Reflexive Fano Polytopes from Feynman integrals

The repository contains three files in Mathematica format containing the list of Feynman integrals that have either Fano or Reflexive polytopes for generic kinematics and internal lines as detailed in the paper Fano and Reflexive Polytopes from Feynman Integrals

• The file ReflexiveCases.m contains the data for the Feynman graphs having a reflexive polytope
• The file FanoCases.m contains the data for the Feynman graphs having a Fano (reflexive or non reflexive) polytopes
• The file FanoNotReflexiveCases.m contains the data for the Feynman graphs which have a Fano but not a reflexive polytope
• The Mathematica notebook results.nb shows how to access the informations in the above files and draw the Feynman graph.

The class of reflexive graph is called by the command

ReflexiveCases[N]

or the class of Fano graph is called by the command

FanoCases[N]

and the class of graph that are Fano but not reflexive is called by the command

FanoNotReflexiveCases[N]

with a number of edges $N$ from $2$ to $10$

The files contains the list of classes of Feynman graphs leading respectively to Reflexive and Fano polytopes with the following format

FanoCases[N] = {{Graph Association, {n_U,n_F}},   }

where Graph Association contains the following data

<|"LoopCount" -> , "Legs" -> {...},   "Edges" -> {...}}|>

and powers is the powers $n_U$ and $n_F$ of the Symanzik polynomial in the finite integral that lead to Fano polytope.

Starting from $N=5$ edges, there exist graphs whose associated Fano polytope is not reflexive. The first case is the one of the two-loop kite graph

The graph is called using

FanoNotReflexiveCases[5]

returns the list of the data of the kite graphs, and the powers $(n_U,n_F)=(1,1)$ for which the polytope is Fano but not reflexive

FanoCases[5] = {{<|"LoopCount" -> 2, 
   "Legs" -> {1 -> p[1], 2 -> p[2], 3 -> p[3], 4 -> p[4]}, 
   "Edges" -> {{3, 1} -> -k[1], {4, 1} -> 
      k[1] + p[1], {3, 2} -> -k[2], {4, 2} -> 
      k[2] + p[2], {4, 3} -> -k[1] - k[2] + p[3]}|>, {1, 1}}}

A table summarising the count of polytopes by edges

n edges	# Fano	# Reflexive	# Fano not Reflexive
2	1	1	0
3	2	2	0
4	3	3	0
5	4	4	1
6	8	7	4
7	11	6	6
8	23	11	16
9	36	14	24
10	104	26	88