Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Phi4tools
This paclet contains a collection of tools for the visualization, simplification and computation of Feynman diagrams in scalar field theories. It also provides the symmetry factors for the and cubic models and the numerical values of the series for the 0, 2 and 4-point functions in three dimensions.
O(Ν)
We consider the models defined by the Hamiltonian
for the -component field . For the theory is super-renormalizable and only a finite number of diagrams are divergent. Within this paclet we adopt the following renormalization scheme, aimed at making the fixed dimensional computations as easy as possible
Ν
ϕ
i
d<4
That is, we choose the mass counterterm to exactly cancel the tadpole diagram as well as the sunset diagram at zero external momentum (that is divergent in ). With this choice, a large number of diagrams, the ones involving tadpoles, are identically zero, greatly simplifying the computations. These diagrams are completely omitted by this paclet. We do not renormalize the fields or the coupling constants . The model is obtained for =λ, while the model with cubic anisotropy is obtained for =v+u. The vacuum-energy renormalization constant is chosen such that it completely cancels all the contributions up to four loops.
d=3
ϕ
i
λ
ijkl
O(Ν)
λ
ijkl
δ
ij
δ
kl
λ
ijkl
δ
ij
δ
kl
δ
jk
ρ
0
More details on the renormalization scheme together with the dimensionally regularized expression for the divergent diagrams in can be found in [1].
d=3
In addition, the paclet allows also the visualization, simplification and computation of Feynman diagrams involving cubic vertices of the type .
∑
ijk
η
ijk
ϕ
i
ϕ
j
ϕ
k
Diagram Anatomy
Symmetry Factors
Feynman Diagrams
Lists and tools specific to the three-dimensional case
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