Wolfram Language Paclet Repository

Community-contributed installable additions to the Wolfram Language

Primary Navigation

    • Cloud & Deployment
    • Core Language & Structure
    • Data Manipulation & Analysis
    • Engineering Data & Computation
    • External Interfaces & Connections
    • Financial Data & Computation
    • Geographic Data & Computation
    • Geometry
    • Graphs & Networks
    • Higher Mathematical Computation
    • Images
    • Knowledge Representation & Natural Language
    • Machine Learning
    • Notebook Documents & Presentation
    • Scientific and Medical Data & Computation
    • Social, Cultural & Linguistic Data
    • Strings & Text
    • Symbolic & Numeric Computation
    • System Operation & Setup
    • Time-Related Computation
    • User Interface Construction
    • Visualization & Graphics
    • Random Paclet
    • Alphabetical List
  • Using Paclets
    • Get Started
    • Download Definition Notebook
  • Learn More about Wolfram Language

Phi4tools

Guides

  • Phi4tools

Tech Notes

  • Feynman Diagram Evaluation
  • Perturbative Series Generation

Symbols

  • BubbleSubdiagram
  • CountLoops
  • DeriveAndWriteExplicit
  • DrawGraph
  • ExternalMomentum
  • InformationDiagram
  • IntegrandDiagram
  • Momentum
  • MomVars
  • NComponents
  • NickelIndex
  • Propagator
  • SquareSubdiagram
  • SunsetSubdiagram
  • SymmetryFactorDiagram
  • TadSunsetSubdiagram
  • TadTriangleBubblesSubdiagram
  • TriangleSubdiagram
  • ValueDiagram
  • VisualizeDiagram
  • WriteExplicit
  • XCubicRatio
Perturbative Series Generation
​
Writing the series for the 1PI functions
References
Writing the β-function in the Parisi scheme
​
In this Tech Note it is presented how to compute some perturbative series for the
O(Ν)
model in three dimensions. In the first part, we focus on the 1PI functions
(2)
Γ
,
(4)
Γ
, and the derivative of the 2-point function with respect to the external momentum
(2)
Γ
'
. Starting from them, the second part presents how to write the β-function for the
O(Ν)
model in three dimensions.
Writing the series for the 1PI functions

The series for
(2)
Γ

Combining the values of the integrals
ValueDiagram
with the
O(Ν)
symmetry factors
SymmetryFactorDiagram
at each order, one writes the coefficients
c2
of the series defined as
(2)
Γ
=
∑
i
c2[i,Ν]
i
g
, where we have denoted with
g
the dimensionless coupling constant
g=
λ
m
. In what follows we use a unit mass scale
m=1
.
In[5]:=
Needs["GSberveglieri`Phi4tools`"]
In[3]:=
c2Nickel[o_,n_?NumberQ]:=Total
ValueDiagram
[2,0,o]
SymmetryFactorDiagram
[2,0,o,"Tensor""O(N)"]/.Νn
It is important to remember that the values follow Nickel normalization (see
ValueDiagram
reference page for additional information) that is order dependent. We then have to multiply them by the factor
o-1
(-1)
-o
(16π)
o
(4!)
, with
o
is the order of the coefficient, to derive the series for the Hamiltonian specified in
Phi4tools
, i.e. using the convention where the quartic vertex is normalized as
2
λϕ
i
2
ϕ
j
. Thus the coefficients
c2[o,n]
are
In[4]:=
c2[o_,n_?NumberQ]:=
o-1
(-1)
o
(4!)
o
(16π)
Total
ValueDiagram
[2,0,o]
SymmetryFactorDiagram
[2,0,o,"Tensor""O(N)"]/.Νn
In[5]:=
c2[4,3]
Out[5]=
-2.94962317713466560096
±
1.7×
-19
10
We define the coefficient at order
0
and the tree level one (recalling that in our renormalization the tadpole is completely canceled by the counter-term)
In[6]:=
c2[o_/;o0,n_?NumberQ]:=1​​c2[o_/;o1,n_?NumberQ]:=0
The series for
(2)
Γ
up to the order
o
and for the model
O(n)
will be
In[8]:=
gamma2series[o_,n_]:=Sum[
i
g
c2[i,n],{i,0,o}]+
o+1
O[g]
In[9]:=
gamma2series[5,1]
Out[9]=
1+(0.81068399736189700137635395177671
±
1.1×
-31
10
)
3
g
+(-1.11245585960605060366
±
7.×
-20
10
)
4
g
+(1.8047374
(
96
±
15
)
)
5
g
+
6
O[g]
In[10]:=
gamma2series[7,1]
Out[10]=
1+(0.81068399736189700137635395177671
±
1.1×
-31
10
)
3
g
+(-1.11245585960605060366
±
7.×
-20
10
)
4
g
+(1.8047374
(
96
±
15
)
)
5
g
+(-3.3715868
(
8
±
7
)
)
6
g
+(7.142756
(
3
±
6
)
)
7
g
+
8
O[g]

