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. In particular, InitialSubstitutions can define rules for completing the square in the loop momenta of the propagator, while IntermediateSubstitutions contains relations for scalar products appearing in those rules.

Another useful option is LoopMomenta which is particularly helpful when converting mixed quadratic-eikonal propagators to quadratic ones.

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. To that aim one should use the option PowerExpand.

See also

Overview, GFAD, SFAD, CFAD, FeynAmpDenominatorExplicit.

Examples

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

$$\text{GFAD}(\text{SPD}(\text{p1}))$$

$$\text{FromGFAD}(\text{GFAD}(\text{SPD}(\text{p1})))$$

ex // StandardForm

(*FromGFAD[GFAD[SPD[p1]]]*)

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

$$\text{GFAD}(2 \;\text{SPD}(\text{p1},\text{p2})+\text{SPD}(\text{p1}))$$

FromGFAD[ex]

$$\text{FromGFAD}(\text{GFAD}(2 \;\text{SPD}(\text{p1},\text{p2})+\text{SPD}(\text{p1})))$$

We can get a proper conversion into a quadratic propagator using the option LoopMomenta. Notice that here p2.p2 is being put into the mass slot

FromGFAD[ex, LoopMomenta -> {p1}]

$$\text{FromGFAD}(\text{GFAD}(2 \;\text{SPD}(\text{p1},\text{p2})+\text{SPD}(\text{p1})),\text{LoopMomenta}\to \{\text{p1}\})$$

ex // StandardForm

(*GFAD[SPD[p1] + 2 SPD[p1, p2]]*)

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

$$\text{GFAD}\left(\left\{\left\{2 \;\text{CSPD}(\text{p1},\text{p2})+\text{CSPD}(\text{p1})+m^2,-1\right\},2\right\}\right)$$

$$\text{FromGFAD}\left(\text{GFAD}\left(\left\{\left\{2 \;\text{CSPD}(\text{p1},\text{p2})+\text{CSPD}(\text{p1})+m^2,-1\right\},2\right\}\right)\right)$$

ex // StandardForm

(*FromGFAD[GFAD[{{m^2 + CSPD[p1] + 2 CSPD[p1, p2], -1}, 2}]]*)

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

$$\text{FeynAmpDenominator}(\text{GenericPropagatorDenominator}(2 \;\text{Pair}(\text{Momentum}(\text{p1},D),\text{Momentum}(q,D))-\text{la} \;\text{Pair}(\text{Momentum}(\text{p1},D),\text{Momentum}(\text{p1},D)),\{1,1\}))$$

ex = FromGFAD[prop]

$$\text{FromGFAD}(\text{FeynAmpDenominator}(\text{GenericPropagatorDenominator}(2 \;\text{Pair}(\text{Momentum}(\text{p1},D),\text{Momentum}(q,D))-\text{la} \;\text{Pair}(\text{Momentum}(\text{p1},D),\text{Momentum}(\text{p1},D)),\{1,1\})))$$

ex = FromGFAD[prop, LoopMomenta -> {p1}]

$$\text{FromGFAD}(\text{FeynAmpDenominator}(\text{GenericPropagatorDenominator}(2 \;\text{Pair}(\text{Momentum}(\text{p1},D),\text{Momentum}(q,D))-\text{la} \;\text{Pair}(\text{Momentum}(\text{p1},D),\text{Momentum}(\text{p1},D)),\{1,1\})),\text{LoopMomenta}\to \{\text{p1}\})$$

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

$$\frac{1}{2} \left(-2 \;\text{mg}^2 \;\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},2\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)-2 \;\text{mg}^2 \;\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},1\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},2\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)-2 \;\text{mg}^2 \;\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},1\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},2\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)\right)+\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},1\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)$$

Notice that FromGFAD does not expand scalar products in the propagators before trying to convert them to SFADs or CFADs. If this is needed, the user should better apply ExpandScalarProduct to the expression by hand.

FromGFAD[ex]

$$\text{FromGFAD}\left(\frac{1}{2} \left(-2 \;\text{mg}^2 \;\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},2\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)-2 \;\text{mg}^2 \;\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},1\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},2\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)-2 \;\text{mg}^2 \;\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},1\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},2\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)\right)+\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},1\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)\right)$$

FromGFAD[ExpandScalarProduct[ex]]

$$\text{FromGFAD}\left(\text{ExpandScalarProduct}\left(\frac{1}{2} \left(-2 \;\text{mg}^2 \;\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},2\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)-2 \;\text{mg}^2 \;\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},1\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},2\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)-2 \;\text{mg}^2 \;\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},1\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},2\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)\right)+\text{GFAD}(\{\{-\text{SPD}(\text{p1},\text{p1}),1\},1\}) \;\text{GFAD}(\{\{-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{GFAD}(\{\{\text{SPD}(\text{p1},2 \;\text{p3}-\text{p1})-\text{SPD}(\text{p3},\text{p3}),1\},1\}) \;\text{SFAD}\left(\left\{\{i (\text{p1}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right) \;\text{SFAD}\left(\left\{\{i (\text{p3}+q),0\},\left\{-\text{mb}^2,1\right\},1\right\}\right)\right)\right)$$

Using the option InitialSubstitutions one can perform certain replacement that might not be found automatically. The values of scalar products can be set using IntermediateSubstitutions

ex = GFAD[{{SPD[k1, k1] - 2*gkin*meta*u0b*SPD[k1, n], 1}, 1}];

Notice that we need to declare the appearing variables as FCVariables

(DataType[#, FCVariable] = True) & /@ {gkin, meta, u0b};

Without these options we get a mixed quadratic-eikonal propagator that will cause us troubles when doing topology minimizations.

