FCFeynmanPrepare
FCFeynmanPrepare[int, {q1, q2, ...}]
is an auxiliary function that returns all necessary building for writing down a Feynman parametrization of the given tensor or scalar multi-loop integral. The integral int can be Lorentzian or Cartesian.
The output of the function is a list given by {U,F, pows, M, Q, J, N, r}
, where U
and F
are the Symanzik polynomials, with U = d e t M U = det M U = d e tM , while pows
contains the powers of the occurring propagators. The vector Q
and the function J
are the usual quantities appearing in the definition of the F`` polynomial.
If the integral has free indices, then N
encodes its tensor structure, while r
gives its tensor rank. For scalar integrals N
is always 1
and r is 0
. In N
the F
-polynomial is not substituted but left as FCGV["F"]
.
To ensure a certain correspondence between propagators and Feynman parameters, it is also possible to enter the integral as a list of propagators, e.g. FCFeynmanPrepare[{FAD[{q,m1}],FAD[{q-p,m2}],SPD[p,q]},{q}]
. In this case the tensor part of the integral should be the very last element of the list.
It is also possible to invoke the function as FCFeynmanPrepare[GLI[...], FCTopology[...]]
or FCFeynmanPrepare[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.
The definitions of M
, Q
, J
and N
follow from Eq. 4.17 in the PhD Thesis of Stefan Jahn and arXiv:1010.1667 .The algorithm for deriving the UF-parametrization of a loop integral was adopted from the UF generator available in multiple codes of Alexander Smirnov, such as FIESTA (arXiv:1511.03614 ) and FIRE (arXiv:1901.07808 ). The code UF.m is also mentioned in the book “Analytic Tools for Feynman Integrals” by Vladimir Smirnov, Chapter 2.3.
See also
Overview , FCFeynmanParametrize , FCFeynmanProjectivize , FCLoopValidTopologyQ .
Examples
One of the simplest examples is the 1-loop tadpole
FCFeynmanPrepare[ FAD[{ q , m1}], { q }]
{ FCGV ( x ) ( 1 ) , m1 2 ( FCGV ( x ) ( 1 ) ) 2 , ( FCGV ( x ) ( 1 ) 1 q 2 − m1 2 1 ) , ( FCGV ( x ) ( 1 ) ) , { 0 } , − m1 2 FCGV ( x ) ( 1 ) , 1 , 0 } \left\{\text{FCGV}(\text{x})(1),\text{m1}^2 (\text{FCGV}(\text{x})(1))^2,\left(
\begin{array}{ccc}
\;\text{FCGV}(\text{x})(1) & \frac{1}{q^2-\text{m1}^2} & 1 \\
\end{array}
\right),\left(
\begin{array}{c}
\;\text{FCGV}(\text{x})(1) \\
\end{array}
\right),\{0\},-\text{m1}^2 \;\text{FCGV}(\text{x})(1),1,0\right\} { FCGV ( x ) ( 1 ) , m1 2 ( FCGV ( x ) ( 1 ) ) 2 , ( FCGV ( x ) ( 1 ) q 2 − m1 2 1 1 ) , ( FCGV ( x ) ( 1 ) ) , { 0 } , − m1 2 FCGV ( x ) ( 1 ) , 1 , 0 }
Use the option Names
to have specific symbols denoting Feynman parameters
FCFeynmanPrepare[ FAD[{ q , m1}], { q }, Names -> x ]
{ x ( 1 ) , m1 2 x ( 1 ) 2 , ( x ( 1 ) 1 q 2 − m1 2 1 ) , ( x ( 1 ) ) , { 0 } , − m1 2 x ( 1 ) , 1 , 0 } \left\{x(1),\text{m1}^2 x(1)^2,\left(
\begin{array}{ccc}
x(1) & \frac{1}{q^2-\text{m1}^2} & 1 \\
\end{array}
\right),\left(
\begin{array}{c}
x(1) \\
\end{array}
\right),\{0\},-\text{m1}^2 x(1),1,0\right\} { x ( 1 ) , m1 2 x ( 1 ) 2 , ( x ( 1 ) q 2 − m1 2 1 1 ) , ( x ( 1 ) ) , { 0 } , − m1 2 x ( 1 ) , 1 , 0 }
It is also possible to obtain e.g. x1, x2, x3, ...
instead of x[1], x[2], x[3], ...
FCFeynmanPrepare[ FAD[{ q , m1}], { q }, Names -> x , Indexed -> False ]
{ x1 , m1 2 x1 2 , ( x1 1 q 2 − m1 2 1 ) , ( x1 ) , { 0 } , − m1 2 x1 , 1 , 0 } \left\{\text{x1},\text{m1}^2 \;\text{x1}^2,\left(
\begin{array}{ccc}
\;\text{x1} & \frac{1}{q^2-\text{m1}^2} & 1 \\
\end{array}
\right),\left(
\begin{array}{c}
\;\text{x1} \\
\end{array}
\right),\{0\},-\text{m1}^2 \;\text{x1},1,0\right\} { x1 , m1 2 x1 2 , ( x1 q 2 − m1 2 1 1 ) , ( x1 ) , { 0 } , − m1 2 x1 , 1 , 0 }
