Added new function FCLoopReplaceQuadraticEikonalPropagators for

rewriting mixed quadratic-eikonal propagators in terms purely quadratic ones.
FeynCalc · Aug 7, 2024 · 8990853 · 8990853
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diff --git a/FeynCalc/Documentation/Markdown/Extra/FeynCalc.md b/FeynCalc/Documentation/Markdown/Extra/FeynCalc.md
@@ -351,6 +351,7 @@ FeynCalc is a Mathematica package for algebraic calculations in Quantum Field Th
 - [FCLoopPropagatorPowersExpand](../FCLoopPropagatorPowersExpand.md) - rewrites propagators raised to integer powers as products of propagators
 - [FCLoopPropagatorsToTopology](../FCLoopPropagatorsToTopology.md) - auxiliary function that generates a list of propagators to describe a topology
 - [FCLoopRemovePropagator](../FCLoopRemovePropagator.md) - removes specific propagators from given `FCTopology`s or `GLI`s
+- [FCLoopReplaceQuadraticEikonalPropagators](../FCLoopReplaceQuadraticEikonalPropagators.md) - tries to eliminate quadratic-eikonal propagators by completing the square
 - [FCLoopSamePropagatorHeadsQ](../FCLoopSamePropagatorHeadsQ.md) - checks if the integral contains different types of propagators
 - [FCLoopRemoveNegativePropagatorPowers](../FCLoopRemoveNegativePropagatorPowers.md) - rewrites propagators rasied to negative integer powers as numerators
 - [FCLoopSelectTopology](../FCLoopSelectTopology.md) - selects topologies belonging for the given `GLI`s

diff --git a/FeynCalc/Documentation/Mathematica/LoopIntegrals/FCLoopReplaceQuadraticEikonalPropagators.m b/FeynCalc/Documentation/Mathematica/LoopIntegrals/FCLoopReplaceQuadraticEikonalPropagators.m
@@ -0,0 +1,49 @@
+(* ::Package:: *)
+
+(* ::Section:: *)
+(*FCLoopReplaceQuadraticEikonalPropagators*)
+
+
+(* ::Text:: *)
+(*`FCLoopReplaceQuadraticEikonalPropagators[topologies]` identifies `SFAD`s and `CFAD`s 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 $[(p-q)^2 - q^2]$.*)
+
+
+(* ::Text:: *)
+(*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.*)
+
+
+(* ::Text:: *)
+(*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 $(p_1+p_2)^2$.*)
+
+
+(* ::Text:: *)
+(*Internally this routine uses `ToGFAD` and `FromGFAD`.*)
+
+
+(* ::Subsection:: *)
+(*See also*)
+
+
+(* ::Text:: *)
+(*[Overview](Extra/FeynCalc.md), [FCTopology](FCTopology.md), [GFAD](GFAD.md), [FromGFAD](FromGFAD.md), [ToGFAD](ToGFAD.md).*)
+
+
+(* ::Subsection:: *)
+(*Examples*)
+
+
+(DataType[#,FCVariable]=True)&/@\!\(TraditionalForm\`{gkin, meta, u0b}\);
+
+
+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},{}]}
+
+
+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}]
diff --git a/FeynCalc/LoopIntegrals/FCLoopReplaceQuadraticEikonalPropagators.m b/FeynCalc/LoopIntegrals/FCLoopReplaceQuadraticEikonalPropagators.m
@@ -0,0 +1,118 @@
+(* ::Package:: *)
+
+(* ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ *)
+
+(* :Title: FCLoopReplaceQuadraticEikonalPropagators							*)
+
+(*
+	This software is covered by the GNU General Public License 3.