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

$$\text{FromGFAD}(\text{GFAD}(\{\{\text{SPD}(\text{k1},\text{k1})-2 \;\text{gkin} \;\text{meta} \;\text{u0b} \;\text{SPD}(\text{k1},n),1\},1\}),\text{FCE}\to \;\text{True})$$

FromGFAD[GFAD[{{SPD[k1, k1] - 2*gkin*meta*u0b*SPD[k1, n], 1}, 1}], 
 FCE -> True]

But when doing everything right we end up with a purely quadratic propagator

FromGFAD[ex, InitialSubstitutions -> {ExpandScalarProduct[SPD[k1 - gkin meta u0b n]] -> SPD[k1 - gkin meta u0b n]}, 
  IntermediateSubstitutions -> {SPD[n] -> 0, SPD[nb] -> 0, SPD[n, nb] -> 2}]

$$\text{FromGFAD}(\text{GFAD}(\{\{\text{SPD}(\text{k1},\text{k1})-2 \;\text{gkin} \;\text{meta} \;\text{u0b} \;\text{SPD}(\text{k1},n),1\},1\}),\text{InitialSubstitutions}\to \{\text{ExpandScalarProduct}(\text{SPD}(\text{k1}-\text{gkin} \;\text{meta} n \;\text{u0b}))\to \;\text{SPD}(\text{k1}-\text{gkin} \;\text{meta} n \;\text{u0b})\},\text{IntermediateSubstitutions}\to \{\text{SPD}(n)\to 0,\text{SPD}(\text{nb})\to 0,\text{SPD}(n,\text{nb})\to 2\})$$

However, in this case the function can also figure out the necessary square completion on its own if we tell it that k1 is a momentum w.r.t which the square should be completed. In this case the option IntermediateSubstitutions is not really needed

FromGFAD[ex, LoopMomenta -> {k1}]

$$\text{FromGFAD}(\text{GFAD}(\{\{\text{SPD}(\text{k1},\text{k1})-2 \;\text{gkin} \;\text{meta} \;\text{u0b} \;\text{SPD}(\text{k1},n),1\},1\}),\text{LoopMomenta}\to \{\text{k1}\})$$

It is still helpful, though

FromGFAD[ex, LoopMomenta -> {k1}, IntermediateSubstitutions -> {SPD[n] -> 0, SPD[nb] -> 0, SPD[n, nb] -> 2}]

$$\text{FromGFAD}(\text{GFAD}(\{\{\text{SPD}(\text{k1},\text{k1})-2 \;\text{gkin} \;\text{meta} \;\text{u0b} \;\text{SPD}(\text{k1},n),1\},1\}),\text{LoopMomenta}\to \{\text{k1}\},\text{IntermediateSubstitutions}\to \{\text{SPD}(n)\to 0,\text{SPD}(\text{nb})\to 0,\text{SPD}(n,\text{nb})\to 2\})$$

If we have multiple loop momenta, we need to first complete the square with respect to them before handling the full expression

ex = GFAD[{{SPD[k1, k1] + 2 SPD[k1, k2] + SPD[k2, k2] + 2 gkin meta (SPD[k1, n] + SPD[k2, n]), 1}, 1}]

$$\text{GFAD}(\{\{2 \;\text{gkin} \;\text{meta} (\text{SPD}(\text{k1},n)+\text{SPD}(\text{k2},n))+2 \;\text{SPD}(\text{k1},\text{k2})+\text{SPD}(\text{k1},\text{k1})+\text{SPD}(\text{k2},\text{k2}),1\},1\})$$

FromGFAD[ex, LoopMomenta -> {k1, k2}]

$$\text{FromGFAD}(\text{GFAD}(\{\{2 \;\text{gkin} \;\text{meta} (\text{SPD}(\text{k1},n)+\text{SPD}(\text{k2},n))+2 \;\text{SPD}(\text{k1},\text{k2})+\text{SPD}(\text{k1},\text{k1})+\text{SPD}(\text{k2},\text{k2}),1\},1\}),\text{LoopMomenta}\to \{\text{k1},\text{k2}\})$$

FromGFAD[ex, LoopMomenta -> {k1, k2}, 
  InitialSubstitutions -> {ExpandScalarProduct[SPD[k1 + k2]] -> SPD[k1 + k2]}, 
  IntermediateSubstitutions -> {SPD[n] -> 0}]

$$\text{FromGFAD}(\text{GFAD}(\{\{2 \;\text{gkin} \;\text{meta} (\text{SPD}(\text{k1},n)+\text{SPD}(\text{k2},n))+2 \;\text{SPD}(\text{k1},\text{k2})+\text{SPD}(\text{k1},\text{k1})+\text{SPD}(\text{k2},\text{k2}),1\},1\}),\text{LoopMomenta}\to \{\text{k1},\text{k2}\},\text{InitialSubstitutions}\to \{\text{ExpandScalarProduct}(\text{SPD}(\text{k1}+\text{k2}))\to \;\text{SPD}(\text{k1}+\text{k2})\},\text{IntermediateSubstitutions}\to \{\text{SPD}(n)\to 0\})$$