# Wolfram Language Paclet Repository

Community-contributed installable additions to the Wolfram Language

Perturbative Series Generation | |

In this Tech Note it is presented how to compute some perturbative series for the model in three dimensions. In the first part, we focus on the 1PI functions , , and the derivative of the 2-point function with respect to the external momentum '. Starting from them, the second part presents how to write the β-function for the model in three dimensions.

O(Ν)

(2)

Γ

(4)

Γ

(2)

Γ

O(Ν)

Writing the series for the 1PI functions

The series for

(2)

Γ

Combining the values of the integrals with the symmetry factors at each order, one writes the coefficients of the series defined as =c2[i,Ν], where we have denoted with the dimensionless coupling constant . In what follows we use a unit mass scale .

O(Ν)

c2

(2)

Γ

∑

i

i

g

g

g=

λ

m

m=1

In[5]:=

Needs["GSberveglieri`Phi4tools`"]

In[3]:=

c2Nickel[o_,n_?NumberQ]:=Total

[2,0,o]

[2,0,o,"Tensor""O(N)"]/.Νn

ValueDiagram |

SymmetryFactorDiagram |

It is important to remember that the values follow Nickel normalization (see reference page for additional information) that is order dependent. We then have to multiply them by the factor , with is the order of the coefficient, to derive the series for the Hamiltonian specified in , i.e. using the convention where the quartic vertex is normalized as . Thus the coefficients are

o-1

(-1)

-o

(16π)

o

(4!)

o

2

λϕ

i

2

ϕ

j

c2[o,n]

In[4]:=

c2[o_,n_?NumberQ]:=Total

[2,0,o]

[2,0,o,"Tensor""O(N)"]/.Νn

o-1

(-1)

o

(4!)

o

(16π)

ValueDiagram |

SymmetryFactorDiagram |

In[5]:=

c2[4,3]

Out[5]=

-2.94962317713466560096

±

1.7×

-19

10

We define the coefficient at order and the tree level one (recalling that in our renormalization the tadpole is completely canceled by the counter-term)

0

In[6]:=

c2[o_/;o0,n_?NumberQ]:=1c2[o_/;o1,n_?NumberQ]:=0

The series for up to the order and for the model will be

(2)

Γ

o

O(n)

In[8]:=

gamma2series[o_,n_]:=Sum[c2[i,n],{i,0,o}]+

i

g

o+1

O[g]

In[9]:=

gamma2series[5,1]

Out[9]=

1+(0.81068399736189700137635395177671+(-1.11245585960605060366+(1.804737496)+

±

1.1×

)-31

10

3

g

±

7.×

)-20

10

4

g

(

±

15

)

5

g

6

O[g]

In[10]:=

gamma2series[7,1]

Out[10]=

1+(0.81068399736189700137635395177671+(-1.11245585960605060366+(1.804737496)+(-3.37158688)+(7.1427563)+

±

1.1×

)-31

10

3

g

±

7.×

)-20

10

4

g

(

±

15

)

5

g

(

±

7

)

6

g

(

±

6

)

7

g

8

O[g]

The series for '

(2)

Γ

With the option in the function it is possible to write the series for '. Combining the values of the integrals with the symmetry factors at each order, one writes the coefficients of the series defined as =c2d[i,Ν].

(2)

Γ

O(Ν)

c2d

(2)'

Γ

∑

i

i

g

In[11]:=

c2dNickel[o_,n_?NumberQ]:=Total

[2,0,o,"Derivative"True]

[2,0,o,"Tensor""O(N)"]/.Νn

ValueDiagram |

SymmetryFactorDiagram |

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]:=Total

[2,0,o,"Derivative"True]

[2,0,o,"Tensor""O(N)"]/.Νn

o-1

(-1)

o

(4!)

o

(16π)

ValueDiagram |

SymmetryFactorDiagram |

We define the coefficient at order 0 and the tree level one

In[13]:=

c2d[o_/;o0,n_?NumberQ]:=1c2d[o_/;o1,n_?NumberQ]:=0

The series for ' up to the order and for the model will be

(2)

Γ

o

O(n)

In[15]:=

gamma2dseries[o_,n_]:=Sum[c2d[i,n],{i,0,o}]+

i

g

o+1

O[g]

In[16]:=

gamma2dseries[3,1]