To fix the correspondence between Feynman parameters and propagators, the latter should be entered as a list
FCFeynmanPrepare[{ FAD[{ q , m }], FAD[{ q - p , m2}], FVD[ q , \ [ Mu]] FVD[ q , \ [ Nu]] FVD[ q , \ [ Rho]]}, { q }, Names -> x ]
{ x ( 1 ) + x ( 2 ) , m 2 x ( 1 ) 2 + m 2 x ( 1 ) x ( 2 ) + m2 2 x ( 2 ) 2 + m2 2 x ( 1 ) x ( 2 ) − p 2 x ( 1 ) x ( 2 ) , ( x ( 1 ) 1 q 2 − m 2 1 x ( 2 ) 1 ( p − q ) 2 − m2 2 1 ) , ( x ( 1 ) + x ( 2 ) ) , { x ( 2 ) p FCGV ( mu ) } , m 2 ( − x ( 1 ) ) − m2 2 x ( 2 ) + p 2 x ( 2 ) , − 1 2 x ( 2 ) Γ ( 1 − D 2 ) FCGV ( F ) p μ g ν ρ − 1 2 x ( 2 ) Γ ( 1 − D 2 ) FCGV ( F ) p ν g μ ρ − 1 2 x ( 2 ) Γ ( 1 − D 2 ) FCGV ( F ) p ρ g μ ν + x ( 2 ) 3 Γ ( 2 − D 2 ) p μ p ν p ρ , 3 } \left\{x(1)+x(2),m^2 x(1)^2+m^2 x(1) x(2)+\text{m2}^2 x(2)^2+\text{m2}^2 x(1) x(2)-p^2 x(1) x(2),\left(
\begin{array}{ccc}
x(1) & \frac{1}{q^2-m^2} & 1 \\
x(2) & \frac{1}{(p-q)^2-\text{m2}^2} & 1 \\
\end{array}
\right),\left(
\begin{array}{c}
x(1)+x(2) \\
\end{array}
\right),\left\{x(2) p^{\text{FCGV}(\text{mu})}\right\},m^2 (-x(1))-\text{m2}^2 x(2)+p^2 x(2),-\frac{1}{2} x(2) \Gamma \left(1-\frac{D}{2}\right) \;\text{FCGV}(\text{F}) p^{\mu } g^{\nu \rho }-\frac{1}{2} x(2) \Gamma \left(1-\frac{D}{2}\right) \;\text{FCGV}(\text{F}) p^{\nu } g^{\mu \rho }-\frac{1}{2} x(2) \Gamma \left(1-\frac{D}{2}\right) \;\text{FCGV}(\text{F}) p^{\rho } g^{\mu \nu }+x(2)^3 \Gamma \left(2-\frac{D}{2}\right) p^{\mu } p^{\nu } p^{\rho },3\right\} { x ( 1 ) + x ( 2 ) , m 2 x ( 1 ) 2 + m 2 x ( 1 ) x ( 2 ) + m2 2 x ( 2 ) 2 + m2 2 x ( 1 ) x ( 2 ) − p 2 x ( 1 ) x ( 2 ) , ( x ( 1 ) x ( 2 ) q 2 − m 2 1 ( p − q ) 2 − m2 2 1 1 1 ) , ( x ( 1 ) + x ( 2 ) ) , { x ( 2 ) p FCGV ( mu ) } , m 2 ( − x ( 1 )) − m2 2 x ( 2 ) + p 2 x ( 2 ) , − 2 1 x ( 2 ) Γ ( 1 − 2 D ) FCGV ( F ) p μ g ν ρ − 2 1 x ( 2 ) Γ ( 1 − 2 D ) FCGV ( F ) p ν g μ ρ − 2 1 x ( 2 ) Γ ( 1 − 2 D ) FCGV ( F ) p ρ g μν + x ( 2 ) 3 Γ ( 2 − 2 D ) p μ p ν p ρ , 3 }
Massless 2-loop self-energy
FCFeynmanPrepare[ FAD[ p1, p2, Q - p1 - p2, Q - p1, Q - p2], { p1, p2}, Names -> x ]
{ x ( 1 ) x ( 2 ) + x ( 3 ) x ( 2 ) + x ( 5 ) x ( 2 ) + x ( 1 ) x ( 4 ) + x ( 3 ) x ( 4 ) + x ( 1 ) x ( 5 ) + x ( 3 ) x ( 5 ) + x ( 4 ) x ( 5 ) , − Q 2 ( x ( 1 ) x ( 2 ) x ( 3 ) + x ( 1 ) x ( 4 ) x ( 3 ) + x ( 2 ) x ( 4 ) x ( 3 ) + x ( 1 ) x ( 5 ) x ( 3 ) + x ( 4 ) x ( 5 ) x ( 3 ) + x ( 1 ) x ( 2 ) x ( 4 ) + x ( 1 ) x ( 2 ) x ( 5 ) + x ( 2 ) x ( 4 ) x ( 5 ) ) , ( x ( 1 ) 1 p1 2 1 x ( 2 ) 1 p2 2 1 x ( 3 ) 1 ( p1 − Q ) 2 1 x ( 4 ) 1 ( p2 − Q ) 2 1 x ( 5 ) 1 ( p1 + p2 − Q ) 2 1 ) , ( x ( 1 ) + x ( 3 ) + x ( 5 ) x ( 5 ) x ( 5 ) x ( 2 ) + x ( 4 ) + x ( 5 ) ) , { ( x ( 3 ) + x ( 5 ) ) Q FCGV ( mu ) , ( x ( 4 ) + x ( 5 ) ) Q FCGV ( mu ) } , Q 2 ( x ( 3 ) + x ( 4 ) + x ( 5 ) ) , 1 , 0 } \left\{x(1) x(2)+x(3) x(2)+x(5) x(2)+x(1) x(4)+x(3) x(4)+x(1) x(5)+x(3) x(5)+x(4) x(5),-Q^2 (x(1) x(2) x(3)+x(1) x(4) x(3)+x(2) x(4) x(3)+x(1) x(5) x(3)+x(4) x(5) x(3)+x(1) x(2) x(4)+x(1) x(2) x(5)+x(2) x(4) x(5)),\left(
\begin{array}{ccc}
x(1) & \frac{1}{\text{p1}^2} & 1 \\
x(2) & \frac{1}{\text{p2}^2} & 1 \\
x(3) & \frac{1}{(\text{p1}-Q)^2} & 1 \\
x(4) & \frac{1}{(\text{p2}-Q)^2} & 1 \\
x(5) & \frac{1}{(\text{p1}+\text{p2}-Q)^2} & 1 \\
\end{array}
\right),\left(
\begin{array}{cc}
x(1)+x(3)+x(5) & x(5) \\
x(5) & x(2)+x(4)+x(5) \\
\end{array}