The series for
(2)
Γ
'

With the option
"Derivative"
True
in the function
ValueDiagram
it is possible to write the series for
(2)
Γ
'
. Combining the values of the integrals with the
O(Ν)
symmetry factors
SymmetryFactorDiagram
at each order, one writes the coefficients
c2d
of the series defined as
(2)'
Γ
=
∑
i
c2d[i,Ν]
i
g
.
In[11]:=
c2dNickel[o_,n_?NumberQ]:=Total
ValueDiagram
[2,0,o,"Derivative"True]
SymmetryFactorDiagram
[2,0,o,"Tensor""O(N)"]/.Νn
Once again, we have to normalize them to pass from the Nickel normalization to a consistent one. We do it as for
(2)
Γ
.
In[12]:=
c2d[o_,n_?NumberQ]:=
o-1
(-1)
o
(4!)
o
(16π)
Total
ValueDiagram
[2,0,o,"Derivative"True]
SymmetryFactorDiagram
[2,0,o,"Tensor""O(N)"]/.Νn
We define the coefficient at order 0 and the tree level one
In[13]:=
c2d[o_/;o0,n_?NumberQ]:=1​​c2d[o_/;o1,n_?NumberQ]:=0
The series for
(2)
Γ
'
up to the order
o
and for the model
O(n)
will be
In[15]:=
gamma2dseries[o_,n_]:=Sum[
i
g
c2d[i,n],{i,0,o}]+
o+1
O[g]
In[16]:=
gamma2dseries[3,1]
Out[16]=
1+
2
g
9
2
π
+
8+3
2
π
+64Log[2]-2Log[3](16+9Log[3])-72PolyLog2,
1
3

3
g
2
3
π
+
4
O[g]

