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Massless one-loop triangle Feynman diagram

Consider the annihilating $D$-ideal of a massless one-loop triangle Feynman integral, appearing in Equation (5.14) in [HPSZ]. In what follows, we write this system of PDEs in connection form.

i1 : D = makeWeylAlgebra(QQ[x,y],{1,1});
i2 : Q1 = x^2*dx^2+2*x*y*dx*dy+(y-1)*y*dy^2+3*x*dx+(3*y-1)*dy+1;
i3 : Q2 = x*dx^2-y*dy^2+dx-dy;
i4 : I = ideal(Q1,Q2);

o4 : Ideal of D

We can check that the ideal has holonomic rank $4$ and compute the standard monomials of the Gröbner basis of $RI$.

i5 : assert( 4 == holonomicRank I )
i6 : standardMonomials I

                   2
o6 = {1, dx, dy, dy }

o6 : List

Finally, we can compute the connection matrices.

i7 : elapsedTime A = connectionMatrices I;
 -- .247168s elapsed
i8 : elapsedTime assert isIntegrable A
 -- .281476s elapsed
i9 : netList A

     +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
o9 = || 0                                                       1                                               0                                                                     0                                                                      ||
     || 0                                                       (-1)/x                                          1/x                                                                   y/x                                                                    ||
     || (-1)/2xy                                                (-1)/y                                          (-x-3y+1)/2xy                                                         (-x-y+1)/2x                                                            ||
     || (x2-4xy+3y2-2x-4y+1)/(2x3y2-4x2y3+2xy4-4x2y2-4xy3+2xy2) (x2-5xy+2y2-2x-3y+1)/(x2y2-2xy3+y4-2xy2-2y3+y2) (x3-7x2y+xy2+9y3-3x2-15y2+3x+7y-1)/(2x3y2-4x2y3+2xy4-4x2y2-4xy3+2xy2) (x3-9x2y+5xy2+3y3-3x2+4xy-7y2+3x+5y-1)/(2x3y-4x2y2+2xy3-4x2y-4xy2+2xy) ||
     +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
     || 0                                  0                                   1                                               0                                             |                                                                                |
     || (-1)/2xy                           (-1)/y                              (-x-3y+1)/2xy                                   (-x-y+1)/2x                                   |                                                                                |
     || 0                                  0                                   0                                               1                                             |                                                                                |
     || (x-y-1)/(x2y2-2xy3+y4-2xy2-2y3+y2) (2x2-2x)/(x2y2-2xy3+y4-2xy2-2y3+y2) (x2+4xy-7y2-2x+2y+1)/(x2y2-2xy3+y4-2xy2-2y3+y2) (-x2+7xy-6y2+2x+7y-1)/(x2y-2xy2+y3-2xy-2y2+y) |                                                                                |
     +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+

References

[HPSZ] J. Henn, E. Pratt, A.-L. Sattelberger, and S. Zoia. $D$-Module Techniques for Solving Differential Equations behind Feynman Integrals. Letters in Mathematical Physics, 114(28), 2024.

See also


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/ConnectionMatrices/examples.m2:100:0.