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Improved FCLoopApplyTopologyMappings to deliver final amplitudes
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writtten in terms of GLIs (provided that tensor decomposition or
projectors have already been applied).
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@vsht
vsht committed Jun 23, 2022
1 parent 20553a3 commit be98252
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25 changes: 16 additions & 9 deletions FeynCalc/Documentation/Mathematica/LoopIntegrals/FCLoopApplyTopologyMappings.m
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Original file line number Diff line number Diff line change
Expand Up @@ -8,7 +8,17 @@


(* ::Text:: *)
(*`FCLoopApplyTopologyMappings[expr, mappings]` applies mappings between topologies obtained using `FCLoopFindTopologyMappings` to the output of `FCLoopFindTopologies` denoted as `expr`.*)
(*`FCLoopApplyTopologyMappings[expr, {mappings, topos}]` applies mappings between topologies obtained using `FCLoopFindTopologyMappings` to the output of `FCLoopFindTopologies` denoted as `expr`. The argument `topos` denotes the final set of topologies present in the expression.*)


(* ::Text:: *)
(*Instead of `{mappings, topos}` one can directly use the output `FCLoopFindTopologyMappings`.*)


(* ::Text:: *)
(*By default the function will attempt to rewrite all the occurring loop integrals as `GLI`s. If you just want to apply the mappings without touching the remaining scalar products, *)
(*set the option `FCLoopCreateRulesToGLI` to `False`. Even when all scalar products depending on loop momenta are rewritten as `GLI`s, you can still suppress the step of multiplying out products*)
(*of `GLI`s by setting the option `GLIMultiply` to `False`.*)


(* ::Subsection:: *)
Expand Down Expand Up @@ -75,21 +85,18 @@
(*`FCLoopApplyTopologyMappings` applies the given mappings to the expression creating an output that is ready to be processed further*)


res=FCLoopApplyTopologyMappings[ex,mappings,Head->gliProduct]
FCLoopApplyTopologyMappings[ex,{mappings,finalTopos},Head->gliProduct,FCVerbose->0]


(* ::Text:: *)
(*Using `FCLoopCreateRulesToGLI` we obtain rules for rewriting scalar products as inverse `GLI`s*)


spToGliRules=FCLoopCreateRulesToGLI[finalTopos]
(*This just applies the mappings without any further simplifications*)


aux=res/.gliProduct[x_,y:GLI[idd_,inds_List]]/;(!FreeQ[spToGliRules,idd]):>gliProduct2[ExpandScalarProduct[x]/.SelectNotFree[spToGliRules,idd][[1]],y]/.gliProduct2[x_,y_]/;FreeQ[x,Pair]:> x y
FCLoopApplyTopologyMappings[ex,{mappings,finalTopos},Head->gliProduct,FCLoopCreateRulesToGLI->False]


(* ::Text:: *)
(*And here is our final expression written only in terms `GLI`s*)
(*This applies the mappings and eliminates the numerators but still keeps products of `GLI`s in the expression*)


resFinal=Collect2[aux/.gliProduct2->Times,GLI]/.GLI->GLIMultiply/.GLIMultiply->GLI
FCLoopApplyTopologyMappings[ex,{mappings,finalTopos},Head->gliProduct,FCLoopCreateRulesToGLI->True,GLIMultiply->False]
97 changes: 88 additions & 9 deletions FeynCalc/LoopIntegrals/FCLoopApplyTopologyMappings.m
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Original file line number Diff line number Diff line change
Expand Up @@ -24,60 +24,139 @@
"Error! FCLoopApplyTopologyMappings has encountered a fatal problem and must abort the computation. \
The problem reads: `1`"

FCLoopApplyTopologyMappings::lmoms =
"Warning! FCLoopApplyTopologyMappings failed to eliminate all loop momenta in the expression."