+	Copyright (C) 1990-2024 Rolf Mertig
+	Copyright (C) 1997-2024 Frederik Orellana
+	Copyright (C) 2014-2024 Vladyslav Shtabovenko
+*)
+
+(* :Summary:	Tries to eliminate quadratic-eikonal propagators by
+				completing the square										*)
+
+(* ------------------------------------------------------------------------ *)
+
+FCLoopReplaceQuadraticEikonalPropagators::usage =
+"FCLoopReplaceQuadraticEikonalPropagators[exp] checks if the integral is free of eikonal
+propagators $\\frac{1}{p \\cdot q+x}$. If the option First is set to False,
+propagators that have both a quadratic and linear piece, e.g. $\\frac{1}{p^2 +
+p \\cdot q+x}$ will also count as eikonal propagators. The option Momentum can
+be used to check for the presence of eikonal propagators only with respect to
+particular momenta. The check is performed only for
+StandardPropagatorDenominator and CartesianPropagatorDenominator.";
+
+FCLoopReplaceQuadraticEikonalPropagators::failmsg =
+"FCLoopReplaceQuadraticEikonalPropagators has encountered a fatal problem and must abort the computation. \
+The problem reads: `1`"
+
+Begin["`Package`"]
+End[]
+
+Begin["`FCLoopReplaceQuadraticEikonalPropagators`Private`"]
+
+rqepVerbose::usage="";
+
+Options[FCLoopReplaceQuadraticEikonalPropagators] = {
+		Check						-> 	True,
+		ExpandScalarProduct			-> 	True,
+		FCE							->	False,
+		FCI							->	False,
+		FCVerbose					-> 	False,
+		FeynAmpDenominatorExplicit	-> 	True,
+		InitialSubstitutions		->	{},
+		IntermediateSubstitutions	->	{},
+		LoopMomenta					-> 	{},
+		MomentumCombine				->  True,
+		PowerExpand					-> 	{}
+};
+
+FCLoopReplaceQuadraticEikonalPropagators[topo_FCTopology, opts:OptionsPattern[]]:=
+	FCLoopReplaceQuadraticEikonalPropagators[{topo}, opts];
+
+FCLoopReplaceQuadraticEikonalPropagators[toposRaw:{__FCTopology}, OptionsPattern[]]:=
+	Block[{	topos, optInitialSubstitutions, optIntermediateSubstitutions,
+			optPowerExpand, optFeynAmpDenominatorExplicit, optLoopMomenta,
+			props, propsConverted, repRule},
+
+		optInitialSubstitutions 		= OptionValue[InitialSubstitutions];
+		optIntermediateSubstitutions	= OptionValue[IntermediateSubstitutions];
+
+		If [OptionValue[FCVerbose]===False,
+			rqepVerbose=$VeryVerbose,
+			If[MatchQ[OptionValue[FCVerbose], _Integer],
+				rqepVerbose=OptionValue[FCVerbose]
+			];
+		];
+
+		FCPrint[1, "FCLoopReplaceQuadraticEikonalPropagators: Entering.", FCDoControl->rqepVerbose];
+		FCPrint[3, "FCLoopReplaceQuadraticEikonalPropagators: Entering with: ", {toposRaw}, FCDoControl->rqepVerbose];
+
+		{optInitialSubstitutions,optIntermediateSubstitutions} = FRH[{optInitialSubstitutions,optIntermediateSubstitutions}];
+
+		If[	!OptionValue[FCI],
+			{topos, optInitialSubstitutions, optIntermediateSubstitutions} =
+				FCI[{toposRaw,optInitialSubstitutions, optIntermediateSubstitutions}],
+			topos = toposRaw
+		];
+
+		props = Union[Cases[topos, FeynAmpDenominator[(StandardPropagatorDenominator|CartesianPropagatorDenominator)[a_ /; a =!= 0, b_ /; b =!= 0, ___]], Infinity]];
+
+		If[props==={},
+			FCPrint[1, "FCLoopReplaceQuadraticEikonalPropagators: Nothing to do.", FCDoControl->rqepVerbose];
+			Return[topos]
+		];
+
+		propsConverted = ToGFAD[props,FCI->True,FinalSubstitutions->optIntermediateSubstitutions];
+
+		FCPrint[3, "FCLoopReplaceQuadraticEikonalPropagators: After ToGFAD: ", propsConverted, FCDoControl->rqepVerbose];
+
+		propsConverted = FromGFAD[propsConverted,FCI->True,
+			InitialSubstitutions->optInitialSubstitutions,
+			IntermediateSubstitutions->optIntermediateSubstitutions,
+			PowerExpand->OptionValue[PowerExpand],
+			FeynAmpDenominatorExplicit->OptionValue[FeynAmpDenominatorExplicit],
+			ExpandScalarProduct->OptionValue[ExpandScalarProduct],
+			LoopMomenta->OptionValue[LoopMomenta],
+			MomentumCombine->OptionValue[MomentumCombine],
+			Check->OptionValue[Check]];
+
+		repRule = Thread[Rule[props,propsConverted]];
+
+		FCPrint[3, "FCLoopReplaceQuadraticEikonalPropagators: Replacement rule: ", repRule, FCDoControl->rqepVerbose];
+
+		topos = topos /. Dispatch[repRule];
+
+		FCPrint[1, "FCLoopReplaceQuadraticEikonalPropagators: Leaving.", FCDoControl->rqepVerbose];
+		FCPrint[3, "FCLoopReplaceQuadraticEikonalPropagators: Leaving with: ", topos, FCDoControl->rqepVerbose];
+
+		topos
+
+
+		]
+
+FCPrint[1,"FCLoopReplaceQuadraticEikonalPropagators.m loaded."];
+End[]
diff --git a/Tests/LoopIntegrals/FCLoopReplaceQuadraticEikonalPropagators.test b/Tests/LoopIntegrals/FCLoopReplaceQuadraticEikonalPropagators.test
@@ -0,0 +1,160 @@
+
+
+(* :Title: FCLoopReplaceQuadraticEikonalPropagators.test										*)
+
+(*
+	This software is covered by the GNU General Public License 3.
