FeynCalc manual (development version)

FromGFAD

FromGFAD[exp] converts all suitable generic propagator denominators into standard and Cartesian propagator denominators.

The options InitialSubstitutions and IntermediateSubstitutions can be used to help the function handle nontrivial propagators.

For propagators containing symbolic variables it might be necessary to tell the function that those are larger than zero (if applicable), so that expressions such as \sqrt{\lambda^2} can be simplified accordingly.

See also

Overview, GFAD, SFAD, CFAD, FeynAmpDenominatorExplicit.

Examples

GFAD[SPD[p1]] 
 
ex = FromGFAD[%]

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

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

ex // StandardForm

(*FeynAmpDenominator[StandardPropagatorDenominator[Momentum[p1, D], 0, 0, {1, 1}]]*)

GFAD[SPD[p1] + 2 SPD[p1, p2]] 
 
ex = FromGFAD[%]

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

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

ex // StandardForm

(*FeynAmpDenominator[StandardPropagatorDenominator[Momentum[p1, D], 2 Pair[Momentum[p1, D], Momentum[p2, D]], 0, {1, 1}]]*)

GFAD[{{CSPD[p1] + 2 CSPD[p1, p2] + m^2, -1}, 2}] 
 
ex = FromGFAD[%]

\frac{1}{(m^2+\text{p1}^2+2 (\text{p1}\cdot \;\text{p2})-i \eta )^2}

\frac{1}{(\text{p1}\cdot (\text{p1}+2 \;\text{p2})+m^2-i \eta )^2}

ex // StandardForm

(*FeynAmpDenominator[CartesianPropagatorDenominator[0, CartesianPair[CartesianMomentum[p1, -1 + D], CartesianMomentum[p1 + 2 p2, -1 + D]], m^2, {2, -1}]]*)

prop = FeynAmpDenominator[GenericPropagatorDenominator[-la Pair[Momentum[p1, D], 
       Momentum[p1, D]] + 2 Pair[Momentum[p1, D], Momentum[q, D]], {1,1}]]

\frac{1}{(2 (\text{p1}\cdot q)-\text{la} \;\text{p1}^2+i \eta )}

ex = FromGFAD[prop]

\frac{1}{(-\text{la} \;\text{p1}^2+2 (\text{p1}\cdot q)+i \eta )}

ex // StandardForm

\text{FeynAmpDenominator}\left[\text{StandardPropagatorDenominator}\left[\sqrt{-\text{la}} \;\text{Momentum}[\text{p1},D],2 \;\text{Pair}[\text{Momentum}[\text{p1},D],\text{Momentum}[q,D]],0,\{1,1\}\right]\right]

ex = FromGFAD[prop, PowerExpand -> {la}]

\frac{1}{(-\text{la} \;\text{p1}^2+2 (\text{p1}\cdot q)+i \eta )}

ex // StandardForm

\text{FeynAmpDenominator}\left[\text{StandardPropagatorDenominator}\left[i \sqrt{\text{la}} \;\text{Momentum}[\text{p1},D],2 \;\text{Pair}[\text{Momentum}[\text{p1},D],\text{Momentum}[q,D]],0,\{1,1\}\right]\right]

ex = GFAD[{{-SPD[p1, p1], 1}, 1}]*GFAD[{{SPD[p1, -p1 + 2*p3] - SPD[p3, p3], 1}, 1}]*
    GFAD[{{-SPD[p3, p3], 1}, 1}]*SFAD[{{I*(p1 + q), 0}, {-mb^2, 1}, 1}]*
    SFAD[{{I*(p3 + q), 0}, {-mb^2, 1}, 1}] +  (-2*mg^2*GFAD[{{-SPD[p1, p1], 1}, 2}]*
       GFAD[{{SPD[p1, -p1 + 2*p3] - SPD[p3, p3], 1}, 1}]*GFAD[{{-SPD[p3, p3], 1}, 1}]*
       SFAD[{{I*(p1 + q), 0}, {-mb^2, 1}, 1}]*SFAD[{{I*(p3 + q), 0}, {-mb^2, 1}, 1}] - 
        2*mg^2*GFAD[{{-SPD[p1, p1], 1}, 1}]*GFAD[{{SPD[p1, -p1 + 2*p3] - SPD[p3, p3], 
            1}, 2}]*GFAD[{{-SPD[p3, p3], 1}, 1}]*SFAD[{{I*(p1 + q), 0}, {-mb^2, 1}, 1}]*
         SFAD[{{I*(p3 + q), 0}, {-mb^2, 1}, 1}] -    2*mg^2*GFAD[{{-SPD[p1, p1], 1}, 
           1}]*GFAD[{{SPD[p1, -p1 + 2*p3] - SPD[p3, p3], 1}, 1}]*GFAD[{{-SPD[p3, p3], 
            1}, 2}]*SFAD[{{I*(p1 + q), 0}, {-mb^2, 1}, 1}]*SFAD[{{I*(p3 + q), 0}, 
           {-mb^2, 1}, 1}])/2

