FeynCalc manual (development version)
+FCLoopFindMomentumShifts
FCLoopFindMomentumShifts[source, target, {p1, p2, ...}]
finds loop momentum shifts that bring loop integrals or topologies in
the list source to the form specified in target. The
@@ -123,7 +124,9 @@
FeynCalc manual (development version)
For topologies involving kinematic constraints some mappings may
require shifts not only in the loop but also in the external momenta.
Such shifts are disabled by default but can be activated by setting the
-option Momentum to All.
Momentum to All. This option can be
+dangerous, because the amplitude does not necessarily have to be
+symmetric under shifts of external momenta!
Normally, FCLoopFindMomentumShifts will abort the
evaluation if it fails to find any suitable shifts. Setting the option
Abort to False will force the function to
@@ -157,15 +160,14 @@
Examples
FCLoopFindMomentumShifts[source, target, {p1, p3, p4}]\{\{\text{p1}\to -\;\text{p1}+\text{p4}+\text{q1},\text{p3}\to \;\text{p3},\text{p4}\to -\;\text{p4}\}\}
+\;\text{p1}+\text{p4}+\text{q1}\}\}FCLoopFindMomentumShifts[{{FAD[r4], FAD[r1], FAD[r1 - p3 - r4],
FAD[{r1 - r4, m1}], FAD[{p3, m1}], FAD[p3 + q1], FAD[r1 + q1]}},
{FAD[p4], FAD[p1 + p4 + q1], FAD[p1 - p3 + q1], FAD[{p1 + q1, m1}],
FAD[{p3, m1}], FAD[p3 + q1], FAD[p1 + p4 + 2 q1]}, {p1, p3, p4, r4,r1}]\{\{\text{p3}\to \;\text{p3},\text{r4}\to -\;\text{p4},\text{r1}\to \;\text{p1}+\text{p4}+\text{q1}\}\}
+\{\{\text{r4}\to \;\text{p4},\text{r1}\to +\;\text{p1}+\text{p4}+\text{q1}\}\}
source1 = {FCTopology[
fctopology3, {SFAD[{{p1, 0}, {0, 1}, 1}],
@@ -222,8 +224,8 @@ Examples
FCLoopFindMomentumShifts[source1, target1]
\{\{\text{p1}\to
--\text{p1}-\text{p3}+Q,\text{p2}\to -\text{p2}-\text{p3}+Q,\text{p3}\to
-\;\text{p3}\},\{\text{p1}\to Q-\text{p2},\text{p2}\to
+-\text{p1}-\text{p3}+Q,\text{p2}\to
+-\text{p2}-\text{p3}+Q\},\{\text{p1}\to Q-\text{p2},\text{p2}\to
Q-\text{p1},\text{p3}\to -\text{p3}\}\}
source2 = {FCTopology[topo1, {
@@ -263,12 +265,146 @@ Examples
external momenta
Quiet[FCLoopFindMomentumShifts[source2, target2, Abort -> False]]
+\text{FCLoopFindMomentumShifts:
+}\;\text{Failed to derive the momentum shifts between topologies topo1
+and topo2. This can be due to the presence of nonquadratic propagators
+or because shifts in external momenta are also necessary.}
\{\{\}\}
Once we allow such shifts, everything works as expected
FCLoopFindMomentumShifts[source2, target2, Momentum -> All]
\{\{\text{l1}\to
-\;\text{l1}-\text{l2}-\text{q2},\text{l2}\to -\text{l2},\text{q1}\to
-\;\text{q2},\text{q2}\to \;\text{q1}\}\}
+-\text{l1}+\text{l2}+\text{q2},\text{q1}\to -\text{q2},\text{q2}\to
+-\text{q1}\}\}
+For equivalent topologies containing mixed quadratic-eikonal
+propagators it’s often not possible to find suitable shifts because the
+function cannot reconstruct the correct momentum flow through such
+propagators
+source3 = {FCTopology["pfrTopo303", {SFAD[{{k2, -2 gkin meta k2 . n + meta u0b k2 . nb},
+ {2 gkin meta^2 u0b, 1}, 1}], SFAD[{{k1 - k2, meta u0b (-k1 + k2) . nb}, {0, 1}, 1}],
+ SFAD[{{k1, -2 gkin meta k1 . n}, {0, 1}, 1}], SFAD[{{k1, -meta u0b k1 . nb}, {0, 1}, 1}],
+ SFAD[{{0, -k2 . nb}, {0, 1}, 1}], SFAD[{{0, -k1 . nb}, {0, 1}, 1}]}, {k1, k2}, {n, nb},
