FeynCalc manual (development version)

FCLoopGLILowerDimension

FCLoopGLILowerDimension[gli, topo] lowers the dimension of the given GLI from D to D-2 and expresses it in terms of D-dimensional loop integrals returned in the output.

The algorithm is based on the code of the function RaisingDRR from R. Lee’s LiteRed

See also

Overview, FCLoopGLIRaiseDimension.

Examples

topo = FCTopology[
   topo1, {SFAD[p1], SFAD[p2], SFAD[Q - p1 - p2], SFAD[Q - p2], 
    SFAD[Q - p1]}, {p1, p2}, {Q}, {Hold[SPD[Q]] -> qq}, {}]

FCTopology(topo1,{1(p12+iη),1(p22+iη),1((−p1−p2+Q)2+iη),1((Q−p2)2+iη),1((Q−p1)2+iη)},{p1,p2},{Q},{Hold[SPD(Q)]→  qq},{})\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+Q)^2+i \eta )},\frac{1}{((Q-\text{p2})^2+i \eta )},\frac{1}{((Q-\text{p1})^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{Q\},\{\text{Hold}[\text{SPD}(Q)]\to \;\text{qq}\},\{\}\right)

FCLoopGLILowerDimension[GLI[topo1, {1, 1, 1, 1, 1}], topo]

Gtopo1(1,1,1,2,2)+Gtopo1(1,1,2,1,2)+Gtopo1(1,1,2,2,1)+Gtopo1(1,2,1,1,2)+Gtopo1(1,2,2,1,1)+Gtopo1(2,1,1,2,1)+Gtopo1(2,1,2,1,1)+Gtopo1(2,2,1,1,1)G^{\text{topo1}}(1,1,1,2,2)+G^{\text{topo1}}(1,1,2,1,2)+G^{\text{topo1}}(1,1,2,2,1)+G^{\text{topo1}}(1,2,1,1,2)+G^{\text{topo1}}(1,2,2,1,1)+G^{\text{topo1}}(2,1,1,2,1)+G^{\text{topo1}}(2,1,2,1,1)+G^{\text{topo1}}(2,2,1,1,1)

FCLoopGLILowerDimension[GLI[topo1, {n1, n2, n3, 1, 1}], topo]

Gtopo1(n1,n2,n3,2,2)+n3Gtopo1(n1,n2,n3+1,1,2)+n3Gtopo1(n1,n2,n3+1,2,1)+n2Gtopo1(n1,n2+1,n3,1,2)+n2  n3Gtopo1(n1,n2+1,n3+1,1,1)+n1Gtopo1(n1+1,n2,n3,2,1)+n1  n3Gtopo1(n1+1,n2,n3+1,1,1)+n1  n2Gtopo1(n1+1,n2+1,n3,1,1)G^{\text{topo1}}(\text{n1},\text{n2},\text{n3},2,2)+\text{n3} G^{\text{topo1}}(\text{n1},\text{n2},\text{n3}+1,1,2)+\text{n3} G^{\text{topo1}}(\text{n1},\text{n2},\text{n3}+1,2,1)+\text{n2} G^{\text{topo1}}(\text{n1},\text{n2}+1,\text{n3},1,2)+\text{n2} \;\text{n3} G^{\text{topo1}}(\text{n1},\text{n2}+1,\text{n3}+1,1,1)+\text{n1} G^{\text{topo1}}(\text{n1}+1,\text{n2},\text{n3},2,1)+\text{n1} \;\text{n3} G^{\text{topo1}}(\text{n1}+1,\text{n2},\text{n3}+1,1,1)+\text{n1} \;\text{n2} G^{\text{topo1}}(\text{n1}+1,\text{n2}+1,\text{n3},1,1)