\frac{1}{2} \left(-\frac{2 \;\text{mg}^2}{(-\text{p1}^2+i \eta )^2 (-\text{p3}^2+i \eta ) (-(\text{p1}+q)^2+\text{mb}^2+i \eta ) (-(\text{p3}+q)^2+\text{mb}^2+i \eta ) (\text{p1}\cdot (2 \;\text{p3}-\text{p1})-\text{p3}^2+i \eta )}-\frac{2 \;\text{mg}^2}{(-\text{p1}^2+i \eta ) (-\text{p3}^2+i \eta ) (-(\text{p1}+q)^2+\text{mb}^2+i \eta ) (-(\text{p3}+q)^2+\text{mb}^2+i \eta ) (\text{p1}\cdot (2 \;\text{p3}-\text{p1})-\text{p3}^2+i \eta )^2}-\frac{2 \;\text{mg}^2}{(-\text{p1}^2+i \eta ) (-\text{p3}^2+i \eta )^2 (-(\text{p1}+q)^2+\text{mb}^2+i \eta ) (-(\text{p3}+q)^2+\text{mb}^2+i \eta ) (\text{p1}\cdot (2 \;\text{p3}-\text{p1})-\text{p3}^2+i \eta )}\right)+\frac{1}{(-\text{p1}^2+i \eta ) (-\text{p3}^2+i \eta ) (-(\text{p1}+q)^2+\text{mb}^2+i \eta ) (-(\text{p3}+q)^2+\text{mb}^2+i \eta ) (\text{p1}\cdot (2 \;\text{p3}-\text{p1})-\text{p3}^2+i \eta )}

FromGFAD[ex]

\frac{1}{2} \left(-\frac{2 \;\text{mg}^2}{(-\text{p1}^2+i \eta )^2 (-\text{p3}^2+i \eta ) (-(\text{p1}+q)^2+\text{mb}^2+i \eta ) (-(\text{p3}+q)^2+\text{mb}^2+i \eta ) (-\text{p3}^2+\text{p1}\cdot (2 \;\text{p3}-\text{p1})+i \eta )}-\frac{2 \;\text{mg}^2}{(-\text{p1}^2+i \eta ) (-\text{p3}^2+i \eta )^2 (-(\text{p1}+q)^2+\text{mb}^2+i \eta ) (-(\text{p3}+q)^2+\text{mb}^2+i \eta ) (-\text{p3}^2+\text{p1}\cdot (2 \;\text{p3}-\text{p1})+i \eta )}-\frac{2 \;\text{mg}^2}{(-\text{p1}^2+i \eta ) (-\text{p3}^2+i \eta ) (-(\text{p1}+q)^2+\text{mb}^2+i \eta ) (-(\text{p3}+q)^2+\text{mb}^2+i \eta ) (-\text{p3}^2+\text{p1}\cdot (2 \;\text{p3}-\text{p1})+i \eta )^2}\right)+\frac{1}{(-\text{p1}^2+i \eta ) (-\text{p3}^2+i \eta ) (-(\text{p1}+q)^2+\text{mb}^2+i \eta ) (-(\text{p3}+q)^2+\text{mb}^2+i \eta ) (-\text{p3}^2+\text{p1}\cdot (2 \;\text{p3}-\text{p1})+i \eta )}

Using the option InitialSubstitutions one can perform certain replacement that might not be found automatically

ex = GFAD[SPD[k1] + 2 SPD[k1, k2] + SPD[k2] + SPD[k1, n]]

\frac{1}{(\text{k1}^2+2 (\text{k1}\cdot \;\text{k2})+\text{k1}\cdot n+\text{k2}^2+i \eta )}

FromGFAD[ex, FCE -> True]
% // InputForm

\frac{1}{(\text{k1}^2+\text{k1}\cdot (2 \;\text{k2}+n)+\text{k2}^2+i \eta )}

SFAD[{{k1, k1 . (2*k2 + n) + k2 . k2}, {0, 1}, 1}]

FromGFAD[ex, FCE -> True, InitialSubstitutions -> {ExpandScalarProduct[SPD[k1 + k2]] -> SPD[k1 + k2]}]
% // InputForm

\frac{1}{((\text{k1}+\text{k2})^2+\text{k1}\cdot n+i \eta )}

SFAD[{{k1 + k2, k1 . n}, {0, 1}, 1}]