FCLoopToPakForm[int, {p1, p2, ...}]
determines a canonical U F UF U F -based representation for the scalar multi-loop integral int
that depend on the loop momenta p1, p2, ...
using the algorithm of Alexey Pak arXiv:1111.0868 .
The current implementation is based on the FindEquivalents
function from FIRE 6 arXiv:1901.07808 . FCLoopToPakForm
is a backend function used in FCLoopPakScalelessQ
, FCLoopFindIntegralMappings
, FCLoopFindTopologyMappings
etc.
It is also possible to invoke the function as FCLoopToPakForm[GLI[...], FCTopology[...]]
or FCLoopToPakForm[FCTopology[…]]. Notice that in this case the value of the option FinalSubstitutions
is ignored, as replacement rules will be extracted directly from the definition of the topology.
See also
Overview , FCTopology , GLI , FCLoopToPakForm , FCLoopPakScalelessQ , FCLoopScalelessQ , FCLoopFindIntegralMappings , FCLoopFindTopologyMappings .
Examples
FCLoopToPakForm[ FAD[ p1, { p3, m1}, { p1 - p4, m1}, p1 + q1, p1 + q1, p3 + q1, p1 - p3 - p4],
{ p1, p3, p4}, Names -> x , Head -> ph, Power -> pow]
{ 1 p1 2 . ( p3 2 − m1 2 ) . ( ( p1 − p4 ) 2 − m1 2 ) . ( p1 + q1 ) 4 . ( p3 + q1 ) 2 . ( p1 − p3 − p4 ) 2 , ph ( m1 2 pow ( 2 ) x ( 2 ) x ( 4 ) 2 x ( 6 ) + m1 2 pow ( 2 ) x ( 3 ) x ( 4 ) 2 x ( 6 ) + m1 2 pow ( 2 ) x ( 2 ) 2 x ( 3 ) x ( 6 ) + m1 2 pow ( 2 ) x ( 2 ) 2 x ( 4 ) x ( 6 ) + 2 m1 2 pow ( 2 ) x ( 2 ) x ( 3 ) x ( 4 ) x ( 6 ) + m1 2 pow ( 2 ) x ( 2 ) 2 x ( 5 ) x ( 6 ) + m1 2 pow ( 2 ) x ( 2 ) x ( 3 ) x ( 5 ) x ( 6 ) + m1 2 pow ( 2 ) x ( 2 ) x ( 4 ) x ( 5 ) x ( 6 ) + m1 2 pow ( 2 ) x ( 3 ) x ( 4 ) x ( 5 ) x ( 6 ) + m1 2 x ( 1 ) x ( 2 ) x ( 4 ) 2 + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) 2 + m1 2 x ( 1 ) x ( 2 ) 2 x ( 3 ) + m1 2 x ( 1 ) x ( 2 ) 2 x ( 4 ) + 2 m1 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 4 ) + m1 2 x ( 1 ) x ( 2 ) 2 x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 5 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 5 ) − pow ( 2 ) q1 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 1 ) x ( 2 ) x ( 5 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 2 ) x ( 3 ) x ( 5 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 2 ) x ( 4 ) x ( 5 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 3 ) x ( 4 ) x ( 5 ) x ( 6 ) + pow ( 2 ) x ( 2 ) x ( 3 ) x ( 6 ) + pow ( 2 ) x ( 2 ) x ( 4 ) x ( 6 ) + pow ( 2 ) x ( 3 ) x ( 4 ) x ( 6 ) + pow ( 2 ) x ( 2 ) x ( 5 ) x ( 6 ) + pow ( 2 ) x ( 3 ) x ( 5 ) x ( 6 ) − q1 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 5 ) − q1 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 5 ) − q1 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 5 ) + x ( 1 ) x ( 2 ) x ( 3 ) + x ( 1 ) x ( 2 ) x ( 4 ) + x ( 1 ) x ( 3 ) x ( 4 ) + x ( 1 ) x ( 2 ) x ( 5 ) + x ( 1 ) x ( 3 ) x ( 5 ) , ( x ( 1 ) x ( 3 ) x ( 4 ) x ( 2 ) x ( 6 ) x ( 5 ) 1 p1 2 1 ( p1 − p4 ) 2 − m1 2 1 ( p1 − p3 − p4 ) 2 1 p3 2 − m1 2 1 ( p3 + q1 ) 2 1 ( p1 + q1 ) 2 1 1 1 1 1 2 ) ) } \left\{\frac{1}{\text{p1}^2.\left(\text{p3}^2-\text{m1}^2\right).\left((\text{p1}-\text{p4})^2-\text{m1}^2\right).(\text{p1}+\text{q1})^4.(\text{p3}+\text{q1})^2.(\text{p1}-\text{p3}-\text{p4})^2},\text{ph}\left(\text{m1}^2 \;\text{pow}(2) x(2) x(4)^2 x(6)+\text{m1}^2 \;\text{pow}(2) x(3) x(4)^2 x(6)+\text{m1}^2 \;\text{pow}(2) x(2)^2 x(3) x(6)+\text{m1}^2 \;\text{pow}(2) x(2)^2 x(4) x(6)+2 \;\text{m1}^2 \;\text{pow}(2) x(2) x(3) x(4) x(6)+\text{m1}^2 \;\text{pow}(2) x(2)^2 x(5) x(6)+\text{m1}^2 \;\text{pow}(2) x(2) x(3) x(5) x(6)+\text{m1}^2 \;\text{pow}(2) x(2) x(4) x(5) x(6)+\text{m1}^2 \;\text{pow}(2) x(3) x(4) x(5) x(6)+\text{m1}^2 x(1) x(2) x(4)^2+\text{m1}^2 x(1) x(3) x(4)^2+\text{m1}^2 x(1) x(2)^2 x(3)+\text{m1}^2 x(1) x(2)^2 x(4)+2 \;\text{m1}^2 x(1) x(2) x(3) x(4)+\text{m1}^2 x(1) x(2)^2 x(5)+\text{m1}^2 x(1) x(2) x(3) x(5)+\text{m1}^2 x(1) x(2) x(4) x(5)+\text{m1}^2 x(1) x(3) x(4) x(5)-\text{pow}(2) \;\text{q1}^2 x(1) x(2) x(3) x(6)-\text{pow}(2) \;\text{q1}^2 x(1) x(2) x(4) x(6)-\text{pow}(2) \;\text{q1}^2 x(1) x(3) x(4) x(6)-\text{pow}(2) \;\text{q1}^2 x(1) x(2) x(5) x(6)-\text{pow}(2) \;\text{q1}^2 x(1) x(3) x(5) x(6)-\text{pow}(2) \;\text{q1}^2 x(2) x(3) x(5) x(6)-\text{pow}(2) \;\text{q1}^2 x(2) x(4) x(5) x(6)-\text{pow}(2) \;\text{q1}^2 x(3) x(4) x(5) x(6)+\text{pow}(2) x(2) x(3) x(6)+\text{pow}(2) x(2) x(4) x(6)+\text{pow}(2) x(3) x(4) x(6)+\text{pow}(2) x(2) x(5) x(6)+\text{pow}(2) x(3) x(5) x(6)-\text{q1}^2 x(1) x(2) x(3) x(5)-\text{q1}^2 x(1) x(2) x(4) x(5)-\text{q1}^2 x(1) x(3) x(4) x(5)+x(1) x(2) x(3)+x(1) x(2) x(4)+x(1) x(3) x(4)+x(1) x(2) x(5)+x(1) x(3) x(5),\left(
\begin{array}{cccccc}
x(1) & x(3) & x(4) & x(2) & x(6) & x(5) \\
