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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
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
O(Ν)
and cubic models and the numerical values of the series for the 0, 2 and 4-point functions at zero external momentum in three dimensions.
We consider the models defined by the Hamiltonian
for the
Ν
-component field
ϕ
i
. For
d<4
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
​​
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
d=3
). 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
ϕ
i
or the coupling constants
λ
ijkl
. The
O(Ν)
model is obtained for
λ
ijkl
=λ
δ
ij
δ
kl
, while the model with cubic anisotropy is obtained for
λ
ijkl
=
δ
ij
δ
kl
v
δ
jk
+u
. The vacuum-energy renormalization constant
ρ
0
is chosen such that it completely cancels all the contributions up to four loops.
More details on the renormalization scheme together with the dimensionally regularized expression for the divergent diagrams in
d=3
can be found in
JHEP 02 (2021) 098
.
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
Propagator
(

) — scalar propagator in Fourier space
Momentum
 ▪
ExternalMomentum
SunsetSubdiagram
(

) — regularized two-loop sunset subdiagram
BubbleSubdiagram
(
ℬ
) — one-loop subdiagram with two vertices
TriangleSubdiagram
(

) — one-loop triangle subdiagram with three vertices
SquareSubdiagram
(

) — one-loop square subdiagram with four vertices
TadSunsetSubdiagram
(

) — tadpole-like subdiagram containing the regularized sunset
TadTriangleBubblesSubdiagram
(
ℬ
) — tadpole-like subdiagram containing a triangle and two bubbles
Symmetry Factors
SymmetryFactorDiagram
— tensorial symmetry factors for the
O(Ν)
and cubic models
NComponents
— number of field components Ν
XCubicRatio
— ratio between the cubic and
O(Ν)
-symmetric coupling constants
Feynman Diagrams
VisualizeDiagram
 ▪
NickelIndex
 ▪
DrawGraph
 ▪
IntegrandDiagram
 ▪
CountLoops
 ▪
MomVars
Lists and tools specific to the three-dimensional case
InformationDiagram
 ▪
ValueDiagram
 ▪
WriteExplicit
 ▪
DeriveAndWriteExplicit
TechNotes
▪
Feynman Diagram Evaluation
▪
Perturbative Series Generation
RelatedLinks
[1]
Giacomo Sberveglieri, Marco Serone, and Gabriele Spada, “Self-Dualities and Renormalization Dependence of the Phase Diagram in 3d O(N ) Vector Models,” JHEP 02, 098 (2021)
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