\right),\left\{(x(3)+x(5)) Q^{\text{FCGV}(\text{mu})},(x(4)+x(5)) Q^{\text{FCGV}(\text{mu})}\right\},Q^2 (x(3)+x(4)+x(5)),1,0\right\} ⎩ ⎨ ⎧ x ( 1 ) x ( 2 ) + x ( 3 ) x ( 2 ) + x ( 5 ) x ( 2 ) + x ( 1 ) x ( 4 ) + x ( 3 ) x ( 4 ) + x ( 1 ) x ( 5 ) + x ( 3 ) x ( 5 ) + x ( 4 ) x ( 5 ) , − Q 2 ( x ( 1 ) x ( 2 ) x ( 3 ) + x ( 1 ) x ( 4 ) x ( 3 ) + x ( 2 ) x ( 4 ) x ( 3 ) + x ( 1 ) x ( 5 ) x ( 3 ) + x ( 4 ) x ( 5 ) x ( 3 ) + x ( 1 ) x ( 2 ) x ( 4 ) + x ( 1 ) x ( 2 ) x ( 5 ) + x ( 2 ) x ( 4 ) x ( 5 )) , x ( 1 ) x ( 2 ) x ( 3 ) x ( 4 ) x ( 5 ) p1 2 1 p2 2 1 ( p1 − Q ) 2 1 ( p2 − Q ) 2 1 ( p1 + p2 − Q ) 2 1 1 1 1 1 1 , ( x ( 1 ) + x ( 3 ) + x ( 5 ) x ( 5 ) x ( 5 ) x ( 2 ) + x ( 4 ) + x ( 5 ) ) , { ( x ( 3 ) + x ( 5 )) Q FCGV ( mu ) , ( x ( 4 ) + x ( 5 )) Q FCGV ( mu ) } , Q 2 ( x ( 3 ) + x ( 4 ) + x ( 5 )) , 1 , 0 ⎭ ⎬ ⎫
Factorizing integrals also work
FCFeynmanPrepare[ FAD[{ p1, m1}, { p2, m2}, Q - p1, Q - p2], { p1, p2}, Names -> x ]
{ ( x ( 1 ) + x ( 3 ) ) ( x ( 2 ) + x ( 4 ) ) , m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) 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 ) 1 p1 2 − m1 2 1 x ( 2 ) 1 p2 2 − m2 2 1 x ( 3 ) 1 ( p1 − Q ) 2 1 x ( 4 ) 1 ( p2 − Q ) 2 1 ) , ( x ( 1 ) + x ( 3 ) 0 0 x ( 2 ) + x ( 4 ) ) , { x ( 3 ) Q FCGV ( mu ) , x ( 4 ) Q FCGV ( mu ) } , m1 2 ( − x ( 1 ) ) − m2 2 x ( 2 ) + Q 2 x ( 3 ) + Q 2 x ( 4 ) , 1 , 0 } \left\{(x(1)+x(3)) (x(2)+x(4)),\text{m1}^2 x(1)^2 x(2)+\text{m1}^2 x(1) x(2) x(3)+\text{m1}^2 x(1)^2 x(4)+\text{m1}^2 x(1) x(3) x(4)+\text{m2}^2 x(1) x(2)^2+\text{m2}^2 x(2)^2 x(3)+\text{m2}^2 x(1) x(2) 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),\left(
\begin{array}{ccc}
x(1) & \frac{1}{\text{p1}^2-\text{m1}^2} & 1 \\
x(2) & \frac{1}{\text{p2}^2-\text{m2}^2} & 1 \\
x(3) & \frac{1}{(\text{p1}-Q)^2} & 1 \\
x(4) & \frac{1}{(\text{p2}-Q)^2} & 1 \\
\end{array}
\right),\left(
\begin{array}{cc}
x(1)+x(3) & 0 \\
0 & x(2)+x(4) \\
\end{array}
\right),\left\{x(3) Q^{\text{FCGV}(\text{mu})},x(4) Q^{\text{FCGV}(\text{mu})}\right\},\text{m1}^2 (-x(1))-\text{m2}^2 x(2)+Q^2 x(3)+Q^2 x(4),1,0\right\} ⎩ ⎨ ⎧ ( x ( 1 ) + x ( 3 )) ( x ( 2 ) + x ( 4 )) , m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) 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 ( 2 ) x ( 3 ) x ( 4 ) p1 2 − m1 2 1 p2 2 − m2 2 1 ( p1 − Q ) 2 1 ( p2 − Q ) 2 1 1 1 1 1 , ( x ( 1 ) + x ( 3 ) 0 0 x ( 2 ) + x ( 4 ) ) , { x ( 3 ) Q FCGV ( mu ) , x ( 4 ) Q FCGV ( mu ) } , m1 2 ( − x ( 1 )) − m2 2 x ( 2 ) + Q 2 x ( 3 ) + Q 2 x ( 4 ) , 1 , 0 ⎭ ⎬ ⎫
Cartesian propagators are equally supported
FCFeynmanPrepare[ CSPD[ q , p ] CFAD[{ q , m }, { q - p , m2}], { q }, Names -> x ]
{ x ( 1 ) + x ( 2 ) , 1 4 ( 4 m x ( 1 ) 2 + 4 m x ( 2 ) x ( 1 ) + 4 m2 x ( 2 ) x ( 1 ) + 4 m2 x ( 2 ) 2 + 4 p 2 x ( 2 ) x ( 1 ) − p 2 x ( 3 ) 2 + 4 p 2 x ( 2 ) x ( 3 ) ) , ( x ( 1 ) 1 ( q 2 + m − i η ) 1 x ( 2 ) 1 ( ( p − q ) 2 + m2 − i η ) 1 x ( 3 ) p ⋅ q − 1 ) , ( x ( 1 ) + x ( 2 ) ) , { 1 2 ( 2 x ( 2 ) − x ( 3 ) ) p FCGV ( i ) } , m x ( 1 ) + m2 x ( 2 ) + p 2 x ( 2 ) , 1 , 0 } \left\{x(1)+x(2),\frac{1}{4} \left(4 m x(1)^2+4 m x(2) x(1)+4 \;\text{m2} x(2) x(1)+4 \;\text{m2} x(2)^2+4 p^2 x(2) x(1)-p^2 x(3)^2+4 p^2 x(2) x(3)\right),\left(
\begin{array}{ccc}
x(1) & \frac{1}{(q^2+m-i \eta )} & 1 \\
x(2) & \frac{1}{((p-q)^2+\text{m2}-i \eta )} & 1 \\
x(3) & p\cdot q & -1 \\
\end{array}
\right),\left(
\begin{array}{c}
x(1)+x(2) \\
\end{array}
\right),\left\{\frac{1}{2} (2 x(2)-x(3)) p^{\text{FCGV}(\text{i})}\right\},m x(1)+\text{m2} x(2)+p^2 x(2),1,0\right\} ⎩ ⎨ ⎧ x ( 1 ) + x ( 2 ) , 4 1 ( 4 m x ( 1 ) 2 + 4 m x ( 2 ) x ( 1 ) + 4 m2 x ( 2 ) x ( 1 ) + 4 m2 x ( 2 ) 2 + 4 p 2 x ( 2 ) x ( 1 ) − p 2 x ( 3 ) 2 + 4 p 2 x ( 2 ) x ( 3 ) ) , x ( 1 ) x ( 2 ) x ( 3 ) ( q 2 + m − i η ) 1 (( p − q ) 2 + m2 − i η ) 1 p ⋅ q 1 1 − 1 , ( x ( 1 ) + x ( 2 ) ) , { 2 1 ( 2 x ( 2 ) − x ( 3 )) p FCGV ( i ) } , m x ( 1 ) + m2 x ( 2 ) + p 2 x ( 2 ) , 1 , 0 ⎭ ⎬ ⎫
FCFeynmanPrepare
also works with FCTopology
and GLI
objects
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 } , { } , { } )
FCFeynmanPrepare[ topo1, Names -> x ]