Begin["`Package`"]
End[]

Begin["`FCLoopApplyTopologyMappings`Private`"]

fclamVerbose::usage = "";
optHead::usage = "";
rule::usage = "";

Options[FCLoopApplyTopologyMappings] = {
Collecting -> True,
ExpandScalarProduct -> True,
FCE -> False,
FCI -> False,
FCLoopCreateRulesToGLI -> True,
FCVerbose -> False,
GLIMultiply -> True,
Head -> FCGV["GLIProduct"],
PreferredTopologies -> {},
Factoring -> {Factor2, 5000},
TimeConstrained -> 3
};

FCLoopApplyTopologyMappings[expr_, mappings_List, OptionsPattern[]] :=
Block[{ex, res, time, uniqueProductsList, tmp, repRule},
FCLoopApplyTopologyMappings[expr_, {mappings_List, toposRaw_List}, OptionsPattern[]] :=
Block[{ ex, res, time, uniqueProductsList, tmp, repRule, optFCLoopCreateRulesToGLI,
topos, optHead, rulesToGLI, topoIDs, rulesToGLIFinal, sps, spsEval, ruleSP,
lmoms, repRuleExtra, aux},

If[ OptionValue[FCVerbose] === False,
fclamVerbose = $VeryVerbose,
If[MatchQ[OptionValue[FCVerbose], _Integer],
fclamVerbose = OptionValue[FCVerbose]];
];

optHead = OptionValue[Head];
optHead = OptionValue[Head];
optFCLoopCreateRulesToGLI = OptionValue[FCLoopCreateRulesToGLI];


If[ !OptionValue[FCI],
(* For large expressions FCI might require a considerable amount of time! *)
time=AbsoluteTime[];
FCPrint[1,"FCLoopApplyTopologyMappings: Applying FCI.", FCDoControl->fclamVerbose];
ex = FCI[expr];
{ex, topos} = FCI[{expr,toposRaw}];
FCPrint[1, "FCLoopApplyTopologyMappings: Done applying FCI, timing: ", N[AbsoluteTime[] - time, 4], FCDoControl->fclamVerbose],
ex = expr
{ex, topos} = {expr,toposRaw}
];

If[ TrueQ[!FreeQ[tmp,optHead]],
tmp = ex;
uniqueProductsList = Cases2[tmp,optHead],

time=AbsoluteTime[];
FCPrint[1,"FCLoopApplyTopologyMappings: Applying Collect2.", FCDoControl->fclamVerbose];
tmp = Collect2[ex,GLI,Factoring->OptionValue[Factoring],TimeConstrained->OptionValue[TimeConstrained]];
uniqueProductsList = Cases2[tmp,optHead]
uniqueProductsList = Cases2[tmp,optHead];
FCPrint[1, "FCLoopApplyTopologyMappings: Collect2 done, timing: ", N[AbsoluteTime[] - time, 4], FCDoControl->fclamVerbose]
];

FCPrint[3,"FCLoopApplyTopologyMappings: Unique products of GLIs and scalar products: ", uniqueProductsList , FCDoControl->fclamVerbose];

(*TODO More checks ...*)
time=AbsoluteTime[];
FCPrint[1,"FCLoopApplyTopologyMappings: Creating initial replacement rules for the products.", FCDoControl->fclamVerbose];
repRule = Map[applyMapping[(SelectNotFree[uniqueProductsList, #[[1]][[1]]]), #] &, mappings];


aux = Complement[uniqueProductsList,First[First[#]]&/@repRule];

(*This is for topologies that remain unchanged*)
repRuleExtra = Thread[Rule[aux,aux]];

repRule = Flatten[repRule] /. rule->Rule;
FCPrint[1, "FCLoopApplyTopologyMappings: Done creating rules, timing: ", N[AbsoluteTime[] - time, 4], FCDoControl->fclamVerbose];
FCPrint[2,"FCLoopApplyTopologyMappings: Generated ", Length[repRule], "initial replacement rules", FCDoControl->fclamVerbose];
FCPrint[3,"FCLoopApplyTopologyMappings: Replacement rules: ",repRule, FCDoControl->fclamVerbose];

