FeynCalc manual (development version)

FCLoopIntegralToPropagators

FCLoopIntegralToPropagators[int, {q1, q2, ...}] is an auxiliary function that converts the loop integral int that depends on the loop momenta q1, q2, ... to a list of propagators and scalar products.

All propagators and scalar products that do not depend on the loop momenta are discarded, unless the Rest option is set to True.

See also

Overview, FCLoopPropagatorsToTopology

Examples

SFAD[p1] 
 
FCLoopIntegralToPropagators[%, {p1}]

\frac{1}{(\text{p1}^2+i \eta )}

\left\{\frac{1}{(\text{p1}^2+i \eta )}\right\}

SFAD[p1, p2] 
 
FCLoopIntegralToPropagators[%, {p1, p2}]

\frac{1}{(\text{p1}^2+i \eta ).(\text{p2}^2+i \eta )}

\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )}\right\}

If the integral contains propagators raised to integer powers, only one propagator will appear in the output.

int = SPD[q, p] SFAD[q, q - p, q - p]

\frac{p\cdot q}{(q^2+i \eta ).((q-p)^2+i \eta )^2}

FCLoopIntegralToPropagators[int, {q}]

\left\{(p\cdot q+i \eta ),\frac{1}{(q^2+i \eta )},\frac{1}{((q-p)^2+i \eta )}\right\}

However, setting the option Tally to True will count the powers of the appearing propagators.

FCLoopIntegralToPropagators[int, {q}, Tally -> True]

\left( \begin{array}{cc} \frac{1}{(q^2+i \eta )} & 1 \\ (p\cdot q+i \eta ) & 1 \\ \frac{1}{((q-p)^2+i \eta )} & 2 \\ \end{array} \right)

Here is a more realistic 3-loop example

int = SFAD[{{-k1, 0}, {mc^2, 1}, 1}] SFAD[{{-k1 - k2, 0}, {mc^2, 1}, 1}] SFAD[{{-k2, 0}, {0, 1}, 1}] SFAD[{{-k2, 0}, {0, 1}, 2}] SFAD[{{-k3, 0}, {mc^2, 1}, 1}] *SFAD[{{k1 - k3 - p1, 0}, {0, 1}, 1}] SFAD[{{-k1 - k2 + k3 + p1, 0}, {0, 1}, 1}] SFAD[{{-k1 - k2 + k3 + p1, 0}, {0, 1}, 2}]

\frac{1}{(\text{k2}^2+i \eta )^3 (\text{k1}^2-\text{mc}^2+i \eta ) (\text{k3}^2-\text{mc}^2+i \eta ) ((-\text{k1}-\text{k2})^2-\text{mc}^2+i \eta ) ((\text{k1}-\text{k3}-\text{p1})^2+i \eta ) ((-\text{k1}-\text{k2}+\text{k3}+\text{p1})^2+i \eta )^3}

FCLoopIntegralToPropagators[int, {k1, k2, k3}, Tally -> True]

\left( \begin{array}{cc} \frac{1}{(\text{k2}^2+i \eta )} & 3 \\ \frac{1}{(\text{k3}^2-\text{mc}^2+i \eta )} & 1 \\ \frac{1}{(\text{k1}^2-\text{mc}^2+i \eta )} & 1 \\ \frac{1}{((\text{k1}-\text{k3}-\text{p1})^2+i \eta )} & 1 \\ \frac{1}{((-\text{k1}-\text{k2}+\text{k3}+\text{p1})^2+i \eta )} & 3 \\ \frac{1}{((-\text{k1}-\text{k2})^2-\text{mc}^2+i \eta )} & 1 \\ \end{array} \right)