{ x ( 1 ) x ( 2 ) + x ( 3 ) x ( 2 ) + x ( 5 ) x ( 2 ) + x ( 1 ) x ( 4 ) + x ( 3 ) x ( 4 ) + x ( 1 ) x ( 5 ) + x ( 3 ) x ( 5 ) + x ( 4 ) x ( 5 ) , m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) + m1 2 x ( 1 ) 2 x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 5 ) + m1 2 x ( 1 ) x ( 3 ) x ( 5 ) + m1 2 x ( 1 ) x ( 4 ) x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) x ( 4 ) + m2 2 x ( 2 ) x ( 3 ) x ( 4 ) + m2 2 x ( 2 ) 2 x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) x ( 5 ) + m2 2 x ( 2 ) x ( 3 ) x ( 5 ) + m2 2 x ( 2 ) x ( 4 ) x ( 5 ) + m3 2 x ( 1 ) x ( 5 ) 2 + m3 2 x ( 2 ) x ( 5 ) 2 + m3 2 x ( 3 ) x ( 5 ) 2 + m3 2 x ( 4 ) x ( 5 ) 2 + m3 2 x ( 1 ) x ( 2 ) x ( 5 ) + m3 2 x ( 2 ) x ( 3 ) x ( 5 ) + m3 2 x ( 1 ) x ( 4 ) x ( 5 ) + m3 2 x ( 3 ) x ( 4 ) x ( 5 ) − 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 ) − q 2 x ( 1 ) x ( 3 ) 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 ) , ( x ( 1 ) 1 ( p1 2 − m1 2 + i η ) 1 x ( 2 ) 1 ( p2 2 − m2 2 + i η ) 1 x ( 3 ) 1 ( ( p1 − q ) 2 + i η ) 1 x ( 4 ) 1 ( ( p2 − q ) 2 + i η ) 1 x ( 5 ) 1 ( ( p1 − p2 ) 2 − m3 2 + i η ) 1 ) , ( x ( 1 ) + x ( 3 ) + x ( 5 ) − x ( 5 ) − x ( 5 ) x ( 2 ) + x ( 4 ) + x ( 5 ) ) , { x ( 3 ) q FCGV ( mu ) , x ( 4 ) q FCGV ( mu ) } , m1 2 ( − x ( 1 ) ) − m2 2 x ( 2 ) − m3 2 x ( 5 ) + q 2 x ( 3 ) + q 2 x ( 4 ) , 1 , 0 } \left\{x(1) x(2)+x(3) x(2)+x(5) x(2)+x(1) x(4)+x(3) x(4)+x(1) x(5)+x(3) x(5)+x(4) x(5),\text{m1}^2 x(1)^2 x(2)+\text{m1}^2 x(1) x(2) x(3)+\text{m1}^2 x(1)^2 x(4)+\text{m1}^2 x(1) x(3) x(4)+\text{m1}^2 x(1)^2 x(5)+\text{m1}^2 x(1) x(2) x(5)+\text{m1}^2 x(1) x(3) x(5)+\text{m1}^2 x(1) x(4) x(5)+\text{m2}^2 x(1) x(2)^2+\text{m2}^2 x(2)^2 x(3)+\text{m2}^2 x(1) x(2) x(4)+\text{m2}^2 x(2) x(3) x(4)+\text{m2}^2 x(2)^2 x(5)+\text{m2}^2 x(1) x(2) x(5)+\text{m2}^2 x(2) x(3) x(5)+\text{m2}^2 x(2) x(4) x(5)+\text{m3}^2 x(1) x(5)^2+\text{m3}^2 x(2) x(5)^2+\text{m3}^2 x(3) x(5)^2+\text{m3}^2 x(4) x(5)^2+\text{m3}^2 x(1) x(2) x(5)+\text{m3}^2 x(2) x(3) x(5)+\text{m3}^2 x(1) x(4) x(5)+\text{m3}^2 x(3) x(4) x(5)-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)-q^2 x(1) x(3) 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),\left(
\begin{array}{ccc}
x(1) & \frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )} & 1 \\
x(2) & \frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )} & 1 \\
x(3) & \frac{1}{((\text{p1}-q)^2+i \eta )} & 1 \\
x(4) & \frac{1}{((\text{p2}-q)^2+i \eta )} & 1 \\
x(5) & \frac{1}{((\text{p1}-\text{p2})^2-\text{m3}^2+i \eta )} & 1 \\
\end{array}
\right),\left(
\begin{array}{cc}
x(1)+x(3)+x(5) & -x(5) \\
-x(5) & x(2)+x(4)+x(5) \\
\end{array}
\right),\left\{x(3) q^{\text{FCGV}(\text{mu})},x(4) q^{\text{FCGV}(\text{mu})}\right\},\text{m1}^2 (-x(1))-\text{m2}^2 x(2)-\text{m3}^2 x(5)+q^2 x(3)+q^2 x(4),1,0\right\} ⎩ ⎨ ⎧ x ( 1 ) x ( 2 ) + x ( 3 ) x ( 2 ) + x ( 5 ) x ( 2 ) + x ( 1 ) x ( 4 ) + x ( 3 ) x ( 4 ) + x ( 1 ) x ( 5 ) + x ( 3 ) x ( 5 ) + x ( 4 ) x ( 5 ) , m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) + m1 2 x ( 1 ) 2 x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 5 ) + m1 2 x ( 1 ) x ( 3 ) x ( 5 ) + m1 2 x ( 1 ) x ( 4 ) x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) x ( 4 ) + m2 2 x ( 2 ) x ( 3 ) x ( 4 ) + m2 2 x ( 2 ) 2 x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) x ( 5 ) + m2 2 x ( 2 ) x ( 3 ) x ( 5 ) + m2 2 x ( 2 ) x ( 4 ) x ( 5 ) + m3 2 x ( 1 ) x ( 5 ) 2 + m3 2 x ( 2 ) x ( 5 ) 2 + m3 2 x ( 3 ) x ( 5 ) 2 + m3 2 x ( 4 ) x ( 5 ) 2 + m3 2 x ( 1 ) x ( 2 ) x ( 5 ) + m3 2 x ( 2 ) x ( 3 ) x ( 5 ) + m3 2 x ( 1 ) x ( 4 ) x ( 5 ) + m3 2 x ( 3 ) x ( 4 ) x ( 5 ) − 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 ) − q 2 x ( 1 ) x ( 3 ) 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 ) , x ( 1 ) x ( 2 ) x ( 3 ) x ( 4 ) x ( 5 ) ( 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 1 1 1 1 1 , ( x ( 1 ) + x ( 3 ) + x ( 5 ) − x ( 5 ) − x ( 5 ) x ( 2 ) + x ( 4 ) + x ( 5 ) ) , { x ( 3 ) q FCGV ( mu ) , x ( 4 ) q FCGV ( mu ) } , m1 2 ( − x ( 1 )) − m2 2 x ( 2 ) − m3 2 x ( 5 ) + q 2 x ( 3 ) + q 2 x ( 4 ) , 1 , 0 ⎭ ⎬ ⎫
