FCLoopGLIExpand
FCLoopGLIExpand[exp, topos, {x, x0, n}]
expands GLI
s defined via the list of topologies topos
in exp
around x=x0
to order n
. Here x
must be a scalar quantity, e.g. a mass or a scalar product.
This routine is particularly useful for doing asymptotic expansions of integrals or amplitudes.
Notice that the series is assumed to be well-defined. The function has no built-in checks against singular behavior.
See also
Overview, FCTopology, GLI, FCLoopGLIDifferentiate, ToGFAD.
Examples
FCLoopGLIExpand[x GLI[tad2l, {1, 1, 1}],
{FCTopology[tad2l, {FAD[{p1, m1}], FAD[{p2, m2}], FAD[{p1 - p2, m3}]},
{p1, p2}, {}, {}, {}]}, {m1, 0, 2}]
{m12xGtad2l(2,1,1)+xGtad2l(1,1,1),{FCTopology(tad2l,{p121,p22−m221,(p1−p2)2−m321},{p1,p2},{},{},{})}}
FCLoopGLIExpand[x GLI[tad2l, {1, 1, 1}],
{FCTopology[tad2l, {FAD[{p1, m1}], FAD[{p2, m2}], FAD[{p1 - p2, m3}]},
{p1, p2}, {}, {}, {}]}, {m1, M, 4}]
{16M8xGtad2l(5,1,1)−64M7m1xGtad2l(5,1,1)+96M6m12xGtad2l(5,1,1)+4M6xGtad2l(4,1,1)−64M5m13xGtad2l(5,1,1)−24M5m1xGtad2l(4,1,1)+16M4m14xGtad2l(5,1,1)+48M4m12xGtad2l(4,1,1)+M4xGtad2l(3,1,1)−40M3m13xGtad2l(4,1,1)+12M2m14xGtad2l(4,1,1)−2M2m12xGtad2l(3,1,1)−M2xGtad2l(2,1,1)+m14xGtad2l(3,1,1)+m12xGtad2l(2,1,1)+xGtad2l(1,1,1),{FCTopology(tad2l,{p12−M21,p22−m221,(p1−p2)2−m321},{p1,p2},{},{},{})}}
FCLoopGLIExpand[m2^2 GLI[tad2l, {1, 1, 1}],
{FCTopology[tad2l, {FAD[{p1, m1}], FAD[{p2, m2}], FAD[{p1 - p2, m3}]},
{p1, p2}, {}, {}, {}]}, {m2, 0, 6}]
{m26Gtad2l(1,3,1)+m24Gtad2l(1,2,1)+m22Gtad2l(1,1,1),{FCTopology(tad2l,{p12−m121,p221,(p1−p2)2−m321},{p1,p2},{},{},{})}}
FCLoopGLIExpand[ GLI[prop1l, {1, 1}] + SPD[q] GLI[prop1l, {1, 0}],
{FCTopology[prop1l, {FAD[{p1, m1}], FAD[{p1 + q, m2}]}, {p1}, {q}, {}, {}]}, {SPD[q], 0, 1}]
{q2Gprop1l(1,0)−q2Gprop1l(1,2)+Gprop1l(1,1),{FCTopology(prop1l,{(p12−m12+iη)1,(−m22+p12+2(p1⋅q)+iη)1},{p1},{q},{},{})}}