\frac{1}{\text{p1}^2} & \frac{1}{(\text{p1}-\text{p4})^2-\text{m1}^2} & \frac{1}{(\text{p1}-\text{p3}-\text{p4})^2} & \frac{1}{\text{p3}^2-\text{m1}^2} & \frac{1}{(\text{p3}+\text{q1})^2} & \frac{1}{(\text{p1}+\text{q1})^2} \\
1 & 1 & 1 & 1 & 1 & 2 \\
\end{array}
\right)\right)\right\} ⎩ ⎨ ⎧ p1 2 . ( p3 2 − m1 2 ) . ( ( p1 − p4 ) 2 − m1 2 ) . ( p1 + q1 ) 4 . ( p3 + q1 ) 2 . ( p1 − p3 − p4 ) 2 1 , ph m1 2 pow ( 2 ) x ( 2 ) x ( 4 ) 2 x ( 6 ) + m1 2 pow ( 2 ) x ( 3 ) x ( 4 ) 2 x ( 6 ) + m1 2 pow ( 2 ) x ( 2 ) 2 x ( 3 ) x ( 6 ) + m1 2 pow ( 2 ) x ( 2 ) 2 x ( 4 ) x ( 6 ) + 2 m1 2 pow ( 2 ) x ( 2 ) x ( 3 ) x ( 4 ) x ( 6 ) + m1 2 pow ( 2 ) x ( 2 ) 2 x ( 5 ) x ( 6 ) + m1 2 pow ( 2 ) x ( 2 ) x ( 3 ) x ( 5 ) x ( 6 ) + m1 2 pow ( 2 ) x ( 2 ) x ( 4 ) x ( 5 ) x ( 6 ) + m1 2 pow ( 2 ) x ( 3 ) x ( 4 ) x ( 5 ) x ( 6 ) + m1 2 x ( 1 ) x ( 2 ) x ( 4 ) 2 + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) 2 + m1 2 x ( 1 ) x ( 2 ) 2 x ( 3 ) + m1 2 x ( 1 ) x ( 2 ) 2 x ( 4 ) + 2 m1 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 4 ) + m1 2 x ( 1 ) x ( 2 ) 2 x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 5 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 5 ) − pow ( 2 ) q1 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 1 ) x ( 2 ) x ( 5 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 1 ) x ( 3 ) x ( 5 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 2 ) x ( 3 ) x ( 5 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 2 ) x ( 4 ) x ( 5 ) x ( 6 ) − pow ( 2 ) q1 2 x ( 3 ) x ( 4 ) x ( 5 ) x ( 6 ) + pow ( 2 ) x ( 2 ) x ( 3 ) x ( 6 ) + pow ( 2 ) x ( 2 ) x ( 4 ) x ( 6 ) + pow ( 2 ) x ( 3 ) x ( 4 ) x ( 6 ) + pow ( 2 ) x ( 2 ) x ( 5 ) x ( 6 ) + pow ( 2 ) x ( 3 ) x ( 5 ) x ( 6 ) − q1 2 x ( 1 ) x ( 2 ) x ( 3 ) x ( 5 ) − q1 2 x ( 1 ) x ( 2 ) x ( 4 ) x ( 5 ) − q1 2 x ( 1 ) x ( 3 ) x ( 4 ) x ( 5 ) + x ( 1 ) x ( 2 ) x ( 3 ) + x ( 1 ) x ( 2 ) x ( 4 ) + x ( 1 ) x ( 3 ) x ( 4 ) + x ( 1 ) x ( 2 ) x ( 5 ) + x ( 1 ) x ( 3 ) x ( 5 ) , x ( 1 ) p1 2 1 1 x ( 3 ) ( p1 − p4 ) 2 − m1 2 1 1 x ( 4 ) ( p1 − p3 − p4 ) 2 1 1 x ( 2 ) p3 2 − m1 2 1 1 x ( 6 ) ( p3 + q1 ) 2 1 1 x ( 5 ) ( p1 + q1 ) 2 1 2 ⎭ ⎬ ⎫
topo1 = FCTopology[ "prop2Lv1" , { SFAD[{ p1, m1^ 2 }], SFAD[{ p2, m2^ 2 }], SFAD[ p1 - q ], SFAD[ p2 - q ], SFAD[{ p1 - p2, m3^ 2 }]}, { p1, p2}, { Q }, {}, {}]
topo2 = FCTopology[ "prop2Lv2" , { SFAD[{ p1, m1^ 2 }], SFAD[{ p2, m2^ 2 }], SFAD[{ p1 - q , M ^ 2 }], SFAD[{ p2 - q , M ^ 2 }], SFAD[ p1 - p2]}, { p1, p2}, { Q }, {}, {}]
