FeynCalc manual (development version)

FCLoopReplaceQuadraticEikonalPropagators

FCLoopReplaceQuadraticEikonalPropagators[topologies] identifies SFADs and CFADs in topologies that represent mixed quadratic-eikonal propagators, e.g. [p2−2p⋅q][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−q2][(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. p12−2p1⋅p2+p22p_1^2 - 2 p_1 \cdot p_2 + p_2^2 goes to (p1+p2)2(p_1+p_2)^2.

Internally this routine uses ToGFAD and FromGFAD.

See also

Overview, FCTopology, GFAD, FromGFAD, ToGFAD.

Examples

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

{FCTopology(preTopoDia1,{1(k22+iη),1(k12+iη),1((k1+k2)2+iη),1(−k1⋅  nb+iη),1(k22−meta  u0b(k2⋅  nb)+iη),1((k1+k2)2−2  gkin  meta  u0b((k1+k2)⋅n)+iη),1(k12+meta  u0b(k1⋅  nb)−2  gkin  meta(k1⋅n)−2  gkin  meta2  u0b+iη),1(k12+meta  u0b(k1⋅  nb)−2  gkin  meta  u0b(k1⋅n)−2  gkin  meta2  u0b2+iη)},{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}]

{FCTopology(preTopoDia1,{1(k22+iη),1(k12+iη),1((k1+k2)2+iη),1(−k1⋅  nb+iη),1((k2−meta  u0b  nb2)2+iη),1((k1+k2−gkin  meta  u0bn)2+iη),1((k1−gkin  metan+meta  u0b  nb2)2−2  gkin  meta2  u0b+iη),1((k1−gkin  meta  u0bn+meta  u0b  nb2)2−2  gkin  meta2  u0b2+iη)},{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}-\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}, {}]

FCTopology(topology1230,{1(k32+iη),1(−k2⋅  nb+iη),1(k12−meta  u0b(k1⋅  nb)+iη),1(k12−2  gkin  meta  u0b(k1⋅n)+iη),1(−(k1+k2)⋅  nb+2  gkin  meta  u0b+iη),1((k1+k2)2−2  gkin  meta  u0b((k1+k2)⋅n)+iη),1(k22+meta  u0b(k2⋅  nb)−2  gkin  meta(k2⋅n)−2  gkin  meta2  u0b+iη),1((k1+k2)2+−2  gkin  meta  u0b((k1+k2)⋅n)+k3⋅(−2  k1−2  k2+k3)+2  gkin  meta  u0b(k3⋅n)+iη),1((k1−k3)⋅  nb+iη),1(k12+meta  u0b((k1−k3)⋅  nb)+k3⋅(k3−2  k1)+iη),1(k12+2  gkin  meta(k1⋅n)+k3⋅(k3−2  k1)−2  gkin  meta(k3⋅n)+iη),1((k1−k2)2+iη)},{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}]

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

FromGFAD: {1(k12+2(k1⋅  k2)−2(k1⋅  k3)+k22−2(k2⋅  k3)+k32−2  gkin  meta  u0b(k1⋅n+k2⋅n−k3⋅n)+iη)}\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\}

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

{FCTopology(topology1230,{1(k32+iη),1(−k2⋅  nb+iη),1((k1−meta  u0b  nb2)2+iη),1((k1−gkin  meta  u0bn)2+iη),1(−(k1+k2)⋅  nb+2  gkin  meta  u0b+iη),1((k1+k2−gkin  meta  u0bn)2+iη),1((k2−gkin  metan+meta  u0b  nb2)2−2  gkin  meta2  u0b+iη),1(k12+2(k1⋅  k2)−2(k1⋅  k3)+k22−2(k2⋅  k3)+k32−2  gkin  meta  u0b(k1⋅n+k2⋅n−k3⋅n)+iη),1((k1−k3)⋅  nb+iη),1((k1−k3+meta  u0b  nb2)2+iη),1((k1−k3+gkin  metan)2+iη),1((k1−k2)2+iη)},{k1,k2,k3},{n,nb},{Hold[SPD][n]→0,Hold[SPD][nb]→0,Hold[SPD][n,nb]→2},{})}\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}]

{FCTopology(topology1230,{1(k32+iη),1(−k2⋅  nb+iη),1((k1−meta  u0b  nb2)2+iη),1((k1−gkin  meta  u0bn)2+iη),1(−(k1+k2)⋅  nb+2  gkin  meta  u0b+iη),1((k1+k2−gkin  meta  u0bn)2+iη),1((k2−gkin  metan+meta  u0b  nb2)2−2  gkin  meta2  u0b+iη),1((k1+k2−k3−gkin  meta  u0bn)2+iη),1((k1−k3)⋅  nb+iη),1((k1−k3+meta  u0b  nb2)2+iη),1((k1−k3+gkin  metan)2+iη),1((k1−k2)2+iη)},{k1,k2,k3},{n,nb},{Hold[SPD][n]→0,Hold[SPD][nb]→0,Hold[SPD][n,nb]→2},{})}\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\}