+	Copyright (C) 1990-2024 Rolf Mertig
+	Copyright (C) 1997-2024 Frederik Orellana
+	Copyright (C) 2014-2024 Vladyslav Shtabovenko
+*)
+
+(* :Summary:  Framework independent unit tests for FCLoopReplaceQuadraticEikonalPropagators	*)
+
+(* ------------------------------------------------------------------------ *)
+
+Tests`LoopIntegrals`FCLoopReplaceQuadraticEikonalPropagators =
+({
+{"fcstFCLoopReplaceQuadraticEikonalPropagators-ID1",
+"FCLoopReplaceQuadraticEikonalPropagators[FCTopology[pfrTopo226, \
+{SFAD[{{k2, 0}, {0, 1}, 1}], SFAD[{{k1, 0}, {0, 1}, 1}], SFAD[{{k1 - \
+(meta*nb*u0b)/2, 0}, {0, 1}, 1}], SFAD[{{k2 + gkin*meta*n*u0b, 0}, \
+{0, 1}, 1}], SFAD[{{k1 - k2 - (meta*nb*u0b)/2, 0}, {0, 1}, 1}], \
+SFAD[{{k1 + (meta*(2*gkin*n - nb)*u0b)/2, 0}, {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[
+pfrTopo226, {FeynAmpDenominator[
+	StandardPropagatorDenominator[Momentum[k2, D], 0, 0, {1, 1}]],
+FeynAmpDenominator[
+	StandardPropagatorDenominator[Momentum[k1, D], 0, 0, {1, 1}]],
+FeynAmpDenominator[
+	StandardPropagatorDenominator[
+	 Momentum[k1, D] - Momentum[meta*nb*u0b, D]/2, 0, 0, {1, 1}]],
+FeynAmpDenominator[
+	StandardPropagatorDenominator[
+	 Momentum[k2, D] + Momentum[gkin*meta*n*u0b, D], 0, 0, {1, 1}]],
+FeynAmpDenominator[
+	StandardPropagatorDenominator[
+	 Momentum[k1, D] - Momentum[k2, D] - Momentum[meta*nb*u0b, D]/2,
+	 0, 0, {1, 1}]],
+FeynAmpDenominator[
+	StandardPropagatorDenominator[
+	 Momentum[k1, D] + Momentum[gkin*meta*n*u0b, D] -
+	  Momentum[meta*nb*u0b, D]/2,
+	 0, -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}, {}]}"},
+{"fcstFCLoopReplaceQuadraticEikonalPropagators-ID2",
+"FCLoopReplaceQuadraticEikonalPropagators[{FCTopology[pfrTopo226, \
+{SFAD[{{k2, 0}, {0, 1}, 1}], SFAD[{{k1, 0}, {0, 1}, 1}], SFAD[{{k1 - \
+(meta*nb*u0b)/2, 0}, {0, 1}, 1}], SFAD[{{k2 + gkin*meta*n*u0b, 0}, \
+{0, 1}, 1}], SFAD[{{k1 - k2 - (meta*nb*u0b)/2, 0}, {0, 1}, 1}], \
+SFAD[{{k1 + (meta*(2*gkin*n - nb)*u0b)/2, 0}, {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[pfrTopo212, {SFAD[{{k2, 0}, \
+{0, 1}, 1}], SFAD[{{k1, 0}, {0, 1}, 1}], SFAD[{{k2 - (meta*nb*u0b)/2, \
+0}, {0, 1}, 1}], SFAD[{{k1 + gkin*meta*n*u0b, 0}, {0, 1}, 1}], \
+SFAD[{{k1 - k2 + gkin*meta*n*u0b, 0}, {0, 1}, 1}], SFAD[{{k1 + \
+(meta*(2*gkin*n - nb)*u0b)/2, 0}, {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}, \
+{}]}]\[IndentingNewLine](DataType[#,FCVariable]=True)&/@\!\(\*FormBox[\
+\({gkin, meta, u0b}\),
+	TraditionalForm]\);FCLoopReplaceQuadraticEikonalPropagators[{\
+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},{}]},\[IndentingNewLine]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, \
+{FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k2, D], 0, \
+0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1, D], 0, \
+0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1, D] + \
+Momentum[k2, D], 0, 0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[0, \
+-Pair[Momentum[k1, D], Momentum[nb, D]], 0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k2, D] - \
+(meta*u0b*Momentum[nb, D])/2, 0, 0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1 + k2, D] \
+- gkin*meta*u0b*Momentum[n, D], 0, 0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1, D] - \
+(meta*Momentum[2*gkin*n - nb*u0b, D])/2, 0, -2*gkin*meta^2*u0b, {1, \