Out[16]=

1+++

2

g

9

2

π

8+3+64Log[2]-2Log[3](16+9Log[3])-72PolyLog2,

2

π

1

3

3

g

2

3

π

4

O[g]

The series for

(4)

Γ

Also for the series for =c4[i,Ν] can be written in a similar way. Combining the values of the integrals with the symmetry factors at each order we have

(4)

Γ

∑

i

i

g

O(Ν)

In[17]:=

c4Nickel[o_,n_?NumberQ]:=Total

[4,0,o]Total/@

[4,0,o,"Tensor""O(N)"]/.Νn

ValueDiagram |

SymmetryFactorDiagram |

Once again, we have to normalize them to pass from the Nickel normalization to a consistent one. We do it as for . This time we have to multiply the coefficients in the Nickel normalization by a factor where 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 consists of a list of three values for each diagram, we simply added them (using ), effectively symmetrizing the factor.Thus the coefficients c4[o,n] are

(2)

Γ

3

o-1

(-1)

-(o-1)

(16π)

o

(4!)

o

In[18]:=

c4[o_,n_?NumberQ]:=Total

[4,0,o]Total/@

[4,0,o,"Tensor""O(N)"]/.Νn

o-1

(-1)

3

o

(4!)

o-1

(16π)

ValueDiagram |

SymmetryFactorDiagram |

In[19]:=

c4[4,2]

Out[19]=

-161.79452882132689254666399697832388427277418275054106

±

9.×

-50

10

We define the coefficient at the tree level

In[20]:=

c4[o_/;o1,n_?NumberQ]:=24

The series for up to the order o and for the model O(n) will be

(4)

Γ

In[21]:=

gamma4series[o_,n_]:=Sum[c4[i,n],{i,1,o}]+

i

g

o+1

O[g]

In[22]:=

gamma4series[6,1]

Out[22]=

24g-++(-120.47872885539742205488409717577414271830651465853232+(281.426456877905)+(-714.9710423)+

108

2

g

π

594

3

g

2

π

±

8.×

)-50

10

4

g

(

±

7

)

5

g

(

±

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 , and , named , and respectively at the order .

(2)

Γ

(2)'

Γ

(4)

Γ

a2[o]

a2d[o]

a4[o]

o

In[23]:=

betaSymbolic[o_]:=Module{order=o,Γ2ns,Γ2,dΓ2,Γ4,m,g,Z,Zg,gofgP,ZP,ZgP,βseries},Γ2=((1+Sum[a2[n],{n,2,order}]))+;dΓ2=(1+Sum[a2d[n],{n,2,order}])+;Γ4=mg(1+Sum[a4[n+1],{n,1,order}])+;Z=//Simplify;Zg=//Simplify;gofgP=InverseSeriesSimplify*Γ2,m>0,gP//Simplify;ZP=Z/.ggofgP//Simplify;ZgP=Zg/.ggofgP//Simplify;βseries=-Log

2

m

n

g

order+1

O[g]

n

g

order+1

O[g]

n

g

order+1

O[g]

1

dΓ2

mg

Γ4

Γ4

3

dΓ2

-1

∂

gP

gP*ZgP

2

ZP

In[24]:=

betaSymbolic[4]

Out[24]=

-gP-a4[2]+(4a2d[2]+2-2a4[3])+6a2d[3]-a2[2]a4[2]-a2d[2]a4[2]-5+8a4[2]a4[3]-3a4[4]+

2

gP

2

a4[2]

3

gP

3

2

17

2

3

a4[2]

4

gP

5

O[gP]

Then we can substitute the values computed in the last section starting from the functions and of the paclet.

In[25]:=

subCoefficients[n_]:=a2[o_],a2d[o_],a4[o_]

c2[o,n]

o

(4!)

c2d[o,n]

o

(4!)

c4[o,n]

o

(4!)

Finally, we normalize the coefficients following the convention introduced by Nickel, .

ℊ=gP

Ν+8

48π

In[26]:=

βfunction[o_,n_]:=NormalbetaSymbolic[o]/.subCoefficients[n]/.gPℊ+

8+n

48π

48π

(8+n)

o+1

O[ℊ]

The output of the function is the perturbative series for the β-function in three dimensions for the up to order .

βfunction[o,N]

O(Ν)

o

In[27]:=

βfunction[8,3]

References