FCLoopAddScalingParameter[topo, la, rules] multiplies
masses and momenta in the propagators of the topology topo
by the scaling parameter la according to the scaling rules
in rules. The id of the topology remains
unchanged. This is useful e.g. for asymptotic expansions of the
corresponding loop integrals given as GLIs.
The scaling variable should be declared as FCVariable
via the DataType mechanism.
Notice that if all terms in a propagator have the same scaling, the scaling variable in the respective propagator will be set to unity.
DataType[la, FCVariable] = True;We declare the external 4-momentum q as our hard scale,
while the mass mc is soft
topoScaled = FCLoopAddScalingParameter[FCTopology[prop1LtopoC11, {SFAD[{{I p1, 0}, {-mc^2, -1}, 1}],
SFAD[{{I (p1 - q), 0}, {-mc^2, -1}, 1}]}, {p1}, {q}, {SPD[q, q] ->mb^2}, {}], la,
{q -> la^0 q, mc -> la^1 mc}]\text{Scalings of momenta and masses in the propagators of }\;\text{prop1LtopoC11}\;\text{ : }\left\{\text{mb}\to \;\text{la}^0 \;\text{mb},\text{mc}\to \;\text{la} \;\text{mc},\text{p1}\to \;\text{la}^0 \;\text{p1},q\to \;\text{la}^0 q\right\}
\text{FCTopology}\left(\text{prop1LtopoC11},\left\{\frac{1}{(\text{la}^2 \;\text{mc}^2-\text{p1}^2-i \eta )},\frac{1}{(-\text{mb}^2+\text{la}^2 \;\text{mc}^2-\text{p1}^2+2 (\text{p1}\cdot q)-i \eta )}\right\},\{\text{p1}\},\{q\},\left\{q^2\to \;\text{mb}^2\right\},\{\}\right)
Having set up the scaling we can now use FCLoopGLIExpand
to expand the loop integrals belonging to this topology up to the
desired order in la. Here we choose \mathcal{O}(\lambda^4)
FCLoopGLIExpand[GLI[prop1LtopoC11, {1, 1}], {topoScaled}, {la, 0, 4}]\left\{\text{la}^4 \;\text{mc}^4 G^{\text{prop1LtopoC11}}(1,3)+\text{la}^4 \;\text{mc}^4 G^{\text{prop1LtopoC11}}(2,2)+\text{la}^4 \;\text{mc}^4 G^{\text{prop1LtopoC11}}(3,1)-\text{la}^2 \;\text{mc}^2 G^{\text{prop1LtopoC11}}(1,2)-\text{la}^2 \;\text{mc}^2 G^{\text{prop1LtopoC11}}(2,1)+G^{\text{prop1LtopoC11}}(1,1),\left\{\text{FCTopology}\left(\text{prop1LtopoC11},\left\{\frac{1}{(-\text{p1}^2-i \eta )},\frac{1}{(-\text{mb}^2-\text{p1}^2+2 (\text{p1}\cdot q)-i \eta )}\right\},\{\text{p1}\},\{q\},\left\{q^2\to \;\text{mb}^2\right\},\{\}\right)\right\}\right\}