Singular

D.4.7.3 propagator

Usage:

propagator(xy,d); xy list, d int
propagator(G,b); G graph, b list

Assume:

xy is a list of two numbers x and y in a rational function field, d non-negative integer.
G is a Feynman graph, a is a list of integers of length equal to the number of edges of G.
We assume that the coefficient ring has one rational variable for each vertex of G.

Return:

number, the propagator associated to the input data.

Theory:

If xy and d are specified, then the function returns x^2*y^2/(x^2-y^2)^2) for d=0, which is a associated to an edge with vertices x and y not passing above the base point. For d>0 it returns the sum of (j*x^(4*j)+j*y^(4*j))/(x*y)^(2*j) over all divisors j of d, which is associated to an edge with vertices x and y passing with multiplicity d above the base point.

Essentially the variables x and y stand for the position of the base points.

In the second way of using this function, G is a Feynman graph and b is a branch type over a fixed base point of a cover with source G and target an elliptic curve. It returns the product of propagator(list(v[i],w[i]),b[i]) over all edges i with multiplicity b[i] over the base point and vertices v[i] and w[i].

LIB "ellipticcovers.lib";
ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp;
graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4)));
propagator(list(x1,x2),0);
==> (x1^2*x2^2)/(x1^4-2*x1^2*x2^2+x2^4)
propagator(list(x1,x2),2);
==> (2*x1^8+x1^6*x2^2+x1^2*x2^6+2*x2^8)/(x1^4*x2^4)
propagator(G,list(1,1,1,0,0,0));
==> (x1^12*x3^2*x4^6+2*x1^8*x2^4*x3^2*x4^6+x1^8*x3^6*x4^6+x1^4*x2^8*x3^2*x4^6\
   +2*x1^4*x2^4*x3^6*x4^6+x2^8*x3^6*x4^6)/(x1^6*x2^6*x3^8-4*x1^6*x2^6*x3^6*x\
   4^2+6*x1^6*x2^6*x3^4*x4^4-4*x1^6*x2^6*x3^2*x4^6+x1^6*x2^6*x4^8-2*x1^6*x2^\
   4*x3^8*x4^2+8*x1^6*x2^4*x3^6*x4^4-12*x1^6*x2^4*x3^4*x4^6+8*x1^6*x2^4*x3^2\
   *x4^8-2*x1^6*x2^4*x4^10+x1^6*x2^2*x3^8*x4^4-4*x1^6*x2^2*x3^6*x4^6+6*x1^6*\
   x2^2*x3^4*x4^8-4*x1^6*x2^2*x3^2*x4^10+x1^6*x2^2*x4^12)