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}]
1 ( p1 2 + i η ) \frac{1}{(\text{p1}^2+i \eta )} ( p1 2 + i η ) 1
{ 1 ( p1 2 + i η ) } \left\{\frac{1}{(\text{p1}^2+i \eta )}\right\} { ( p1 2 + i η ) 1 }
SFAD[ p1, p2]
FCLoopIntegralToPropagators[ % , { p1, p2}]
1 ( p1 2 + i η ) . ( p2 2 + i η ) \frac{1}{(\text{p1}^2+i \eta ).(\text{p2}^2+i \eta )} ( p1 2 + i η ) . ( p2 2 + i η ) 1
{ 1 ( p1 2 + i η ) , 1 ( p2 2 + i η ) } \left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )}\right\} { ( p1 2 + i η ) 1 , ( p2 2 + i η ) 1 }
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 ]
p ⋅ q ( q 2 + i η ) . ( ( q − p ) 2 + i η ) 2 \frac{p\cdot q}{(q^2+i \eta ).((q-p)^2+i \eta )^2} ( q 2 + i η ) . (( q − p ) 2 + i η ) 2 p ⋅ q
FCLoopIntegralToPropagators[ int, { q }]
{ ( p ⋅ q + i η ) , 1 ( q 2 + i η ) , 1 ( ( q − p ) 2 + i η ) } \left\{(p\cdot q+i \eta ),\frac{1}{(q^2+i \eta )},\frac{1}{((q-p)^2+i \eta )}\right\} { ( p ⋅ q + i η ) , ( q 2 + i η ) 1 , (( q − p ) 2 + i η ) 1 }
However, setting the option Tally
to True
will count the powers of the appearing propagators.
FCLoopIntegralToPropagators[ int, { q }, Tally -> True ]
( 1 ( q 2 + i η ) 1 ( p ⋅ q + i η ) 1 1 ( ( q − p ) 2 + i η ) 2 ) \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) ( q 2 + i η ) 1 ( p ⋅ q + i η ) (( q − p ) 2 + i η ) 1 1 1 2
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 }]
1 ( k2 2 + i η ) 3 ( k1 2 − mc 2 + i η ) ( k3 2 − mc 2 + i η ) ( ( − k1 − k2 ) 2 − mc 2 + i η ) ( ( k1 − k3 − p1 ) 2 + i η ) ( ( − k1 − k2 + k3 + p1 ) 2 + i η ) 3 \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} ( k2 2 + i η ) 3 ( k1 2 − mc 2 + i η ) ( k3 2 − mc 2 + i η ) (( − k1 − k2 ) 2 − mc 2 + i η ) (( k1 − k3 − p1 ) 2 + i η ) (( − k1 − k2 + k3 + p1 ) 2 + i η ) 3 1
FCLoopIntegralToPropagators[ int, { k1, k2, k3}, Tally -> True ]
( 1 ( k2 2 + i η ) 3 1 ( k3 2 − mc 2 + i η ) 1 1 ( k1 2 − mc 2 + i η ) 1 1 ( ( k1 − k3 − p1 ) 2 + i η ) 1 1 ( ( − k1 − k2 + k3 + p1 ) 2 + i η ) 3 1 ( ( − k1 − k2 ) 2 − mc 2 + i η ) 1 ) \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) ( k2 2 + i η ) 1 ( k3 2 − mc 2 + i η ) 1 ( k1 2 − mc 2 + i η ) 1 (( k1 − k3 − p1 ) 2 + i η ) 1 (( − k1 − k2 + k3 + p1 ) 2 + i η ) 1 (( − k1 − k2 ) 2 − mc 2 + i η ) 1 3 1 1 1 3 1