Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
GSberveglieri`Phi4tools`
SunsetSubdiagram  | |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
Needs["GSberveglieri`Phi4tools`"]
Let's look at the sunset diagram, i.e. diagram for for the theory with =2 quartic vertices, let's visualize it and print its integrand:
(2)
Γ
4
ϕ
v4
In[2]:=
[2,0,2,1],
[2,0,2,1,"ExternalMomentum"True]
 | VisualizeDiagram |
| IntegrandDiagram |
Out[2]=
,[[1]][[1]-[1]-[2]][[2]]
1
6
The integrand in is divergent. Let's write it in the case of zero external momentum
d=3
In[3]:=
 | WriteExplicit |
| IntegrandDiagram |
Out[3]=
1
61+1+1++2Cos[]+
2
[1]
ρ
2
[2]
ρ
2
[1]
ρ
[2]
θ
[1]
ρ
[2]
ρ
2
[2]
ρ
The divergence is cured by substituting the sunset subdiagrams with regularized by zero-momentum subtraction.
[q]
In[4]:=
[2,0,2,1,"Substitutions""Analytics"],
[2,0,2,1,"ExternalMomentum"True,"Substitutions""Analytics"]
| VisualizeDiagram |
| IntegrandDiagram |
Out[4]=
,
[[1]]
6
In[5]:=
| WriteExplicit |
[[1]]
6
Out[5]=
-
-2++Log1+
6ArcTan
2
[1]
ρ
3
2
[1]
ρ
2
[1]
ρ
9
192
2
π
In[6]:=
| WriteExplicit |
[0]
6
Out[6]=
0
The substitution not only allows us to cure these divergences but also reduces the number of effective loops. Every sunset subdiagram substitution reduces the number of loop of the diagram by two. Let's look at another example: the diagram for for the theory with vertices =6 quartic vertices, without and with the sunset subdiagram substitutions.
th
69
(2)
Γ
4
ϕ
v4
In[7]:=
[2,0,6,69],
[2,0,6,69,"Substitutions""Analytics"]
| VisualizeDiagram |
| VisualizeDiagram |
Out[7]=
,
Thus we are left with a diagram with 2 loops instead of 6.
In[8]:=
| IntegrandDiagram |
Out[8]=
1
72
2
[[1]]
2
[[2]]
In[9]:=
| WriteExplicit |
| IntegrandDiagram |
Out[9]=
-2+Log9++-2+Log9++737281++2Cos[]+
1
9
2
[1]
ρ
6ArcTan
[1]
ρ
3
[1]
ρ
1
9
2
[2]
ρ
6ArcTan
[2]
ρ
3
[2]
ρ
4
π
2
1+
2
[1]
ρ
2
1+
2
[2]
ρ
2
[1]
ρ
[2]
θ
[1]
ρ
[2]
ρ
2
[2]
ρ
In[10]:=
| MomVars |
 | WriteExplicit |
 | IntegrandDiagram |
Out[10]=
{,,}
[1]
ρ
[2]
θ
[2]
ρ
 |
|
 |
|
""