FeynCalc manual (development version)

FCLoopGLIExpand

FCLoopGLIExpand[exp, topos, {x, x0, n}] expands GLIs 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,{1p12,1p22m22,1(p1p2)2m32},{p1,p2},{},{},{})}}\left\{\text{m1}^2 x G^{\text{tad2l}}(2,1,1)+x G^{\text{tad2l}}(1,1,1),\left\{\text{FCTopology}\left(\text{tad2l},\left\{\frac{1}{\text{p1}^2},\frac{1}{\text{p2}^2-\text{m2}^2},\frac{1}{(\text{p1}-\text{p2})^2-\text{m3}^2}\right\},\{\text{p1},\text{p2}\},\{\},\{\},\{\}\right)\right\}\right\}

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)64M7  m1xGtad2l(5,1,1)+96M6  m12xGtad2l(5,1,1)+4M6xGtad2l(4,1,1)64M5  m13xGtad2l(5,1,1)24M5  m1xGtad2l(4,1,1)+16M4  m14xGtad2l(5,1,1)+48M4  m12xGtad2l(4,1,1)+M4xGtad2l(3,1,1)40M3  m13xGtad2l(4,1,1)+12M2  m14xGtad2l(4,1,1)2M2  m12xGtad2l(3,1,1)M2xGtad2l(2,1,1)+m14xGtad2l(3,1,1)+m12xGtad2l(2,1,1)+xGtad2l(1,1,1),{FCTopology(tad2l,{1p12M2,1p22m22,1(p1p2)2m32},{p1,p2},{},{},{})}}\left\{16 M^8 x G^{\text{tad2l}}(5,1,1)-64 M^7 \;\text{m1} x G^{\text{tad2l}}(5,1,1)+96 M^6 \;\text{m1}^2 x G^{\text{tad2l}}(5,1,1)+4 M^6 x G^{\text{tad2l}}(4,1,1)-64 M^5 \;\text{m1}^3 x G^{\text{tad2l}}(5,1,1)-24 M^5 \;\text{m1} x G^{\text{tad2l}}(4,1,1)+16 M^4 \;\text{m1}^4 x G^{\text{tad2l}}(5,1,1)+48 M^4 \;\text{m1}^2 x G^{\text{tad2l}}(4,1,1)+M^4 x G^{\text{tad2l}}(3,1,1)-40 M^3 \;\text{m1}^3 x G^{\text{tad2l}}(4,1,1)+12 M^2 \;\text{m1}^4 x G^{\text{tad2l}}(4,1,1)-2 M^2 \;\text{m1}^2 x G^{\text{tad2l}}(3,1,1)-M^2 x G^{\text{tad2l}}(2,1,1)+\text{m1}^4 x G^{\text{tad2l}}(3,1,1)+\text{m1}^2 x G^{\text{tad2l}}(2,1,1)+x G^{\text{tad2l}}(1,1,1),\left\{\text{FCTopology}\left(\text{tad2l},\left\{\frac{1}{\text{p1}^2-M^2},\frac{1}{\text{p2}^2-\text{m2}^2},\frac{1}{(\text{p1}-\text{p2})^2-\text{m3}^2}\right\},\{\text{p1},\text{p2}\},\{\},\{\},\{\}\right)\right\}\right\}

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,{1p12m12,1p22,1(p1p2)2m32},{p1,p2},{},{},{})}}\left\{\text{m2}^6 G^{\text{tad2l}}(1,3,1)+\text{m2}^4 G^{\text{tad2l}}(1,2,1)+\text{m2}^2 G^{\text{tad2l}}(1,1,1),\left\{\text{FCTopology}\left(\text{tad2l},\left\{\frac{1}{\text{p1}^2-\text{m1}^2},\frac{1}{\text{p2}^2},\frac{1}{(\text{p1}-\text{p2})^2-\text{m3}^2}\right\},\{\text{p1},\text{p2}\},\{\},\{\},\{\}\right)\right\}\right\}

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,{1(p12m12+iη),1(m22+p12+2(p1q)+iη)},{p1},{q},{},{})}}\left\{q^2 G^{\text{prop1l}}(1,0)-q^2 G^{\text{prop1l}}(1,2)+G^{\text{prop1l}}(1,1),\left\{\text{FCTopology}\left(\text{prop1l},\left\{\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )},\frac{1}{(-\text{m2}^2+\text{p1}^2+2 (\text{p1}\cdot q)+i \eta )}\right\},\{\text{p1}\},\{q\},\{\},\{\}\right)\right\}\right\}