FCTopology ( prop2Lv1 , { 1 ( p1 2 − m1 2 + i η ) , 1 ( p2 2 − m2 2 + i η ) , 1 ( ( p1 − q ) 2 + i η ) , 1 ( ( p2 − q ) 2 + i η ) , 1 ( ( p1 − p2 ) 2 − m3 2 + i η ) } , { p1 , p2 } , { Q } , { } , { } ) \text{FCTopology}\left(\text{prop2Lv1},\left\{\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )},\frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )},\frac{1}{((\text{p1}-q)^2+i \eta )},\frac{1}{((\text{p2}-q)^2+i \eta )},\frac{1}{((\text{p1}-\text{p2})^2-\text{m3}^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{Q\},\{\},\{\}\right) FCTopology ( prop2Lv1 , { ( p1 2 − m1 2 + i η ) 1 , ( p2 2 − m2 2 + i η ) 1 , (( p1 − q ) 2 + i η ) 1 , (( p2 − q ) 2 + i η ) 1 , (( p1 − p2 ) 2 − m3 2 + i η ) 1 } , { p1 , p2 } , { Q } , { } , { } )
FCTopology ( prop2Lv2 , { 1 ( p1 2 − m1 2 + i η ) , 1 ( p2 2 − m2 2 + i η ) , 1 ( ( p1 − q ) 2 − M 2 + i η ) , 1 ( ( p2 − q ) 2 − M 2 + i η ) , 1 ( ( p1 − p2 ) 2 + i η ) } , { p1 , p2 } , { Q } , { } , { } ) \text{FCTopology}\left(\text{prop2Lv2},\left\{\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )},\frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )},\frac{1}{((\text{p1}-q)^2-M^2+i \eta )},\frac{1}{((\text{p2}-q)^2-M^2+i \eta )},\frac{1}{((\text{p1}-\text{p2})^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{Q\},\{\},\{\}\right) FCTopology ( prop2Lv2 , { ( p1 2 − m1 2 + i η ) 1 , ( p2 2 − m2 2 + i η ) 1 , (( p1 − q ) 2 − M 2 + i η ) 1 , (( p2 − q ) 2 − M 2 + i η ) 1 , (( p1 − p2 ) 2 + i η ) 1 } , { p1 , p2 } , { Q } , { } , { } )
FCLoopToPakForm[ topo1, Names -> x , Head -> ph, Power -> pow]
{ FCTopology ( prop2Lv1 , { 1 ( p1 2 − m1 2 + i η ) , 1 ( p2 2 − m2 2 + i η ) , 1 ( ( p1 − q ) 2 + i η ) , 1 ( ( p2 − q ) 2 + i η ) , 1 ( ( p1 − p2 ) 2 − m3 2 + i η ) } , { p1 , p2 } , { Q } , { } , { } ) , ph ( m1 2 x ( 1 ) x ( 2 ) 2 + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 2 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 2 ) x ( 4 ) + m1 2 x ( 2 ) x ( 3 ) x ( 4 ) + m1 2 x ( 2 ) 2 x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 5 ) + m1 2 x ( 2 ) x ( 3 ) x ( 5 ) + m2 2 x ( 1 ) x ( 4 ) 2 + m2 2 x ( 2 ) x ( 4 ) 2 + m2 2 x ( 3 ) x ( 4 ) 2 + m2 2 x ( 1 ) x ( 2 ) x ( 4 ) + m2 2 x ( 1 ) x ( 3 ) x ( 4 ) + m2 2 x ( 1 ) x ( 4 ) x ( 5 ) + m2 2 x ( 2 ) x ( 4 ) x ( 5 ) + m2 2 x ( 3 ) x ( 4 ) x ( 5 ) + m3 2 x ( 1 ) 2 x ( 2 ) + m3 2 x ( 1 ) 2 x ( 3 ) + m3 2 x ( 1 ) 2 x ( 4 ) + m3 2 x ( 1 ) x ( 2 ) x ( 4 ) + m3 2 x ( 1 ) x ( 3 ) x ( 4 ) + m3 2 x ( 1 ) 2 x ( 5 ) + m3 2 x ( 1 ) x ( 2 ) x ( 5 ) + m3 2 x ( 1 ) x ( 3 ) x ( 5 ) − q 2 x ( 1 ) x ( 2 ) x ( 3 ) − q 2 x ( 1 ) x ( 3 ) x ( 4 ) − q 2 x ( 2 ) x ( 3 ) x ( 4 ) − q 2 x ( 1 ) x ( 2 ) x ( 5 ) − q 2 x ( 2 ) x ( 3 ) x ( 5 ) − q 2 x ( 1 ) x ( 4 ) x ( 5 ) − q 2 x ( 2 ) x ( 4 ) x ( 5 ) − q 2 x ( 3 ) x ( 4 ) x ( 5 ) + x ( 1 ) x ( 2 ) + x ( 1 ) x ( 3 ) + x ( 1 ) x ( 4 ) + x ( 2 ) x ( 4 ) + x ( 3 ) x ( 4 ) + x ( 1 ) x ( 5 ) + x ( 2 ) x ( 5 ) + x ( 3 ) x ( 5 ) , ( x ( 5 ) x ( 1 ) x ( 3 ) x ( 2 ) x ( 4 ) 1 ( ( p1 − p2 ) 2 − m3 2 + i η ) 1 ( p1 2 − m1 2 + i η ) 1 ( ( p1 − q ) 2 + i η ) 1 ( p2 2 − m2 2 + i η ) 1 ( ( p2 − q ) 2 + i η ) 1 1 1 1 1 ) ) } \left\{\text{FCTopology}\left(\text{prop2Lv1},\left\{\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )},\frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )},\frac{1}{((\text{p1}-q)^2+i \eta )},\frac{1}{((\text{p2}-q)^2+i \eta )},\frac{1}{((\text{p1}-\text{p2})^2-\text{m3}^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{Q\},\{\},\{\}\right),\text{ph}\left(\text{m1}^2 x(1) x(2)^2+\text{m1}^2 x(1) x(2) x(3)+\text{m1}^2 x(2)^2 x(4)+\text{m1}^2 x(1) x(2) x(4)+\text{m1}^2 x(2) x(3) x(4)+\text{m1}^2 x(2)^2 x(5)+\text{m1}^2 x(1) x(2) x(5)+\text{m1}^2 x(2) x(3) x(5)+\text{m2}^2 x(1) x(4)^2+\text{m2}^2 x(2) x(4)^2+\text{m2}^2 x(3) x(4)^2+\text{m2}^2 x(1) x(2) x(4)+\text{m2}^2 x(1) x(3) x(4)+\text{m2}^2 x(1) x(4) x(5)+\text{m2}^2 x(2) x(4) x(5)+\text{m2}^2 x(3) x(4) x(5)+\text{m3}^2 x(1)^2 x(2)+\text{m3}^2 x(1)^2 x(3)+\text{m3}^2 x(1)^2 x(4)+\text{m3}^2 x(1) x(2) x(4)+\text{m3}^2 x(1) x(3) x(4)+\text{m3}^2 x(1)^2 x(5)+\text{m3}^2 x(1) x(2) x(5)+\text{m3}^2 x(1) x(3) x(5)-q^2 x(1) x(2) x(3)-q^2 x(1) x(3) x(4)-q^2 x(2) x(3) x(4)-q^2 x(1) x(2) x(5)-q^2 x(2) x(3) x(5)-q^2 x(1) x(4) x(5)-q^2 x(2) x(4) x(5)-q^2 x(3) x(4) x(5)+x(1) x(2)+x(1) x(3)+x(1) x(4)+x(2) x(4)+x(3) x(4)+x(1) x(5)+x(2) x(5)+x(3) x(5),\left(
\begin{array}{ccccc}
x(5) & x(1) & x(3) & x(2) & x(4) \\
\frac{1}{((\text{p1}-\text{p2})^2-\text{m3}^2+i \eta )} & \frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )} & \frac{1}{((\text{p1}-q)^2+i \eta )} & \frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )} & \frac{1}{((\text{p2}-q)^2+i \eta )} \\
1 & 1 & 1 & 1 & 1 \\
\end{array}