+ {Hold[SPD][n] -> 0, Hold[SPD][nb] -> 0, Hold[SPD][n, nb] -> 2}, {}]}
+\left\{\text{FCTopology}\left(\text{pfrTopo303},\left\{\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+\text{meta} \;\text{u0b}
+((\text{k2}-\text{k1})\cdot \;\text{nb})+i \eta
+)},\frac{1}{(\text{k1}^2-2 \;\text{gkin} \;\text{meta} (\text{k1}\cdot
+n)+i \eta )},\frac{1}{(\text{k1}^2-\text{meta} \;\text{u0b}
+(\text{k1}\cdot \;\text{nb})+i \eta )},\frac{1}{(-\text{k2}\cdot
+\;\text{nb}+i \eta )},\frac{1}{(-\text{k1}\cdot \;\text{nb}+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\}
+target3 = FCTopology["pfrTopo267", {SFAD[{{k2, 0}, {0, 1}, 1}], SFAD[{{k1 - k2, 2 gkin meta k1 . n - 2 gkin meta u0b k1 . n - 2 gkin meta k2 . n + 2 gkin meta u0b k2 . n + meta u0b (-k1 + k2) . nb}, {2 gkin meta^2 u0b - 2 gkin meta^2 u0b^2, 1}, 1}], SFAD[{{k1, 2 gkin meta k1 . n - 2 gkin meta u0b k1 . n - meta u0b k1 . nb}, {2 gkin meta^2 u0b - 2 gkin meta^2 u0b^2, 1}, 1}], SFAD[{{k1, -2 gkin meta u0b k1 . n}, {0, 1}, 1}], SFAD[{{0, k2 . nb}, {2 gkin meta, 1}, 1}], SFAD[{{0, k1 . nb}, {2 gkin meta u0b, 1}, 1}]}, {k1, k2}, {n, nb}, {Hold[SPD][n] -> 0, Hold[SPD][nb] -> 0, Hold[SPD][n, nb] -> 2}, {}]
+\text{FCTopology}\left(\text{pfrTopo267},\left\{\frac{1}{(\text{k2}^2+i
+\eta )},1\left/\left(((\text{k1}-\text{k2})^2+2 \;\text{gkin}
+\;\text{meta} (\text{k1}\cdot n)-2 \;\text{gkin} \;\text{meta}
+\;\text{u0b} (\text{k1}\cdot n)-2 \;\text{gkin} \;\text{meta}
+(\text{k2}\cdot n)+2 \;\text{gkin} \;\text{meta} \;\text{u0b}
+(\text{k2}\cdot n)+\text{meta} \;\text{u0b} ((\text{k2}-\text{k1})\cdot
+\;\text{nb})+2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}^2-2
+\;\text{gkin} \;\text{meta}^2 \;\text{u0b}+i \eta
+)\right)\right.,\frac{1}{(\text{k1}^2+2 \;\text{gkin} \;\text{meta}
+(\text{k1}\cdot n)-2 \;\text{gkin} \;\text{meta} \;\text{u0b}
+(\text{k1}\cdot n)-\text{meta} \;\text{u0b} (\text{k1}\cdot
+\;\text{nb})+2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}^2-2
+\;\text{gkin} \;\text{meta}^2 \;\text{u0b}+i \eta
+)},\frac{1}{(\text{k1}^2-2 \;\text{gkin} \;\text{meta} \;\text{u0b}
+(\text{k1}\cdot n)+i \eta )},\frac{1}{(\text{k2}\cdot \;\text{nb}-2
+\;\text{gkin} \;\text{meta}+i \eta )},\frac{1}{(\text{k1}\cdot
+\;\text{nb}-2 \;\text{gkin} \;\text{meta} \;\text{u0b}+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)
+(\text{DataType}[\#,\text{FCVariable}]=\text{True})\&\text{/@}\{\text{gkin},\text{meta},\text{u0b}\};
+FCLoopFindMomentumShifts[source3, target3]
+\text{FCLoopFindMomentumShifts:
+}\;\text{The topologies contain following mixed quadratic-eikonal
+propagators that complicate the determination of the shifts:
+}\left\{\frac{1}{(\text{k1}^2-2 \;\text{gkin} \;\text{meta}
+(\text{k1}\cdot n)+i \eta )},\frac{1}{(\text{k1}^2-2 \;\text{gkin}
+\;\text{meta} \;\text{u0b} (\text{k1}\cdot n)+i \eta
+)},\frac{1}{(\text{k1}^2-\text{meta} \;\text{u0b} (\text{k1}\cdot
+\;\text{nb})+i \eta )},\frac{1}{(\text{k1}^2+n\cdot (2 \;\text{gkin}
+\;\text{meta} \;\text{k1}-2 \;\text{gkin} \;\text{meta} \;\text{u0b}
+\;\text{k1})-\text{meta} \;\text{u0b} (\text{k1}\cdot \;\text{nb})+2
+\;\text{gkin} \;\text{meta}^2 \;\text{u0b}^2-2 \;\text{gkin}
+\;\text{meta}^2 \;\text{u0b}+i \eta
