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\}

Notice that the ordering of scalar products in the InitialSubstitutions option is important. It is recommended to put the longest and most complicated rules first and all simpler rules thereafter. Otherwise, it might happen that a simple rule will be first applied to a complicated expression making it impossible to apply the actually needed complicated rule later on. For example, this fails, because the most complicated rule containing 3 loop momenta comes last

testTopo = FCTopology["topology1230", {SFAD[{{k3, 0}, {0, 1}, 1}], SFAD[{{0, -k2 . nb}, {0, 1}, 1}], 
    SFAD[{{k1, -(meta*u0b*k1 . nb)}, {0, 1}, 1}], SFAD[{{k1, -2*gkin*meta*u0b*k1 . n}, {0, 1}, 1}], 
    SFAD[{{0, -(k1 + k2) . nb}, {-2*gkin*meta*u0b, 1}, 1}], SFAD[{{k1 + k2, -2*gkin*meta*u0b*(k1 + k2) . n}, {0, 1}, 1}], 
     SFAD[{{k2, -2*gkin*meta*k2 . n + meta*u0b*k2 . nb}, {2*gkin*meta^2*u0b, 1}, 1}], 
     SFAD[{{k1 + k2, -2*gkin*meta*u0b*(k1 + k2) . n + k3 . (-2*k1 - 2*k2 + k3) + 2*gkin*meta*u0b*k3 . n}, {0, 1}, 1}], 
     SFAD[{{0, (k1 - k3) . nb}, {0, 1}, 1}], 
     SFAD[{{k1, meta*u0b*(k1 - k3) . nb + k3 . (-2*k1 + k3)}, {0, 1}, 1}], SFAD[{{k1, 2*gkin*meta*k1 . n + k3 . (-2*k1 + k3) - 2*gkin*meta*k3 . n}, {0, 1}, 1}], 
     SFAD[{{k1 - k2, 0}, {0, 1}, 1}]}, {k1, k2, k3}, {n, nb}, {Hold[SPD][n] -> 0, Hold[SPD][nb] -> 0, Hold[SPD][n, nb] -> 2}, {}]

\text{FCTopology}\left(\text{topology1230},\left\{\frac{1}{(\text{k3}^2+i \eta )},\frac{1}{(-\text{k2}\cdot \;\text{nb}+i \eta )},\frac{1}{(\text{k1}^2-\text{meta} \;\text{u0b} (\text{k1}\cdot \;\text{nb})+i \eta )},\frac{1}{(\text{k1}^2-2 \;\text{gkin} \;\text{meta} \;\text{u0b} (\text{k1}\cdot n)+i \eta )},\frac{1}{(-(\text{k1}+\text{k2})\cdot \;\text{nb}+2 \;\text{gkin} \;\text{meta} \;\text{u0b}+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{k2}^2+\text{meta} \;\text{u0b} (\text{k2}\cdot \;\text{nb})-2 \;\text{gkin} \;\text{meta} (\text{k2}\cdot n)-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2+-2 \;\text{gkin} \;\text{meta} \;\text{u0b} ((\text{k1}+\text{k2})\cdot n)+\text{k3}\cdot (-2 \;\text{k1}-2 \;\text{k2}+\text{k3})+2 \;\text{gkin} \;\text{meta} \;\text{u0b} (\text{k3}\cdot n)+i \eta )},\frac{1}{((\text{k1}-\text{k3})\cdot \;\text{nb}+i \eta )},\frac{1}{(\text{k1}^2+\text{meta} \;\text{u0b} ((\text{k1}-\text{k3})\cdot \;\text{nb})+\text{k3}\cdot (\text{k3}-2 \;\text{k1})+i \eta )},\frac{1}{(\text{k1}^2+2 \;\text{gkin} \;\text{meta} (\text{k1}\cdot n)+\text{k3}\cdot (\text{k3}-2 \;\text{k1})-2 \;\text{gkin} \;\text{meta} (\text{k3}\cdot n)+i \eta )},\frac{1}{((\text{k1}-\text{k2})^2+i \eta )}\right\},\{\text{k1},\text{k2},\text{k3}\},\{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)

FCLoopReplaceQuadraticEikonalPropagators[testTopo, LoopMomenta -> {k1, k2, k3}, InitialSubstitutions -> {
    ExpandScalarProduct[SPD[k2 - k3]] -> SPD[k2 - k3], ExpandScalarProduct[SPD[k1 - k3]] -> SPD[k1 - k3], 
    ExpandScalarProduct[SPD[k1 + k3]] -> SPD[k1 + k3], 
    ExpandScalarProduct[SPD[k1 + k2]] -> SPD[k1 + k2], 
    ExpandScalarProduct[SPD[k1 + k2 - k3]] -> SPD[k1 + k2 - k3] 
   }, IntermediateSubstitutions -> {SPD[n] -> 0, SPD[nb] -> 0, SPD[n, nb] -> 0}]