FCFeynmanPrepare[{ topo1, topo2}, Names -> x ]
( x ( 1 ) x ( 2 ) + x ( 3 ) x ( 2 ) + x ( 5 ) x ( 2 ) + x ( 1 ) x ( 4 ) + x ( 3 ) x ( 4 ) + x ( 1 ) x ( 5 ) + x ( 3 ) x ( 5 ) + x ( 4 ) x ( 5 ) m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) + m1 2 x ( 1 ) 2 x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 5 ) + m1 2 x ( 1 ) x ( 3 ) x ( 5 ) + m1 2 x ( 1 ) x ( 4 ) x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) x ( 4 ) + m2 2 x ( 2 ) x ( 3 ) x ( 4 ) + m2 2 x ( 2 ) 2 x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) x ( 5 ) + m2 2 x ( 2 ) x ( 3 ) x ( 5 ) + m2 2 x ( 2 ) x ( 4 ) x ( 5 ) + m3 2 x ( 1 ) x ( 5 ) 2 + m3 2 x ( 2 ) x ( 5 ) 2 + m3 2 x ( 3 ) x ( 5 ) 2 + m3 2 x ( 4 ) x ( 5 ) 2 + m3 2 x ( 1 ) x ( 2 ) x ( 5 ) + m3 2 x ( 2 ) x ( 3 ) x ( 5 ) + m3 2 x ( 1 ) x ( 4 ) x ( 5 ) + m3 2 x ( 3 ) x ( 4 ) x ( 5 ) − 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 ) − q 2 x ( 1 ) x ( 3 ) 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 ) ( x ( 1 ) 1 ( p1 2 − m1 2 + i η ) 1 x ( 2 ) 1 ( p2 2 − m2 2 + i η ) 1 x ( 3 ) 1 ( ( p1 − q ) 2 + i η ) 1 x ( 4 ) 1 ( ( p2 − q ) 2 + i η ) 1 x ( 5 ) 1 ( ( p1 − p2 ) 2 − m3 2 + i η ) 1 ) ( x ( 1 ) + x ( 3 ) + x ( 5 ) − x ( 5 ) − x ( 5 ) x ( 2 ) + x ( 4 ) + x ( 5 ) ) { x ( 3 ) q FCGV ( mu ) , x ( 4 ) q FCGV ( mu ) } m1 2 ( − x ( 1 ) ) − m2 2 x ( 2 ) − m3 2 x ( 5 ) + q 2 x ( 3 ) + q 2 x ( 4 ) 1 0 x ( 1 ) x ( 2 ) + x ( 3 ) x ( 2 ) + x ( 5 ) x ( 2 ) + x ( 1 ) x ( 4 ) + x ( 3 ) x ( 4 ) + x ( 1 ) x ( 5 ) + x ( 3 ) x ( 5 ) + x ( 4 ) x ( 5 ) M 2 x ( 2 ) x ( 3 ) 2 + M 2 x ( 1 ) x ( 4 ) 2 + M 2 x ( 3 ) x ( 4 ) 2 + M 2 x ( 1 ) x ( 2 ) x ( 3 ) + M 2 x ( 3 ) 2 x ( 4 ) + M 2 x ( 1 ) x ( 2 ) x ( 4 ) + M 2 x ( 1 ) x ( 3 ) x ( 4 ) + M 2 x ( 2 ) x ( 3 ) x ( 4 ) + M 2 x ( 3 ) 2 x ( 5 ) + M 2 x ( 4 ) 2 x ( 5 ) + M 2 x ( 1 ) x ( 3 ) x ( 5 ) + M 2 x ( 2 ) x ( 3 ) x ( 5 ) + M 2 x ( 1 ) x ( 4 ) x ( 5 ) + M 2 x ( 2 ) x ( 4 ) x ( 5 ) + 2 M 2 x ( 3 ) x ( 4 ) x ( 5 ) + m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) + m1 2 x ( 1 ) 2 x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 5 ) + m1 2 x ( 1 ) x ( 3 ) x ( 5 ) + m1 2 x ( 1 ) x ( 4 ) x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) x ( 4 ) + m2 2 x ( 2 ) x ( 3 ) x ( 4 ) + m2 2 x ( 2 ) 2 x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) x ( 5 ) + m2 2 x ( 2 ) x ( 3 ) x ( 5 ) + m2 2 x ( 2 ) x ( 4 ) x ( 5 ) − 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 ) − q 2 x ( 1 ) x ( 3 ) 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 ) ( x ( 1 ) 1 ( p1 2 − m1 2 + i η ) 1 x ( 2 ) 1 ( p2 2 − m2 2 + i η ) 1 x ( 3 ) 1 ( ( p1 − q ) 2 − M 2 + i η ) 1 x ( 4 ) 1 ( ( p2 − q ) 2 − M 2 + i η ) 1 x ( 5 ) 1 ( ( p1 − p2 ) 2 + i η ) 1 ) ( x ( 1 ) + x ( 3 ) + x ( 5 ) − x ( 5 ) − x ( 5 ) x ( 2 ) + x ( 4 ) + x ( 5 ) ) { x ( 3 ) q FCGV ( mu ) , x ( 4 ) q FCGV ( mu ) } M 2 ( − x ( 3 ) ) − M 2 x ( 4 ) − m1 2 x ( 1 ) − m2 2 x ( 2 ) + q 2 x ( 3 ) + q 2 x ( 4 ) 1 0 ) \left(
\begin{array}{cccccccc}