\right)\right)\right\} ⎩ ⎨ ⎧ FCTopology ( prop2Lv1 , { ( p1 2 − m1 2 + i η ) 1 , ( p2 2 − m2 2 + i η ) 1 , (( p1 − q ) 2 + i η ) 1 , (( p2 − q ) 2 + i η ) 1 , (( p1 − p2 ) 2 − m3 2 + i η ) 1 } , { p1 , p2 } , { Q } , { } , { } ) , ph m1 2 x ( 1 ) x ( 2 ) 2 + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 2 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 2 ) x ( 4 ) + m1 2 x ( 2 ) x ( 3 ) x ( 4 ) + m1 2 x ( 2 ) 2 x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 5 ) + m1 2 x ( 2 ) x ( 3 ) x ( 5 ) + m2 2 x ( 1 ) x ( 4 ) 2 + m2 2 x ( 2 ) x ( 4 ) 2 + m2 2 x ( 3 ) x ( 4 ) 2 + m2 2 x ( 1 ) x ( 2 ) x ( 4 ) + m2 2 x ( 1 ) x ( 3 ) x ( 4 ) + m2 2 x ( 1 ) x ( 4 ) x ( 5 ) + m2 2 x ( 2 ) x ( 4 ) x ( 5 ) + m2 2 x ( 3 ) x ( 4 ) x ( 5 ) + m3 2 x ( 1 ) 2 x ( 2 ) + m3 2 x ( 1 ) 2 x ( 3 ) + m3 2 x ( 1 ) 2 x ( 4 ) + m3 2 x ( 1 ) x ( 2 ) x ( 4 ) + m3 2 x ( 1 ) x ( 3 ) x ( 4 ) + m3 2 x ( 1 ) 2 x ( 5 ) + m3 2 x ( 1 ) x ( 2 ) x ( 5 ) + m3 2 x ( 1 ) x ( 3 ) x ( 5 ) − q 2 x ( 1 ) x ( 2 ) x ( 3 ) − q 2 x ( 1 ) x ( 3 ) x ( 4 ) − q 2 x ( 2 ) x ( 3 ) x ( 4 ) − q 2 x ( 1 ) x ( 2 ) x ( 5 ) − q 2 x ( 2 ) x ( 3 ) x ( 5 ) − q 2 x ( 1 ) x ( 4 ) x ( 5 ) − q 2 x ( 2 ) x ( 4 ) x ( 5 ) − q 2 x ( 3 ) x ( 4 ) x ( 5 ) + x ( 1 ) x ( 2 ) + x ( 1 ) x ( 3 ) + x ( 1 ) x ( 4 ) + x ( 2 ) x ( 4 ) + x ( 3 ) x ( 4 ) + x ( 1 ) x ( 5 ) + x ( 2 ) x ( 5 ) + x ( 3 ) x ( 5 ) , x ( 5 ) (( p1 − p2 ) 2 − m3 2 + i η ) 1 1 x ( 1 ) ( p1 2 − m1 2 + i η ) 1 1 x ( 3 ) (( p1 − q ) 2 + i η ) 1 1 x ( 2 ) ( p2 2 − m2 2 + i η ) 1 1 x ( 4 ) (( p2 − q ) 2 + i η ) 1 1 ⎭ ⎬ ⎫
FCLoopToPakForm[{ GLI[ "prop2Lv1" , { 1 , 1 , 1 , 1 , 0 }], GLI[ "prop2Lv2" , { 1 , 1 , 0 , 0 , 1 }]}, { topo1, topo2}, Names -> x , Head -> ph, Power -> pow]
( G prop2Lv1 ( 1 , 1 , 1 , 1 , 0 ) ph ( m1 2 x ( 1 ) 2 x ( 3 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 2 ) x ( 4 ) + m2 2 x ( 1 ) x ( 3 ) 2 + m2 2 x ( 2 ) x ( 3 ) 2 + m2 2 x ( 1 ) x ( 3 ) x ( 4 ) + m2 2 x ( 2 ) x ( 3 ) x ( 4 ) − q 2 x ( 1 ) x ( 2 ) x ( 3 ) − q 2 x ( 1 ) x ( 2 ) x ( 4 ) − q 2 x ( 1 ) x ( 3 ) x ( 4 ) − q 2 x ( 2 ) x ( 3 ) x ( 4 ) + x ( 1 ) x ( 3 ) + x ( 2 ) x ( 3 ) + x ( 1 ) x ( 4 ) + x ( 2 ) x ( 4 ) , ( x ( 1 ) x ( 3 ) x ( 2 ) x ( 4 ) 1 ( p1 2 − m1 2 + i η ) 1 ( ( p1 − q ) 2 + i η ) 1 ( p2 2 − m2 2 + i η ) 1 ( ( p2 − q ) 2 + i η ) 1 1 1 1 ) ) G prop2Lv2 ( 1 , 1 , 0 , 0 , 1 ) ph ( m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) 2 x ( 3 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) x ( 3 ) + x ( 1 ) x ( 2 ) + x ( 1 ) x ( 3 ) + x ( 2 ) x ( 3 ) , ( x ( 1 ) x ( 3 ) x ( 2 ) 1 ( p1 2 − m1 2 + i η ) 1 ( p2 2 − m2 2 + i η ) 1 ( ( p1 − p2 ) 2 + i η ) 1 1 1 ) ) ) \left(
