]
 | WriteExplicit |
| IntegrandDiagram |
Out[3]=
1
1+1+-2Cos[]+
2
[1]
ρ
2
1+
2
[2]
ρ
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 bubble subdiagram substituted
In[5]:=
[2,2,1,3,"Substitutions""Analytics"],
[2,2,1,3,"Substitutions""Analytics"]
| VisualizeDiagram |
| IntegrandDiagram |
Out[5]=
,ℬ[[1]]
2
[[1]]
In[6]:=
integrand2213bubble=
@
[2,2,1,3,"Substitutions""Analytics"]
| WriteExplicit |
| IntegrandDiagram |
Out[6]=
ArcTan
[1]
ρ
2
4π
[1]
ρ
2
1+
2
[1]
ρ
The integral to perform is just one-dimensional. Notice how the writes as its expression in .
ℬ[q]
d=3
Every bubble 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 =5 quartic vertices, without and with the bubble subdiagram substitutions.
rd
3
(0)
Γ
4
ϕ
v4
In[7]:=
[0,0,5,3],
[0,0,5,3]
| VisualizeDiagram |
| IntegrandDiagram |
Out[7]=
,[[1]][[2]][-[1]-[2]-[3]][[3]][[2]+[3]-[4]][[4]][-[2]-[3]-[5]][[5]][-[2]-[3]-[6]][[6]]
1
320
In[8]:=
[0,0,5,3,"Substitutions""Analytics"],
[0,0,5,3,"Substitutions""Analytics"]
| VisualizeDiagram |
| IntegrandDiagram |
Out[8]=
,
5
ℬ[[1]]
320
The 5 substitutions reduced the number of loops from 6 to 1, as we can see printed using the function on the respective integrands
In[9]:=
| CountLoops |
 | IntegrandDiagram |
| IntegrandDiagram |
Out[9]=
{6,1}
 |
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 |
|
""