x(1) x(2)+x(3) x(2)+x(5) x(2)+x(1) x(4)+x(3) x(4)+x(1) x(5)+x(3) x(5)+x(4) x(5) & \;\text{m1}^2 x(1)^2 x(2)+\text{m1}^2 x(1) x(2) x(3)+\text{m1}^2 x(1)^2 x(4)+\text{m1}^2 x(1) x(3) x(4)+\text{m1}^2 x(1)^2 x(5)+\text{m1}^2 x(1) x(2) x(5)+\text{m1}^2 x(1) x(3) x(5)+\text{m1}^2 x(1) x(4) x(5)+\text{m2}^2 x(1) x(2)^2+\text{m2}^2 x(2)^2 x(3)+\text{m2}^2 x(1) x(2) x(4)+\text{m2}^2 x(2) x(3) x(4)+\text{m2}^2 x(2)^2 x(5)+\text{m2}^2 x(1) x(2) x(5)+\text{m2}^2 x(2) x(3) x(5)+\text{m2}^2 x(2) x(4) x(5)+\text{m3}^2 x(1) x(5)^2+\text{m3}^2 x(2) x(5)^2+\text{m3}^2 x(3) x(5)^2+\text{m3}^2 x(4) x(5)^2+\text{m3}^2 x(1) x(2) x(5)+\text{m3}^2 x(2) x(3) x(5)+\text{m3}^2 x(1) x(4) x(5)+\text{m3}^2 x(3) x(4) x(5)-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)-q^2 x(1) x(3) 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) & \left(
\begin{array}{ccc}
x(1) & \frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )} & 1 \\
x(2) & \frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )} & 1 \\
x(3) & \frac{1}{((\text{p1}-q)^2+i \eta )} & 1 \\
x(4) & \frac{1}{((\text{p2}-q)^2+i \eta )} & 1 \\
x(5) & \frac{1}{((\text{p1}-\text{p2})^2-\text{m3}^2+i \eta )} & 1 \\
\end{array}
\right) & \left(
\begin{array}{cc}
x(1)+x(3)+x(5) & -x(5) \\
-x(5) & x(2)+x(4)+x(5) \\
\end{array}
\right) & \left\{x(3) q^{\text{FCGV}(\text{mu})},x(4) q^{\text{FCGV}(\text{mu})}\right\} & \;\text{m1}^2 (-x(1))-\text{m2}^2 x(2)-\text{m3}^2 x(5)+q^2 x(3)+q^2 x(4) & 1 & 0 \\
x(1) x(2)+x(3) x(2)+x(5) x(2)+x(1) x(4)+x(3) x(4)+x(1) x(5)+x(3) x(5)+x(4) x(5) & M^2 x(2) x(3)^2+M^2 x(1) x(4)^2+M^2 x(3) x(4)^2+M^2 x(1) x(2) x(3)+M^2 x(3)^2 x(4)+M^2 x(1) x(2) x(4)+M^2 x(1) x(3) x(4)+M^2 x(2) x(3) x(4)+M^2 x(3)^2 x(5)+M^2 x(4)^2 x(5)+M^2 x(1) x(3) x(5)+M^2 x(2) x(3) x(5)+M^2 x(1) x(4) x(5)+M^2 x(2) x(4) x(5)+2 M^2 x(3) x(4) x(5)+\text{m1}^2 x(1)^2 x(2)+\text{m1}^2 x(1) x(2) x(3)+\text{m1}^2 x(1)^2 x(4)+\text{m1}^2 x(1) x(3) x(4)+\text{m1}^2 x(1)^2 x(5)+\text{m1}^2 x(1) x(2) x(5)+\text{m1}^2 x(1) x(3) x(5)+\text{m1}^2 x(1) x(4) x(5)+\text{m2}^2 x(1) x(2)^2+\text{m2}^2 x(2)^2 x(3)+\text{m2}^2 x(1) x(2) x(4)+\text{m2}^2 x(2) x(3) x(4)+\text{m2}^2 x(2)^2 x(5)+\text{m2}^2 x(1) x(2) x(5)+\text{m2}^2 x(2) x(3) x(5)+\text{m2}^2 x(2) x(4) x(5)-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)-q^2 x(1) x(3) 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) & \left(
\begin{array}{ccc}
x(1) & \frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )} & 1 \\
x(2) & \frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )} & 1 \\
x(3) & \frac{1}{((\text{p1}-q)^2-M^2+i \eta )} & 1 \\
x(4) & \frac{1}{((\text{p2}-q)^2-M^2+i \eta )} & 1 \\
x(5) & \frac{1}{((\text{p1}-\text{p2})^2+i \eta )} & 1 \\
\end{array}
\right) & \left(
\begin{array}{cc}
x(1)+x(3)+x(5) & -x(5) \\
-x(5) & x(2)+x(4)+x(5) \\
\end{array}
\right) & \left\{x(3) q^{\text{FCGV}(\text{mu})},x(4) q^{\text{FCGV}(\text{mu})}\right\} & M^2 (-x(3))-M^2 x(4)-\text{m1}^2 x(1)-\text{m2}^2 x(2)+q^2 x(3)+q^2 x(4) & 1 & 0 \\
\end{array}
\right) x ( 1 ) x ( 2 ) + x ( 3 ) x ( 2 ) + x ( 5 ) x ( 2 ) + x ( 1 ) x ( 4 ) + x ( 3 ) x ( 4 ) + x ( 1 ) x ( 5 ) + x ( 3 ) x ( 5 ) + x ( 4 ) x ( 5 ) x ( 1 ) x ( 2 ) + x ( 3 ) x ( 2 ) + x ( 5 ) x ( 2 ) + x ( 1 ) x ( 4 ) + x ( 3 ) x ( 4 ) + x ( 1 ) x ( 5 ) + x ( 3 ) x ( 5 ) + x ( 4 ) x ( 5 ) m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) + m1 2 x ( 1 ) 2 x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 5 ) + m1 2 x ( 1 ) x ( 3 ) x ( 5 ) + m1 2 x ( 1 ) x ( 4 ) x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) x ( 4 ) + m2 2 x ( 2 ) x ( 3 ) x ( 4 ) + m2 2 x ( 2 ) 2 x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) x ( 5 ) + m2 2 x ( 2 ) x ( 3 ) x ( 5 ) + m2 2 x ( 2 ) x ( 4 ) x ( 5 ) + m3 2 x ( 1 ) x ( 5 ) 2 + m3 2 x ( 2 ) x ( 5 ) 2 + m3 2 x ( 3 ) x ( 5 ) 2 + m3 2 x ( 4 ) x ( 5 ) 2 + m3 2 x ( 1 ) x ( 2 ) x ( 5 ) + m3 2 x ( 2 ) x ( 3 ) x ( 5 ) + m3 2 x ( 1 ) x ( 4 ) x ( 5 ) + m3 2 x ( 3 ) x ( 4 ) x ( 5 ) − 