FeynCalc manual (development version)

SFAD

SFAD[{{q1 +..., p1 . q2 +...,} {m^2, s}, n}, ...] denotes a standard Lorentzian propagator given by $\frac{1}{[(q_1+\ldots)^2 + p_1 \cdot q_2 ... + m^2 + s i \eta]^n}$ , where $q_1^2$ and $p_1 \cdot q_2$ are Lorentzian scalar products in $D$ dimensions.

For brevity one can also use shorter forms such as SFAD[{q1+ ..., m^2}, ...], SFAD[{q1+ ..., m^2 , n}, ...], SFAD[{q1+ ..., {m^2, -1}}, ...], SFAD[q1,...] etc.

If s is not explicitly specified, its value is determined by the option EtaSign, which has the default value +1 and corresponds to $+ i \eta$

If n is not explicitly specified, then the default value 1 is assumed. Translation into the FeynCalc internal form is performed by FeynCalcInternal, where an SFAD is encoded using the special head StandardPropagatorDenominator.

SFAD can represent more versatile propagators as compared to the old FAD. In particular, FAD does not allow one to enter eikonal propagators, track the sign of the $i \eta$ or change the sign and the form of the mass term.

Examples

SFAD[{{p, 0}, m^2}]

$\frac{1}{(p^2-m^2+i \eta )}$

SFAD[{{p, 0}, {m^2, -1}}]

$\frac{1}{(p^2-m^2-i \eta )}$

SFAD[{{p, 0}, {-m^2, -1}}]

$\frac{1}{(p^2+m^2-i \eta )}$

SFAD[{{0, p . q}, m^2}]

$\frac{1}{(p\cdot q-m^2+i \eta )}$

SFAD[{{0, n . q}}]

$\frac{1}{(n\cdot q+i \eta )}$

SFAD[{{p, p . q}, m^2}]

$\frac{1}{(p^2+p\cdot q-m^2+i \eta )}$

The so called Smirnov-notation for propagators can be achieved by multiplying the quadratic part by I and switching the sign of the mass term.

SFAD[{{I*p, 0}, -m^2}]

$\frac{1}{(-p^2+m^2+i \eta )}$

If one wants to have additional variables multiplying loop or external momenta, those need to be declared to be of the FCVariable type

DataType[la, FCVariable] = True

$\text{True}$

SFAD[{{0, la p . q}, m^2}]

$\frac{1}{(\text{la} (p\cdot q)-m^2+i \eta )}$

% // FCI // StandardForm

(*FeynAmpDenominator[StandardPropagatorDenominator[0, la Pair[Momentum[p, D], Momentum[q, D]], -m^2, {1, 1}]]*)

SFAD

See also

Examples