+)},\frac{1}{((\text{k1}-\text{k2})^2+\text{meta} \;\text{u0b}
+((\text{k2}-\text{k1})\cdot \;\text{nb})+i \eta
+)},\frac{1}{((\text{k1}-\text{k2})^2+\text{meta} \;\text{u0b}
+((\text{k2}-\text{k1})\cdot \;\text{nb})+n\cdot (2 \;\text{gkin}
+\;\text{meta} \;\text{k1}-2 \;\text{gkin} \;\text{meta} \;\text{u0b}
+\;\text{k1}-2 \;\text{gkin} \;\text{meta} \;\text{k2}+2 \;\text{gkin}
+\;\text{meta} \;\text{u0b} \;\text{k2})+2 \;\text{gkin} \;\text{meta}^2
+\;\text{u0b}^2-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}+i \eta
+)},\frac{1}{(\text{k2}^2+\text{k2}\cdot (\text{meta} \;\text{u0b}
+\;\text{nb}-2 \;\text{gkin} \;\text{meta} n)-2 \;\text{gkin}
+\;\text{meta}^2 \;\text{u0b}+i \eta )}\right\}
+\text{FCLoopFindMomentumShifts:
+}\;\text{You can try to trade them for purely quadratic propagators
+using FCLoopReplaceQuadraticEikonalPropagators.}
+
+\text{FCLoopFindMomentumShifts:
+}\;\text{Failed to derive the momentum shifts between topologies
+pfrTopo303 and pfrTopo267. This can be due to the presence of
+nonquadratic propagators or because shifts in external momenta are also
+necessary.}
+\text{\$Aborted}
+To this aim one can try converting those mixed propagators to purely
+quadratic ones using
+FCLoopReplaceQuadraticEikonalPropagators
+source3New = FCLoopReplaceQuadraticEikonalPropagators[source3, LoopMomenta -> {k1, k2},
+ InitialSubstitutions -> {ExpandScalarProduct[SPD[k1 - k2]] -> SPD[k1 - k2]},
+ IntermediateSubstitutions -> {SPD[n] -> 0, SPD[nb] -> 0, SPD[n, nb] -> 2}]
+\left\{\text{FCTopology}\left(\text{pfrTopo303},\left\{\frac{1}{((\text{k2}-\text{gkin}
+\;\text{meta} n+\frac{\text{meta} \;\text{u0b} \;\text{nb}}{2})^2+i \eta
+)},\frac{1}{((\text{k1}-\text{k2}-\frac{\text{meta} \;\text{u0b}
+\;\text{nb}}{2})^2+i \eta )},\frac{1}{((\text{k1}-\text{gkin}
+\;\text{meta} n)^2+i \eta )},\frac{1}{((\text{k1}-\frac{\text{meta}
+\;\text{u0b} \;\text{nb}}{2})^2+i \eta )},\frac{1}{(-\text{k2}\cdot
+\;\text{nb}+i \eta )},\frac{1}{(-\text{k1}\cdot \;\text{nb}+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\}
+target3New = FCLoopReplaceQuadraticEikonalPropagators[target3, LoopMomenta -> {k1, k2},
+ InitialSubstitutions -> {ExpandScalarProduct[SPD[k1 - k2]] -> SPD[k1 - k2]},
+ IntermediateSubstitutions -> {SPD[n] -> 0, SPD[nb] -> 0, SPD[n, nb] -> 2}] // First
+\text{FCTopology}\left(\text{pfrTopo267},\left\{\frac{1}{(\text{k2}^2+i
+\eta )},\frac{1}{((\text{k1}-\text{k2}+\text{gkin} \;\text{meta}
+n-\text{gkin} \;\text{meta} \;\text{u0b} n-\frac{\text{meta}
+\;\text{u0b} \;\text{nb}}{2})^2+i \eta
+)},\frac{1}{((\text{k1}+\text{gkin} \;\text{meta} n-\text{gkin}
+\;\text{meta} \;\text{u0b} n-\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{k2}\cdot
+\;\text{nb}-2 \;\text{gkin} \;\text{meta}+i \eta
+)},\frac{1}{(\text{k1}\cdot \;\text{nb}-2 \;\text{gkin} \;\text{meta}
+\;\text{u0b}+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)
+With the new topologies everything works as expected
+FCLoopFindMomentumShifts[source3New, target3New]
+\left\{\left\{\text{k1}\to \;\text{gkin}
+\;\text{meta} n \;\text{u0b}-\text{k1}+\frac{\text{meta} \;\text{nb}
+\;\text{u0b}}{2},\text{k2}\to \frac{1}{2} (2 \;\text{gkin} \;\text{meta}
+n-2 \;\text{k2}-\text{meta} \;\text{nb}
+\;\text{u0b})\right\}\right\}