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 = det M, 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.
Overview, FCFeynmanParametrize, FCFeynmanProjectivize, FCLoopValidTopologyQ.
One of the simplest examples is the 1-loop tadpole
FCFeynmanPrepare[FAD[{q, m1}], {q}]\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\}
Use the option Names to have specific symbols denoting
Feynman parameters
FCFeynmanPrepare[FAD[{q, m1}], {q}, Names -> x]\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\}
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]\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\}
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]\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\}
Massless 2-loop self-energy
FCFeynmanPrepare[FAD[p1, p2, Q - p1 - p2, Q - p1, Q - p2], {p1, p2}, Names -> x]\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\}
Factorizing integrals also work
FCFeynmanPrepare[FAD[{p1, m1}, {p2, m2}, Q - p1, Q - p2], {p1, p2}, Names -> x]\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\}
Cartesian propagators are equally supported
FCFeynmanPrepare[CSPD[q, p] CFAD[{q, m}, {q - p, m2}], {q}, Names -> x]\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\}
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}, {}, {}]\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{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)
FCFeynmanPrepare[topo1, Names -> x]\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\}
FCFeynmanPrepare[{topo1, topo2}, Names -> x]\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)
FCFeynmanPrepare[{GLI["prop2Lv1", {1, 1, 1, 1, 0}], GLI["prop2Lv2", {1, 1, 0, 0, 1}]},
{topo1, topo2}, Names -> x]\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)
FCFeynmanPrepare can also handle products of
GLIs. 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}, {}]\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)
FCFeynmanPrepare[GLI[prop2Ltopo13311, {1, 0, 0, 0, 0}]^2, topo, Names -> x, FCE -> True,
LoopMomenta -> Function[{x, y}, lmom[x, y]]]\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\}