\begin{array}{cc}
G^{\text{prop2Lv1}}(1,1,1,1,0) & \;\text{ph}\left(\text{m1}^2 x(1)^2 x(3)+\text{m1}^2 x(1) x(2) x(3)+\text{m1}^2 x(1)^2 x(4)+\text{m1}^2 x(1) x(2) x(4)+\text{m2}^2 x(1) x(3)^2+\text{m2}^2 x(2) x(3)^2+\text{m2}^2 x(1) x(3) x(4)+\text{m2}^2 x(2) x(3) x(4)-q^2 x(1) x(2) x(3)-q^2 x(1) x(2) x(4)-q^2 x(1) x(3) x(4)-q^2 x(2) x(3) x(4)+x(1) x(3)+x(2) x(3)+x(1) x(4)+x(2) x(4),\left(
\begin{array}{cccc}
x(1) & x(3) & x(2) & x(4) \\
\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )} & \frac{1}{((\text{p1}-q)^2+i \eta )} & \frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )} & \frac{1}{((\text{p2}-q)^2+i \eta )} \\
1 & 1 & 1 & 1 \\
\end{array}
\right)\right) \\
G^{\text{prop2Lv2}}(1,1,0,0,1) & \;\text{ph}\left(\text{m1}^2 x(1)^2 x(2)+\text{m1}^2 x(1)^2 x(3)+\text{m1}^2 x(1) x(2) x(3)+\text{m2}^2 x(1) x(2)^2+\text{m2}^2 x(2)^2 x(3)+\text{m2}^2 x(1) x(2) x(3)+x(1) x(2)+x(1) x(3)+x(2) x(3),\left(
\begin{array}{ccc}
x(1) & x(3) & x(2) \\
\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )} & \frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )} & \frac{1}{((\text{p1}-\text{p2})^2+i \eta )} \\
1 & 1 & 1 \\
\end{array}
\right)\right) \\
\end{array}
\right) G prop2Lv1 ( 1 , 1 , 1 , 1 , 0 ) G prop2Lv2 ( 1 , 1 , 0 , 0 , 1 ) ph m1 2 x ( 1 ) 2 x ( 3 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 2 ) x ( 4 ) + m2 2 x ( 1 ) x ( 3 ) 2 + m2 2 x ( 2 ) x ( 3 ) 2 + m2 2 x ( 1 ) x ( 3 ) x ( 4 ) + m2 2 x ( 2 ) x ( 3 ) x ( 4 ) − q 2 x ( 1 ) x ( 2 ) x ( 3 ) − q 2 x ( 1 ) x ( 2 ) x ( 4 ) − q 2 x ( 1 ) x ( 3 ) x ( 4 ) − q 2 x ( 2 ) x ( 3 ) x ( 4 ) + x ( 1 ) x ( 3 ) + x ( 2 ) x ( 3 ) + x ( 1 ) x ( 4 ) + x ( 2 ) x ( 4 ) , x ( 1 ) ( p1 2 − m1 2 + i η ) 1 1 x ( 3 ) (( p1 − q ) 2 + i η ) 1 1 x ( 2 ) ( p2 2 − m2 2 + i η ) 1 1 x ( 4 ) (( p2 − q ) 2 + i η ) 1 1 ph m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) 2 x ( 3 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) x ( 3 ) + x ( 1 ) x ( 2 ) + x ( 1 ) x ( 3 ) + x ( 2 ) x ( 3 ) , x ( 1 ) ( p1 2 − m1 2 + i η ) 1 1 x ( 3 ) ( p2 2 − m2 2 + i η ) 1 1 x ( 2 ) (( p1 − p2 ) 2 + i η ) 1 1
Products of GLI
s are also supported.