The series for
(4)
Γ

Also for the series for
(4)
Γ
=
∑
i
c4[i,Ν]
i
g
can be written in a similar way. Combining the values of the integrals
ValueDiagram
with the
O(Ν)
symmetry factors
SymmetryFactorDiagram
at each order we have
In[17]:=
c4Nickel[o_,n_?NumberQ]:=Total
ValueDiagram
[4,0,o]Total/@
SymmetryFactorDiagram
[4,0,o,"Tensor""O(N)"]/.Νn
Once again, we have to normalize them to pass from the Nickel normalization to a consistent one. We do it as for
(2)
Γ
. This time we have to multiply the coefficients in the Nickel normalization by a factor
3
o-1
(-1)
-(o-1)
(16π)
o
(4!)
where
o
is the order of the coefficient. The different exponent is due to the fact that for the same order, the diagrams have a number of loops lower by 1, and the factor 3 is due to the presence of three possible channels for each diagram. Because of the presence of 3 different channels, the output of
SymmetryFactorDiagram
consists of a list of three values for each diagram, we simply added them (using
Total
/@
), effectively symmetrizing the factor.Thus the coefficients c4[o,n] are
In[18]:=
c4[o_,n_?NumberQ]:=
o-1
(-1)
3
o
(4!)
o-1
(16π)
Total
ValueDiagram
[4,0,o]Total/@
SymmetryFactorDiagram
[4,0,o,"Tensor""O(N)"]/.Νn
In[19]:=
c4[4,2]
Out[19]=
-161.79452882132689254666399697832388427277418275054106
±
9.×
-50
10
We define the coefficient at the tree level
In[20]:=
c4[o_/;o1,n_?NumberQ]:=24
The series for
(4)
Γ
up to the order o and for the model O(n) will be
In[21]:=
gamma4series[o_,n_]:=Sum[
i
g
c4[i,n],{i,1,o}]+
o+1
O[g]
In[22]:=
gamma4series[6,1]
Out[22]=
24g-
108
2
g
π
+
594
3
g
2
π
+(-120.47872885539742205488409717577414271830651465853232
±
8.×
-50
10
)
4
g
+(281.42645687790
(
5
±
7
)
)
5
g
+(-714.97104
(
23
±
33
)
)
6
g
+
7
O[g]
Writing the β-function in the Parisi scheme
In this section, we show how starting from the series of the 1PI function, written in the last section, it is possible to write the series for the β-function, reproducing and extending the series of refs. [1,2].
First of all write a function that gives as output the coefficients of the β-function in terms of those of
(2)
Γ
,
(2)'
Γ
and
(4)
Γ
, named
a2[o]
,
a2d[o]
and
a4[o]
respectively at the order
o
.
In[23]:=
betaSymbolic[o_]:=Module{order=o,Γ2ns,Γ2,dΓ2,Γ4,m,g,Z,Zg,gofgP,ZP,ZgP,βseries},​​​​Γ2=(
2
m
(1+Sum[a2[n]
n
g
,{n,2,order}]))+
order+1
O[g]
;​​dΓ2=(1+Sum[a2d[n]
n
g
,{n,2,order}])+
order+1
O[g]
;​​Γ4=mg(1+Sum[a4[n+1]
n
g
,{n,1,order}])+
order+1
O[g]
;​​​​Z=
1
dΓ2
//Simplify;​​Zg=
mg
Γ4
//Simplify;​​​​gofgP=InverseSeriesSimplify
Γ4
3
dΓ2
*Γ2
,m>0,gP//Simplify;​​​​ZP=Z/.ggofgP//Simplify;​​ZgP=Zg/.ggofgP//Simplify;​​βseries=-
-1
∂
gP
Log
gP*ZgP
2
ZP

​​
In[24]:=
betaSymbolic[4]
Out[24]=
-gP-a4[2]
2
gP
+(4a2d[2]+2
2
a4[2]
-2a4[3])
3
gP
+6a2d[3]-
3
2
a2[2]a4[2]-
17
2
a2d[2]a4[2]-5
3
a4[2]
+8a4[2]a4[3]-3a4[4]
4
gP
+
5
O[gP]
Then we can substitute the values computed in the last section starting from the functions
ValueDiagram
and
SymmetryFactorDiagram
of the paclet.
In[25]:=
subCoefficients[n_]:=a2[o_]
c2[o,n]
o
(4!)
,a2d[o_]
c2d[o,n]
o
(4!)
,a4[o_]
c4[o,n]
o
(4!)

Finally, we normalize the coefficients following the convention introduced by Nickel,
ℊ=
Ν+8
48π
gP
.
In[26]:=
βfunction[o_,n_]:=Normal
8+n
48π
betaSymbolic[o]/.subCoefficients[n]/.gP
48π
(8+n)
ℊ+
o+1
O[ℊ]
The output of the function
βfunction[o,N]
is the perturbative series for the β-function in three dimensions for the
O(Ν)
up to order
o
.
In[27]:=
βfunction[8,3]
References

© 2025 Wolfram. All rights reserved.

  • Legal & Privacy Policy
  • Contact Us
  • WolframAlpha.com
  • WolframCloud.com