FCLoopGLIDifferentiate
FCLoopGLIDifferentiate[exp, topos, inv]
calculates the partial derivative of GLIs present in exp
with respect to the scalar quantity inv
. Here inv
can be a constant (e.g. mass), a scalar product of some momenta or a 4-vector.
The list topos must contain the topologies describing all of the occurring GLIs.
To calculate multiple derivatives, use the notation FCLoopGLIDifferentiate[exp , topos, {inv,n}]
for scalars and FCLoopGLIDifferentiate[exp , topos, {vec1, vec2, ...}]
for vectors.
See also
Overview, FCTopology, GLI, FCLoopGLIExpand.
Examples
FCLoopGLIDifferentiate[x GLI[tad2l, {1, 1, 1}],
{FCTopology[tad2l, {FAD[{p1, m1}], FAD[{p2, m2}], FAD[{p1 - p2, m3}]}, {p1, p2}, {}, {}, {}]}, m1]
2m1xGtad2l(2,1,1)
FCLoopGLIDifferentiate[x GLI[tad2l, {1, 1, 1}],
{FCTopology[tad2l, {FAD[{p1, m1}], FAD[{p2, m2}], FAD[{p1 - p2, m3}]}, {p1, p2}, {}, {}, {}]}, {m1, 5}]
3840m15xGtad2l(6,1,1)+3840m13xGtad2l(5,1,1)+720m1xGtad2l(4,1,1)
FCLoopGLIDifferentiate[m2^2 GLI[tad2l, {1, 1, 1}],
{FCTopology[tad2l, {FAD[{p1, m1}], FAD[{p2, m2}], FAD[{p1 - p2, m3}]}, {p1, p2}, {}, {}, {}]}, m2]
2m23Gtad2l(1,2,1)+2m2Gtad2l(1,1,1)
FCLoopGLIDifferentiate[ GLI[prop1l, {1, 1}] + SPD[q] GLI[prop1l, {1, 0}],
{FCTopology[prop1l, {FAD[{p1, m1}], FAD[{p1 + q, m2}]}, {p1}, {q}, {}, {}]}, SPD[q]]
Gprop1l(1,0)−Gprop1l(1,2)
FCLoopGLIDifferentiate[SPD[p1, p2] GLI[topo1, {1, 1, 1}],
{FCTopology[topo1, {SFAD[{p1, m1^2}], SFAD[{p2, m2^2}], SFAD[p1 - p2]}, {p1, p2}, {}, {}, {}]},
FVD[p1, mu], FCE -> True]
−2Gtopo1(1,1,2)(p1mu−p2mu)(p1⋅p2)−2p1muGtopo1(2,1,1)(p1⋅p2)+p2muGtopo1(1,1,1)
FCLoopGLIDifferentiate[SPD[p1, p2] GLI[topo1, {1, 1, 1}], {FCTopology[topo1,
{SFAD[{p1, m1^2}], SFAD[{p2, m2^2}], SFAD[p1 - p2]}, {p1, p2}, {}, {}, {}]}
, {FVD[p1, mu], FVD[p2, nu]}, FCE -> True]
−2Gtopo1(1,1,2)(−gmunu(p1⋅p2)−2p2mup1nu+p1mup1nu+p2mup2nu)+Gtopo1(1,1,1)gmunu−8Gtopo1(1,1,3)(p1mu−p2mu)(p1nu−p2nu)(p1⋅p2)+4p2nuGtopo1(1,2,2)(p1mu−p2mu)(p1⋅p2)−4p1muGtopo1(2,1,2)(p1nu−p2nu)(p1⋅p2)+4p1mup2nuGtopo1(2,2,1)(p1⋅p2)−2p1mup1nuGtopo1(2,1,1)−2p2mup2nuGtopo1(1,2,1)