Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
GSberveglieri`Phi4tools`
TadTriangleBubblesSubdiagram  | |
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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 =4 quartic vertices, let's visualize it and print its integrand:
nd
2
(2)
Γ
4
ϕ
v4
In[2]:=
[2,0,4,2],
[2,0,4,2]
 | VisualizeDiagram |
| IntegrandDiagram |
Out[2]=
,[[1]][[2]][[1]-[2]-[3]][[3]][-[1]+[2]+[3]][-[2]-[3]-[4]][[4]]
1
8
The function prints the integrand in in terms of the three-dimensional components in spherical coordinates ready to be integrated.
d=3
In[3]:=
integrand2042=
@
[2,0,4,2]
 | WriteExplicit |
| IntegrandDiagram |
Out[3]=
181+1+1+1+1+++
2
[1]
ρ
2
[2]
ρ
2
[3]
ρ
2
[4]
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
2
(Cos[]+Cos[]+Cos[])
[2]
θ
[2]
ρ
[3]
θ
[3]
ρ
[4]
θ
[4]
ρ
2
Sin[]+CosSin[]+CosSin[]
[2]
ρ
[2]
θ
[3]
ϕ
[3]
ρ
[3]
θ
[4]
ϕ
[4]
ρ
[4]
θ
2
Sin[]Sin+Sin[]Sin
[3]
ρ
[3]
θ
[3]
ϕ
[4]
ρ
[4]
θ
[4]
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 nine-dimensional integral to perform.
In[4]:=
 | MomVars |
Out[4]=
,,,,,,,,
[1]
ρ
[2]
θ
[2]
ρ
[3]
θ
[3]
ρ
[3]
ϕ
[4]
θ
[4]
ρ
[4]
ϕ
Let's look at the case with the tadpole-like subdiagram substituted
In[5]:=
[2,0,4,2,"Substitutions""Analytics"],
[2,0,4,2,"Substitutions""Analytics"]
| VisualizeDiagram |
| IntegrandDiagram |
Out[5]=
,
ℬ
8
In[6]:=
| WriteExplicit |
| IntegrandDiagram |
Out[6]=
1
98304
2
π
This is directly the value of this Feynman diagram in .
d=3
The substitution of this tadpole-like subdiagram with a triangle and two bubbles reduces the number of loop of the diagram by four. Let's look at an example: the diagram for for the theory with vertices =7 quartic vertices, without and with the subdiagrams substitutions.
nd
2
(0)
Γ
4
ϕ
v4
In[7]:=
[0,0,7,2],
[0,0,7,2,"Substitutions""Analytics"]
| VisualizeDiagram |
| VisualizeDiagram |
Out[7]=
,
In[8]:=
| IntegrandDiagram |
| IntegrandDiagram |
Out[8]=
1
192
2
[[3]]
2
[[4]]
2
[[5]]
Out[8]=
1
192
The 3 substitutions reduced the number of loops from 8 to 0, as we can see printed using the function on the respective integrands
In[9]:=
| CountLoops |
 | IntegrandDiagram |
| IntegrandDiagram |
Out[9]=
{8,0}
 |
|
 |
|
""