\text{FromGFAD: }\;\text{Some of the converted propagators are not strictly quadratic or eikonal.}

\text{FromGFAD: }\left\{\frac{1}{(\text{k1}^2+2 (\text{k1}\cdot \;\text{k2})-2 (\text{k1}\cdot \;\text{k3})+\text{k2}^2-2 (\text{k2}\cdot \;\text{k3})+\text{k3}^2-2 \;\text{gkin} \;\text{meta} \;\text{u0b} (\text{k1}\cdot n+\text{k2}\cdot n-\text{k3}\cdot n)+i \eta )}\right\}

\text{FromGFAD: }\;\text{These propagators may later cause issues with topology minimization routines.}

\left\{\text{FCTopology}\left(\text{topology1230},\left\{\frac{1}{(\text{k3}^2+i \eta )},\frac{1}{(-\text{k2}\cdot \;\text{nb}+i \eta )},\frac{1}{((\text{k1}-\frac{\text{meta} \;\text{u0b} \;\text{nb}}{2})^2+i \eta )},\frac{1}{((\text{k1}-\text{gkin} \;\text{meta} \;\text{u0b} n)^2+i \eta )},\frac{1}{(-(\text{k1}+\text{k2})\cdot \;\text{nb}+2 \;\text{gkin} \;\text{meta} \;\text{u0b}+i \eta )},\frac{1}{((\text{k1}+\text{k2}-\text{gkin} \;\text{meta} \;\text{u0b} n)^2+i \eta )},\frac{1}{((\text{k2}-\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}^2+2 (\text{k1}\cdot \;\text{k2})-2 (\text{k1}\cdot \;\text{k3})+\text{k2}^2-2 (\text{k2}\cdot \;\text{k3})+\text{k3}^2-2 \;\text{gkin} \;\text{meta} \;\text{u0b} (\text{k1}\cdot n+\text{k2}\cdot n-\text{k3}\cdot n)+i \eta )},\frac{1}{((\text{k1}-\text{k3})\cdot \;\text{nb}+i \eta )},\frac{1}{((\text{k1}-\text{k3}+\frac{\text{meta} \;\text{u0b} \;\text{nb}}{2})^2+i \eta )},\frac{1}{((\text{k1}-\text{k3}+\text{gkin} \;\text{meta} n)^2+i \eta )},\frac{1}{((\text{k1}-\text{k2})^2+i \eta )}\right\},\{\text{k1},\text{k2},\text{k3}\},\{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\}

Rearranging the rules accordingly, we can make the conversion succeed

FCLoopReplaceQuadraticEikonalPropagators[testTopo, LoopMomenta -> {k1, k2, k3}, InitialSubstitutions -> {
    ExpandScalarProduct[SPD[k1 + k2 - k3]] -> SPD[k1 + k2 - k3], 
    ExpandScalarProduct[SPD[k2 - k3]] -> SPD[k2 - k3], ExpandScalarProduct[SPD[k1 - k3]] -> SPD[k1 - k3], 
    ExpandScalarProduct[SPD[k1 + k3]] -> SPD[k1 + k3], 
    ExpandScalarProduct[SPD[k1 + k2]] -> SPD[k1 + k2] 
   }, IntermediateSubstitutions -> {SPD[n] -> 0, SPD[nb] -> 0, SPD[n, nb] -> 0}]

\left\{\text{FCTopology}\left(\text{topology1230},\left\{\frac{1}{(\text{k3}^2+i \eta )},\frac{1}{(-\text{k2}\cdot \;\text{nb}+i \eta )},\frac{1}{((\text{k1}-\frac{\text{meta} \;\text{u0b} \;\text{nb}}{2})^2+i \eta )},\frac{1}{((\text{k1}-\text{gkin} \;\text{meta} \;\text{u0b} n)^2+i \eta )},\frac{1}{(-(\text{k1}+\text{k2})\cdot \;\text{nb}+2 \;\text{gkin} \;\text{meta} \;\text{u0b}+i \eta )},\frac{1}{((\text{k1}+\text{k2}-\text{gkin} \;\text{meta} \;\text{u0b} n)^2+i \eta )},\frac{1}{((\text{k2}-\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{k2}-\text{k3}-\text{gkin} \;\text{meta} \;\text{u0b} n)^2+i \eta )},\frac{1}{((\text{k1}-\text{k3})\cdot \;\text{nb}+i \eta )},\frac{1}{((\text{k1}-\text{k3}+\frac{\text{meta} \;\text{u0b} \;\text{nb}}{2})^2+i \eta )},\frac{1}{((\text{k1}-\text{k3}+\text{gkin} \;\text{meta} n)^2+i \eta )},\frac{1}{((\text{k1}-\text{k2})^2+i \eta )}\right\},\{\text{k1},\text{k2},\text{k3}\},\{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\}