Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
GSberveglieri`Phi4tools`
ExternalMomentum  |
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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,"ExternalMomentum"True]
 | VisualizeDiagram |
| IntegrandDiagram |
Out[2]=
,[[1]][[1]-[2]][[2]][-[1]+[2]]
Here is the and and are the internal momenta of this two-loop diagram.
[1]
[1]
[2]
With the function we can write explicitly the integrand in in terms of the three dimensional components in spherical coordinates.
d=3
In[3]:=
 | WriteExplicit |
Out[3]=
11+1+1+-2Cos[]+1++-2Cos[]Cos[]+CosSin[]Sin[]
2
[1]
ρ
2
[2]
ρ
2
[1]
ρ
[2]
θ
[1]
ρ
[2]
ρ
2
[2]
ρ
2
[1]
ρ
2
[2]
ρ
[1]
ρ
[2]
ρ
[1]
θ
[2]
θ
[1]
ϕ
[1]
θ
[2]
θ
With the function we derive the integrand with respect to the external momentum squared and write the result at in in terms of the three-dimensional components in spherical coordinates.
2
[1]
[1]=0
d=3
In[4]:=
Simplify@
[[[1]][[1]-[2]][[2]][-[1]+[2]]]
 | DeriveAndWriteExplicit |
Out[4]=
-3+
2
[2]
ρ
31+1+-2Cos[]+
2
[1]
ρ
4
1+
2
[2]
ρ
2
[1]
ρ
[2]
θ
[1]
ρ
[2]
ρ
2
[2]
ρ
With the analytics substitutions in place we have
In[5]:=
[2,2,1,3,"Substitutions""Analytics"],
[2,2,1,3,"ExternalMomentum"True,"Substitutions""Analytics"]
| VisualizeDiagram |
| IntegrandDiagram |
Out[5]=
,ℬ[[1]][[1]-[1]][[1]]
In[6]:=
Simplify@
[ℬ[[1]][[1]-[1]][[1]]]
| DeriveAndWriteExplicit |
Out[6]=
ArcTan-3+
[1]
ρ
2
2
[1]
ρ
12π
[1]
ρ
4
1+
2
[1]
ρ
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""