FCFeynmanParameterJoin[{{{prop1,prop2,x},prop3,y},...}, {p1,p2,...}] joins all propagators in int using Feynman parameters but does not integrate over the loop momenta p_i. The function returns {fpInt,pref,vars}, where fpInt is the piece of the integral that contains a single GFAD-type propagator and pref is the part containing the res. The introduced Feynman parameters are listed in vars. The overall Dirac delta is omitted.
Notice that each inner list must contain exactly thee elements, the first two being propagators (or products of propagators) and the last one denoting the head of Feynman parameter variables used to join those propagators. For example, {FAD[{p1,m1}],FAD[{p2,m2}],x}, but also {FAD[{p1,m1}],FAD[{p2,m2}]FAD[{p3,m3}],x} or even {FAD[{p1,m1}]FAD[{p2,m2}],FAD[{p3,m3}]FAD[{p4,m4}],x} represent valid examples of such lists, while something like {FAD[{p1,m1}],FAD[{p2,m2}],FAD[{p3,m3}],x} (4 instead of 3 list elements) is invalid and will not work.
Having obtained an output of FCFeynmanParameterJoin (e.g. called intT), you should use the following syntax to pass this to FCFeynmanParametrize: FCFeynmanParametrize[intT[[1]],intT[[2]],{lmom1, lmom2, ...},Variables->intT[[3]]]
Overview, FCFeynmanParametrize.
testProps = {FAD[{p1, m1}], FAD[{p2, m2}], FAD[{p3, m3}], FAD[{p4, m4}]}\left\{\frac{1}{\text{p1}^2-\text{m1}^2},\frac{1}{\text{p2}^2-\text{m2}^2},\frac{1}{\text{p3}^2-\text{m3}^2},\frac{1}{\text{p4}^2-\text{m4}^2}\right\}
Let us first join two propagators with each other using Feynman parameters x[i]
FCFeynmanParameterJoin[{testProps[[1]], testProps[[2]], x}, {p1, p2, p3, p4}]\left\{\frac{1}{(\left(\text{p1}^2-\text{m1}^2\right) x(1)+\left(\text{p2}^2-\text{m2}^2\right) x(2)+i \eta )^2},1,\{x(1),x(2)\}\right\}
Now we can join the resulting propagator with another propagator by introducing another set of Feynman parameters y[i]
FCFeynmanParameterJoin[{{testProps[[1]], testProps[[2]], x}, testProps[[3]], y},
{p1, p2, p3, p4}]\left\{\frac{1}{(\left(-x(1) \;\text{m1}^2+\text{p1}^2 x(1)-\text{m2}^2 x(2)+\text{p2}^2 x(2)\right) y(1)+\left(\text{p3}^2-\text{m3}^2\right) y(2)+i \eta )^3},2 y(1),\{x(1),x(2),y(1),y(2)\}\right\}
If needed, this procedure can be nested even further
FCFeynmanParameterJoin[{{{testProps[[1]], testProps[[2]], x}, testProps[[3]], y},
testProps[[4]], z}, {p1, p2, p3, p4}]\left\{\frac{1}{(\left(-x(1) y(1) \;\text{m1}^2+\text{p1}^2 x(1) y(1)-\text{m2}^2 x(2) y(1)+\text{p2}^2 x(2) y(1)-\text{m3}^2 y(2)+\text{p3}^2 y(2)\right) z(1)+\left(\text{p4}^2-\text{m4}^2\right) z(2)+i \eta )^4},6 y(1) z(1)^2,\{x(1),x(2),y(1),y(2),z(1),z(2)\}\right\}
Notice that FCFeynmanParametrizeknows how to deal with the output produced by FCFeynmanParameterJoin
intT = FCFeynmanParameterJoin[{{SFAD[{p1, mg^2}] SFAD[{p3 - p1, mg^2}], 1, x},
SFAD[{{0, -2 p1 . q}}] SFAD[{{0, -2 p3 . q}}], y}, {p1, p3}]\left\{\frac{1}{(\left(-x(1) \;\text{mg}^2-x(2) \;\text{mg}^2+\text{p1}^2 x(1)+\text{p1}^2 x(2)-2 (\text{p1}\cdot \;\text{p3}) x(2)+\text{p3}^2 x(2)\right) y(1)-2 (\text{p1}\cdot q) y(2)-2 (\text{p3}\cdot q) y(3)+i \eta )^4},6 y(1),\{x(1),x(2),y(1),y(2),y(3)\}\right\}
FCFeynmanParametrize[intT[[1]], intT[[2]], {p1, p3}, Names -> z, Indexed -> True,
FCReplaceD -> {D -> 4 - 2 ep}, Simplify -> True, Assumptions -> {mg > 0, ep > 0},
FinalSubstitutions -> {FCI@SPD[q] -> qq, mg^2 -> mg2}, Variables -> intT[[3]]]\left\{\frac{\left(x(1) x(2) y(1)^2\right)^{3 \;\text{ep}} \left(\text{mg2} x(1) x(2) (x(1)+x(2)) y(1)^3+\text{qq} y(1) \left(x(1) y(3)^2+x(2) (y(2)+y(3))^2\right)\right)^{-2 \;\text{ep}}}{x(1)^2 x(2)^2 y(1)^3},\Gamma (2 \;\text{ep}),\{x(1),x(2),y(1),y(2),y(3)\}\right\}