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 ) − q 2 x ( 1 ) x ( 3 ) 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 ) M 2 x ( 2 ) x ( 3 ) 2 + M 2 x ( 1 ) x ( 4 ) 2 + M 2 x ( 3 ) x ( 4 ) 2 + M 2 x ( 1 ) x ( 2 ) x ( 3 ) + M 2 x ( 3 ) 2 x ( 4 ) + M 2 x ( 1 ) x ( 2 ) x ( 4 ) + M 2 x ( 1 ) x ( 3 ) x ( 4 ) + M 2 x ( 2 ) x ( 3 ) x ( 4 ) + M 2 x ( 3 ) 2 x ( 5 ) + M 2 x ( 4 ) 2 x ( 5 ) + M 2 x ( 1 ) x ( 3 ) x ( 5 ) + M 2 x ( 2 ) x ( 3 ) x ( 5 ) + M 2 x ( 1 ) x ( 4 ) x ( 5 ) + M 2 x ( 2 ) x ( 4 ) x ( 5 ) + 2 M 2 x ( 3 ) x ( 4 ) x ( 5 ) + m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) + m1 2 x ( 1 ) 2 x ( 5 ) + m1 2 x ( 1 ) x ( 2 ) x ( 5 ) + m1 2 x ( 1 ) x ( 3 ) x ( 5 ) + m1 2 x ( 1 ) x ( 4 ) x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) x ( 4 ) + m2 2 x ( 2 ) x ( 3 ) x ( 4 ) + m2 2 x ( 2 ) 2 x ( 5 ) + m2 2 x ( 1 ) x ( 2 ) x ( 5 ) + m2 2 x ( 2 ) x ( 3 ) x ( 5 ) + m2 2 x ( 2 ) x ( 4 ) x ( 5 ) − 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 ) − q 2 x ( 1 ) x ( 3 ) 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 ) x ( 1 ) x ( 2 ) x ( 3 ) x ( 4 ) x ( 5 ) ( 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 1 1 1 1 1 x ( 1 ) x ( 2 ) x ( 3 ) x ( 4 ) x ( 5 ) ( 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 1 1 1 1 1 ( x ( 1 ) + x ( 3 ) + x ( 5 ) − x ( 5 ) − x ( 5 ) x ( 2 ) + x ( 4 ) + x ( 5 ) ) ( x ( 1 ) + x ( 3 ) + x ( 5 ) − x ( 5 ) − x ( 5 ) x ( 2 ) + x ( 4 ) + x ( 5 ) ) { x ( 3 ) q FCGV ( mu ) , x ( 4 ) q FCGV ( mu ) } { x ( 3 ) q FCGV ( mu ) , x ( 4 ) q FCGV ( mu ) } m1 2 ( − x ( 1 )) − m2 2 x ( 2 ) − m3 2 x ( 5 ) + q 2 x ( 3 ) + q 2 x ( 4 ) M 2 ( − x ( 3 )) − M 2 x ( 4 ) − m1 2 x ( 1 ) − m2 2 x ( 2 ) + q 2 x ( 3 ) + q 2 x ( 4 ) 1 1 0 0
FCFeynmanPrepare[{ GLI[ "prop2Lv1" , { 1 , 1 , 1 , 1 , 0 }], GLI[ "prop2Lv2" , { 1 , 1 , 0 , 0 , 1 }]},
{ topo1, topo2}, Names -> x ]
( ( x ( 1 ) + x ( 3 ) ) ( x ( 2 ) + x ( 4 ) ) m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) 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 ) 1 ( p1 2 − m1 2 + i η ) 1 x ( 2 ) 1 ( p2 2 − m2 2 + i η ) 1 x ( 3 ) 1 ( ( p1 − q ) 2 + i η ) 1 x ( 4 ) 1 ( ( p2 − q ) 2 + i η ) 1 ) ( x ( 1 ) + x ( 3 ) 0 0 x ( 2 ) + x ( 4 ) ) { x ( 3 ) q FCGV ( mu ) , x ( 4 ) q FCGV ( mu ) } m1 2 ( − x ( 1 ) ) − m2 2 x ( 2 ) + q 2 x ( 3 ) + q 2 x ( 4 ) 1 0 x ( 1 ) x ( 2 ) + x ( 3 ) x ( 2 ) + x ( 1 ) x ( 3 ) ( x ( 1 ) x ( 2 ) + x ( 3 ) x ( 2 ) + x ( 1 ) x ( 3 ) ) ( m1 2 x ( 1 ) + m2 2 x ( 3 ) ) ( x ( 1 ) 1 ( p1 2 − m1 2 + i η ) 1 x ( 2 ) 1 ( ( p1 − p2 ) 2 + i η ) 1 x ( 3 ) 1 ( p2 2 − m2 2 + i η ) 1 ) ( x ( 1 ) + x ( 2 ) − x ( 2 ) − x ( 2 ) x ( 2 ) + x ( 3 ) ) { 0 , 0 } m1 2 ( − x ( 1 ) ) − m2 2 x ( 3 ) 1 0 ) \left(
\begin{array}{cccccccc}
(x(1)+x(3)) (x(2)+x(4)) & \;\text{m1}^2 x(1)^2 x(2)+\text{m1}^2 x(1) x(2) x(3)+\text{m1}^2 x(1)^2 x(4)+\text{m1}^2 x(1) x(3) x(4)+\text{m2}^2 x(1) x(2)^2+\text{m2}^2 x(2)^2 x(3)+\text{m2}^2 x(1) x(2) 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) & \left(
\begin{array}{ccc}
x(1) & \frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )} & 1 \\
x(2) & \frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )} & 1 \\
x(3) & \frac{1}{((\text{p1}-q)^2+i \eta )} & 1 \\
x(4) & \frac{1}{((\text{p2}-q)^2+i \eta )} & 1 \\
\end{array}
\right) & \left(
\begin{array}{cc}
x(1)+x(3) & 0 \\
0 & x(2)+x(4) \\
\end{array}
\right) & \left\{x(3) q^{\text{FCGV}(\text{mu})},x(4) q^{\text{FCGV}(\text{mu})}\right\} & \;\text{m1}^2 (-x(1))-\text{m2}^2 x(2)+q^2 x(3)+q^2 x(4) & 1 & 0 \\
x(1) x(2)+x(3) x(2)+x(1) x(3) & (x(1) x(2)+x(3) x(2)+x(1) x(3)) \left(\text{m1}^2 x(1)+\text{m2}^2 x(3)\right) & \left(
\begin{array}{ccc}
x(1) & \frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )} & 1 \\
