FeynCalc manual (development version)

FCLoopReplaceQuadraticEikonalPropagators

FCLoopReplaceQuadraticEikonalPropagators[topologies] identifies SFADs and CFADs in topologies that represent mixed quadratic-eikonal propagators, e.g. $[p^2 - 2 p \cdot q]$. Using the information on loop momenta provided by the user the routine will try to rewrite those denominators by completing the square, e.g. as in $[(p-q)^2 - q^2]$.

This procedure is useful because one cannot easily determine the momentum flow from looking at quadratic-eikonal propagators as it is possible in the case of purely quadratic ones.

For this to work it is crucial to specify the loop momenta via the LoopMomenta option as well as the kinematics (IntermediateSubstitutions) and the rules for completing the square (InitialSubstitutions) on the purely loop-momentum dependent piece of the propagator (e.g. $p_1^2 - 2 p_1 \cdot p_2 + p_2^2$ goes to $(p_1+p_2)^2$.

Internally this routine uses ToGFAD and FromGFAD.

See also

Overview, FCTopology, GFAD, FromGFAD, ToGFAD.

Examples

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

topos = {FCTopology[preTopoDia1, {SFAD[{{k2, 0}, {0, 1}, 1}], SFAD[{{k1, 0}, {0, 1}, 1}], 
     SFAD[{{k1 + k2, 0}, {0, 1}, 1}], SFAD[{{0, -k1 . nb}, {0, 1}, 1}], SFAD[{{k2, -(meta*u0b*k2 . nb)}, {0, 1}, 1}], 
     SFAD[{{k1 + k2, -2*gkin*meta*u0b*(k1 + k2) . n}, {0, 1}, 1}], SFAD[{{k1, -2*gkin*meta*k1 . n + meta*u0b*k1 . nb}, 
       {2*gkin*meta^2*u0b, 1}, 1}], SFAD[{{k1, -2*gkin*meta*u0b*k1 . n + meta*u0b*k1 . nb}, {2*gkin*meta^2*u0b^2, 1}, 1}]}, 
    {k1, k2}, {n, nb}, {Hold[SPD][n] -> 0, Hold[SPD][nb] -> 0, Hold[SPD][n, nb] -> 2}, {}]}

$$\left\{\text{FCTopology}\left(\text{preTopoDia1},\left\{\frac{1}{(\text{k2}^2+i \eta )},\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2+i \eta )},\frac{1}{(-\text{k1}\cdot \;\text{nb}+i \eta )},\frac{1}{(\text{k2}^2-\text{meta} \;\text{u0b} (\text{k2}\cdot \;\text{nb})+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2-2 \;\text{gkin} \;\text{meta} \;\text{u0b} ((\text{k1}+\text{k2})\cdot n)+i \eta )},\frac{1}{(\text{k1}^2+\text{meta} \;\text{u0b} (\text{k1}\cdot \;\text{nb})-2 \;\text{gkin} \;\text{meta} (\text{k1}\cdot n)-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}+i \eta )},\frac{1}{(\text{k1}^2+\text{meta} \;\text{u0b} (\text{k1}\cdot \;\text{nb})-2 \;\text{gkin} \;\text{meta} \;\text{u0b} (\text{k1}\cdot n)-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}^2+i \eta )}\right\},\{\text{k1},\text{k2}\},\{n,\text{nb}\},\{\text{Hold}[\text{SPD}][n]\to 0,\text{Hold}[\text{SPD}][\text{nb}]\to 0,\text{Hold}[\text{SPD}][n,\text{nb}]\to 2\},\{\}\right)\right\}$$

FCLoopReplaceQuadraticEikonalPropagators[topos, LoopMomenta -> {k1, k2}, 
  InitialSubstitutions -> {
    ExpandScalarProduct[SPD[k1 - k2]] -> SPD[k1 - k2], 
    ExpandScalarProduct[SPD[k1 + k2]] -> SPD[k1 + k2]}, 
  IntermediateSubstitutions -> {SPD[n] -> 0, SPD[nb] -> 0, SPD[n, nb] -> 0}]

$$\left\{\text{FCTopology}\left(\text{preTopoDia1},\left\{\frac{1}{(\text{k2}^2+i \eta )},\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2+i \eta )},\frac{1}{(-\text{k1}\cdot \;\text{nb}+i \eta )},\frac{1}{((\text{k2}-\frac{\text{meta} \;\text{u0b} \;\text{nb}}{2})^2+i \eta )},\frac{1}{((\text{k1}+\text{k2}-\text{gkin} \;\text{meta} \;\text{u0b} n)^2+i \eta )},\frac{1}{((\text{k1}-\text{gkin} \;\text{meta} n+\frac{\text{meta} \;\text{u0b} \;\text{nb}}{2})^2-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}+i \eta )},\frac{1}{((\text{k1}-\text{gkin} \;\text{meta} \;\text{u0b} n+\frac{\text{meta} \;\text{u0b} \;\text{nb}}{2})^2-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}^2+i \eta )}\right\},\{\text{k1},\text{k2}\},\{n,\text{nb}\},\{\text{Hold}[\text{SPD}][n]\to 0,\text{Hold}[\text{SPD}][\text{nb}]\to 0,\text{Hold}[\text{SPD}][n,\text{nb}]\to 2\},\{\}\right)\right\}$$