+1}]], FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1, \
+D] - (meta*u0b*Momentum[2*gkin*n - nb, D])/2, 0, \
+-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}, {}]}"},
+{"fcstFCLoopReplaceQuadraticEikonalPropagators-ID3",
+"FCLoopReplaceQuadraticEikonalPropagators[{FCTopology[preTopoDia2,{\
+SFAD[{{k2,0},{0,1},1}],SFAD[{{k1,0},{0,1},1}],SFAD[{{0,k2.nb},{2*gkin*\
+meta,1},1}],SFAD[{{0,(k1+k2).nb},{2*gkin*meta,1},1}],SFAD[{{k1,-(meta*\
+u0b*k1.nb)},{0,1},1}],SFAD[{{k1,-2*gkin*meta*u0b*k1.n},{0,1},1}],SFAD[\
+{{k1+k2,-2*gkin*meta*(k1+k2).n},{0,1},1}],SFAD[{{k2,-2*gkin*meta*k2.n+\
+2*gkin*meta*u0b*k2.n},{0,1},1}]},{k1,k2},{n,nb},{Hold[SPD][n]->0,Hold[\
+SPD][nb]->0,Hold[SPD][n,nb]->2},{}]},\[IndentingNewLine]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[preTopoDia2, \
+{FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k2, D], 0, \
+0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1, D], 0, \
+0, {1, 1}]], FeynAmpDenominator[StandardPropagatorDenominator[0, \
+Pair[Momentum[k2, D], Momentum[nb, D]], -2*gkin*meta, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[0, Pair[Momentum[k1, \
+D] + Momentum[k2, D], Momentum[nb, D]], -2*gkin*meta, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1, D] - \
+(meta*u0b*Momentum[nb, D])/2, 0, 0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1, D] - \
+gkin*meta*u0b*Momentum[n, D], 0, 0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1 + k2, D] \
+- gkin*meta*Momentum[n, D], 0, 0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k2, D] - \
+gkin*meta*Momentum[n*(1 - u0b), D], 0, 0, {1, 1}]]}, {k1, k2}, {n, \
+nb}, {Hold[SPD][n] -> 0, Hold[SPD][nb] -> 0, Hold[SPD][n, nb] -> 2}, \
+{}]}"},
+{"fcstFCLoopReplaceQuadraticEikonalPropagators-ID4",
+"FCLoopReplaceQuadraticEikonalPropagators[{FCTopology[preTopoDia3,{\
+SFAD[{{k2,0},{0,1},1}],SFAD[{{k1,0},{0,1},1}],SFAD[{{0,k2.nb},{2*gkin*\
+meta,1},1}],SFAD[{{0,k1.nb},{2*gkin*meta*u0b,1},1}],SFAD[{{k1,-(meta*\
+u0b*k1.nb)},{0,1},1}],SFAD[{{k1,-2*gkin*meta*u0b*k1.n},{0,1},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-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}]},{k1,k2},{n,\
+nb},{Hold[SPD][n]->0,Hold[SPD][nb]->0,Hold[SPD][n,nb]->2},{}]},\
+\[IndentingNewLine]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[preTopoDia3, \
+{FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k2, D], 0, \
+0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1, D], 0, \
+0, {1, 1}]], FeynAmpDenominator[StandardPropagatorDenominator[0, \
+Pair[Momentum[k2, D], Momentum[nb, D]], -2*gkin*meta, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[0, Pair[Momentum[k1, \
+D], Momentum[nb, D]], -2*gkin*meta*u0b, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1, D] - \
+(meta*u0b*Momentum[nb, D])/2, 0, 0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1, D] - \
+gkin*meta*u0b*Momentum[n, D], 0, 0, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1, D] + \
+(meta*Momentum[2*gkin*n - 2*gkin*n*u0b - nb*u0b, D])/2, 0, \
+-2*gkin*meta^2*u0b + 2*gkin*meta^2*u0b^2, {1, 1}]], \
+FeynAmpDenominator[StandardPropagatorDenominator[Momentum[k1 - k2, D] \
++ (meta*Momentum[2*gkin*n - 2*gkin*n*u0b - nb*u0b, D])/2, 0, \
+-2*gkin*meta^2*u0b + 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}, \
+{}]}"}
+})