FCLoopToPakForm[{ GLI[ "prop2Lv1" , { 1 , 1 , 0 , 0 , 0 }] ^ 2 }, { topo1, topo2}, Names -> x , Head -> ph, Power -> pow]
( G prop2Lv1 ( 1 , 1 , 0 , 0 , 0 ) 2 ph ( m1 2 x ( 2 ) x ( 3 ) x ( 4 ) x ( 1 ) 2 + m1 2 x ( 2 ) 2 x ( 3 ) x ( 4 ) x ( 1 ) + m2 2 x ( 2 ) x ( 3 ) x ( 4 ) 2 x ( 1 ) + m2 2 x ( 2 ) x ( 3 ) 2 x ( 4 ) x ( 1 ) + x ( 2 ) x ( 3 ) x ( 4 ) x ( 1 ) , ( x ( 1 ) x ( 3 ) x ( 2 ) x ( 4 ) 1 ( FCGV ( lmom ) ( 1 , 1 ) 2 − m1 2 + i η ) 1 ( FCGV ( lmom ) ( 2 , 1 ) 2 − m1 2 + i η ) 1 ( FCGV ( lmom ) ( 1 , 2 ) 2 − m2 2 + i η ) 1 ( FCGV ( lmom ) ( 2 , 2 ) 2 − m2 2 + i η ) 1 1 1 1 ) ) ) \left(
\begin{array}{cc}
G^{\text{prop2Lv1}}(1,1,0,0,0)^2 & \;\text{ph}\left(\text{m1}^2 x(2) x(3) x(4) x(1)^2+\text{m1}^2 x(2)^2 x(3) x(4) x(1)+\text{m2}^2 x(2) x(3) x(4)^2 x(1)+\text{m2}^2 x(2) x(3)^2 x(4) x(1)+x(2) x(3) x(4) x(1),\left(
\begin{array}{cccc}
x(1) & x(3) & x(2) & x(4) \\
\frac{1}{(\text{FCGV}(\text{lmom})(1,1)^2-\text{m1}^2+i \eta )} & \frac{1}{(\text{FCGV}(\text{lmom})(2,1)^2-\text{m1}^2+i \eta )} & \frac{1}{(\text{FCGV}(\text{lmom})(1,2)^2-\text{m2}^2+i \eta )} & \frac{1}{(\text{FCGV}(\text{lmom})(2,2)^2-\text{m2}^2+i \eta )} \\
1 & 1 & 1 & 1 \\
\end{array}
\right)\right) \\
\end{array}
\right) G prop2Lv1 ( 1 , 1 , 0 , 0 , 0 ) 2 ph m1 2 x ( 2 ) x ( 3 ) x ( 4 ) x ( 1 ) 2 + m1 2 x ( 2 ) 2 x ( 3 ) x ( 4 ) x ( 1 ) + m2 2 x ( 2 ) x ( 3 ) x ( 4 ) 2 x ( 1 ) + m2 2 x ( 2 ) x ( 3 ) 2 x ( 4 ) x ( 1 ) + x ( 2 ) x ( 3 ) x ( 4 ) x ( 1 ) , x ( 1 ) ( FCGV ( lmom ) ( 1 , 1 ) 2 − m1 2 + i η ) 1 1 x ( 3 ) ( FCGV ( lmom ) ( 2 , 1 ) 2 − m1 2 + i η ) 1 1 x ( 2 ) ( FCGV ( lmom ) ( 1 , 2 ) 2 − m2 2 + i η ) 1 1 x ( 4 ) ( FCGV ( lmom ) ( 2 , 2 ) 2 − m2 2 + i η ) 1 1