FCMellinJoin[int, {q1, q2, ...}, {prop1, prop2, ...}]
applies the standard formula for splitting propagators
prop1, prop2, ...
into summands by introducing integrations
along a contour in the complex space.
The main purpose of this routine is to convert massive propagators into massless ones when using Mellin-Barnes integration techniques.
The output consists of a list containing two elements, the first one
being the prefactor and the second one the product of remaining
propagators. The second element (or, alternatively, the product of both
elements) can be then further processed using
FCFeynmanParametrize
. Setting the option List
to False
will return a product instead of a list.
The option FCSplit
can be used to split a propagators in
more than 2 terms as it is done by default.
Overview, FCFeynmanParametrize.
[FAD[q, {-p1 + q, m}, {-p1 + q, m}, {-p2 + q, m}], {q}, {SFAD[{q - p1, m^2}],
FCMellinJoin[{q - p2, m^2}]}, Names -> z] SFAD
\left\{-\frac{\Gamma (-z(1)) \Gamma (z(1)+2) \Gamma (-z(2)) \Gamma (z(2)+1)}{4 \pi ^2},\frac{\left(-m^2+i \eta \right)^{z(1)+z(2)}}{(q^2+i \eta ).(\text{p1}^2-2 (\text{p1}\cdot q)+q^2+i \eta )^{z(1)+2}.(\text{p2}^2-2 (\text{p2}\cdot q)+q^2+i \eta )^{z(2)+1}}\right\}
[FAD[q, {-p1 + q, m}, {-p1 + q, m}, {-p2 + q, m}], {q}, {SFAD[{q - p1, m^2}],
FCMellinJoin[{q - p2, m^2}]}, Names -> z, FCSplit -> {{q, m, p1}, {q, m, p2}}] SFAD
\left\{-\frac{\Gamma (-z(1)(1)) \Gamma (-z(1)(2)) \Gamma (-z(1)(3)) \Gamma (z(1)(1)+z(1)(2)+z(1)(3)+2) \Gamma (-z(2)(1)) \Gamma (-z(2)(2)) \Gamma (-z(2)(3)) \Gamma (z(2)(1)+z(2)(2)+z(2)(3)+1)}{64 \pi ^6},\frac{\left(-m^2+i \eta \right)^{z(1)(3)+z(2)(3)} \left(q^2+i \eta \right)^{z(1)(2)+z(2)(2)} (-2 (\text{p1}\cdot q)+i \eta )^{z(1)(1)} (-2 (\text{p2}\cdot q)+i \eta )^{z(2)(1)}}{(q^2+i \eta ).(\text{p1}^2+i \eta )^{z(1)(1)+z(1)(2)+z(1)(3)+2}.(\text{p2}^2+i \eta )^{z(2)(1)+z(2)(2)+z(2)(3)+1}}\right\}
[SFAD[{k, m^2, nu1}, {p - k, 0, nu2}], {k}, {SFAD[{k, m^2}]},
FCMellinJoin-> -1, Names -> z] FCLoopSwitchEtaSign
\left\{-\frac{i (-1)^{-\text{nu1}-\text{nu2}} \Gamma (-z(1)) \Gamma (\text{nu1}+z(1))}{2 \pi \Gamma (\text{nu1})},\left(m^2-i \eta \right)^{z(1)} (-k^2-i \eta )^{-\text{nu1}-z(1)} \frac{1}{(-(p-k)^2-i \eta )}^{\text{nu2}}\right\}