x(2) & \frac{1}{((\text{p1}-\text{p2})^2+i \eta )} & 1 \\
x(3) & \frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )} & 1 \\
\end{array}
\right) & \left(
\begin{array}{cc}
x(1)+x(2) & -x(2) \\
-x(2) & x(2)+x(3) \\
\end{array}
\right) & \{0,0\} & \;\text{m1}^2 (-x(1))-\text{m2}^2 x(3) & 1 & 0 \\
\end{array}
\right) ( x ( 1 ) + x ( 3 )) ( x ( 2 ) + x ( 4 )) x ( 1 ) x ( 2 ) + x ( 3 ) x ( 2 ) + x ( 1 ) x ( 3 ) m1 2 x ( 1 ) 2 x ( 2 ) + m1 2 x ( 1 ) x ( 2 ) x ( 3 ) + m1 2 x ( 1 ) 2 x ( 4 ) + m1 2 x ( 1 ) x ( 3 ) x ( 4 ) + m2 2 x ( 1 ) x ( 2 ) 2 + m2 2 x ( 2 ) 2 x ( 3 ) + m2 2 x ( 1 ) x ( 2 ) 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 ( 2 ) + x ( 3 ) x ( 2 ) + x ( 1 ) x ( 3 )) ( m1 2 x ( 1 ) + m2 2 x ( 3 ) ) x ( 1 ) x ( 2 ) x ( 3 ) x ( 4 ) ( 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 1 1 1 1 x ( 1 ) x ( 2 ) x ( 3 ) ( p1 2 − m1 2 + i η ) 1 (( p1 − p2 ) 2 + i η ) 1 ( p2 2 − m2 2 + i η ) 1 1 1 1 ( x ( 1 ) + x ( 3 ) 0 0 x ( 2 ) + x ( 4 ) ) ( x ( 1 ) + x ( 2 ) − x ( 2 ) − x ( 2 ) x ( 2 ) + x ( 3 ) ) { x ( 3 ) q FCGV ( mu ) , x ( 4 ) q FCGV ( mu ) } { 0 , 0 } m1 2 ( − x ( 1 )) − m2 2 x ( 2 ) + q 2 x ( 3 ) + q 2 x ( 4 ) m1 2 ( − x ( 1 )) − m2 2 x ( 3 ) 1 1 0 0
FCFeynmanPrepare
can also handle products of GLI
s. In this case it will automatically introduce dummy names for the loop momenta (the name generation is controlled by the LoopMomentum
option).
topo = FCTopology[
prop2Ltopo13311, { SFAD[{{ I * p1, 0 }, { - m1^ 2 , - 1 }, 1 }], SFAD[{{ I * (p1 + q1), 0 }, { -
m3^ 2 , - 1 }, 1 }], SFAD[{{ I * p3, 0 }, { - m3^ 2 , - 1 }, 1 }], SFAD[{{ I * (p3 + q1), 0 }, { - m1^ 2 ,
- 1 }, 1 }], SFAD[{{ I * (p1 - p3), 0 }, { - m1^ 2 , - 1 }, 1 }]}, { p1, p3}, { q1}, { SPD[ q1, q1] -> m1^ 2 }, {}]
FCTopology ( prop2Ltopo13311 , { 1 ( − p1 2 + m1 2 − i η ) , 1 ( − ( p1 + q1 ) 2 + m3 2 − i η ) , 1 ( − p3 2 + m3 2 − i η ) , 1 ( − ( p3 + q1 ) 2 + m1 2 − i η ) , 1 ( − ( p1 − p3 ) 2 + m1 2 − i η ) } , { p1 , p3 } , { q1 } , { q1 2 → m1 2 } , { } ) \text{FCTopology}\left(\text{prop2Ltopo13311},\left\{\frac{1}{(-\text{p1}^2+\text{m1}^2-i \eta )},\frac{1}{(-(\text{p1}+\text{q1})^2+\text{m3}^2-i \eta )},\frac{1}{(-\text{p3}^2+\text{m3}^2-i \eta )},\frac{1}{(-(\text{p3}+\text{q1})^2+\text{m1}^2-i \eta )},\frac{1}{(-(\text{p1}-\text{p3})^2+\text{m1}^2-i \eta )}\right\},\{\text{p1},\text{p3}\},\{\text{q1}\},\left\{\text{q1}^2\to \;\text{m1}^2\right\},\{\}\right) FCTopology ( prop2Ltopo13311 , { ( − p1 2 + m1 2 − i η ) 1 , ( − ( p1 + q1 ) 2 + m3 2 − i η ) 1 , ( − p3 2 + m3 2 − i η ) 1 , ( − ( p3 + q1 ) 2 + m1 2 − i η ) 1 , ( − ( p1 − p3 ) 2 + m1 2 − i η ) 1 } , { p1 , p3 } , { q1 } , { q1 2 → m1 2 } , { } )
FCFeynmanPrepare[ GLI[ prop2Ltopo13311, { 1 , 0 , 0 , 0 , 0 }] ^ 2 , topo, Names -> x , FCE -> True ,
LoopMomenta -> Function [{ x , y }, lmom[ x , y ]]]
{ x ( 1 ) x ( 2 ) , − m1 2 x ( 1 ) x ( 2 ) ( x ( 1 ) + x ( 2 ) ) , ( x ( 1 ) 1 ( − lmom ( 1 , 1 ) 2 + m1 2 − i η ) 1 x ( 2 ) 1 ( − lmom ( 2 , 1 ) 2 + m1 2 − i η ) 1 ) , ( − x ( 1 ) 0 0 − x ( 2 ) ) , { 0 , 0 } , m1 2 ( x ( 1 ) + x ( 2 ) ) , 1 , 0 } \left\{x(1) x(2),-\text{m1}^2 x(1) x(2) (x(1)+x(2)),\left(
\begin{array}{ccc}
x(1) & \frac{1}{(-\text{lmom}(1,1)^2+\text{m1}^2-i \eta )} & 1 \\
x(2) & \frac{1}{(-\text{lmom}(2,1)^2+\text{m1}^2-i \eta )} & 1 \\
\end{array}
\right),\left(
\begin{array}{cc}
-x(1) & 0 \\
0 & -x(2) \\
\end{array}
\right),\{0,0\},\text{m1}^2 (x(1)+x(2)),1,0\right\} { x ( 1 ) x ( 2 ) , − m1 2 x ( 1 ) x ( 2 ) ( x ( 1 ) + x ( 2 )) , ( x ( 1 ) x ( 2 ) ( − lmom ( 1 , 1 ) 2 + m1 2 − i η ) 1 ( − lmom ( 2 , 1 ) 2 + m1 2 − i η ) 1 1 1 ) , ( − x ( 1 ) 0 0 − x ( 2 ) ) , { 0 , 0 } , m1 2 ( x ( 1 ) + x ( 2 )) , 1 , 0 }