If[ TrueQ[optFCLoopCreateRulesToGLI=!=False],
time=AbsoluteTime[];
FCPrint[1,"FCLoopApplyTopologyMappings: Rewriting numerators with loop momenta as GLIs.", FCDoControl->fclamVerbose];

repRule = Join[repRule,repRuleExtra];
rulesToGLI = FCLoopCreateRulesToGLI[topos,FCI->True];
topoIDs= First[Last[Last[#]]]&/@repRule;


If[ Union[First/@topos]=!=Union[topoIDs],
Message[FCLoopApplyTopologyMappings::failmsg,"Missing topologies present in the input expression"];
Abort[]
];

res = tmp /. Dispatch[repRule] (*/. optHead -> Times*);
rulesToGLIFinal= Map[First[SelectNotFree[rulesToGLI,#]]&,topoIDs];

If[ OptionValue[ExpandScalarProduct],
sps = Cases2[Last/@repRule, Pair,CartesianPair];
spsEval = ExpandScalarProduct[sps,FCI->True];
ruleSP = Thread[Rule[sps,spsEval]],
ruleSP = {}
];

repRule = MapThread[Rule[First[#1],Last[#1]/. Dispatch[ruleSP] /. #2 ]&, {repRule,rulesToGLIFinal}];

FCPrint[1, "FCLoopApplyTopologyMappings: Done rewriting numerators, timing: ", N[AbsoluteTime[] - time, 4], FCDoControl->fclamVerbose];
FCPrint[3, "FCLoopApplyTopologyMappings: Replacement rule: ", repRule, FCDoControl->fclamVerbose];

If[ OptionValue[GLIMultiply],
time=AbsoluteTime[];
FCPrint[1,"FCLoopApplyTopologyMappings: Rewriting remaining products of GLIs.", FCDoControl->fclamVerbose];
repRule = Map[Rule[First[#],Expand2[Last[#]/. optHead->Times,GLI]/.GLI->GLIMultiply/.GLIMultiply->GLI]&,repRule];
FCPrint[1, "FCLoopApplyTopologyMappings: Done rewriting products of GLIs, timing: ", N[AbsoluteTime[] - time, 4], FCDoControl->fclamVerbose]
];

];


FCPrint[3, "FCLoopApplyTopologyMappings: Final replacement rule: ", repRule, FCDoControl->fclamVerbose];
time=AbsoluteTime[];
FCPrint[1,"FCLoopApplyTopologyMappings: Applying the final replacement rule.", FCDoControl->fclamVerbose];
res = tmp /. Dispatch[repRule] ;
FCPrint[1, "FCLoopApplyTopologyMappings: Done applying the final replacement rule, timing: ", N[AbsoluteTime[] - time, 4], FCDoControl->fclamVerbose];

If[ OptionValue[GLIMultiply] && optFCLoopCreateRulesToGLI,

lmoms = Union[Flatten[#[[3]]&/@topos]];
If[ !FreeQ2[res,lmoms],
Message[FCLoopApplyTopologyMappings::lmoms]
]
];

If[ OptionValue[Collecting],
res = Collect2[res,GLI,Factoring->OptionValue[Factoring],TimeConstrained->OptionValue[TimeConstrained]];
];

If[ OptiopnValue[FCE],
res = FCE[res]
Expand All @@ -88,7 +167,7 @@
];

applyMapping[terms_List, mappingRules_List] :=
Map[rule[#, # /. mappingRules[[2]] /. mappingRules[[3]] (*/. optHead -> Times*)] &, terms]
Map[rule[#, # /. mappingRules[[2]] /. mappingRules[[3]]] &, terms]

FCPrint[1,"FCLoopApplyTopologyMappings.m loaded."];
End[]
154 changes: 138 additions & 16 deletions Tests/LoopIntegrals/FCLoopApplyTopologyMappings.test
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Original file line number Diff line number Diff line change
Expand Up @@ -23,9 +23,9 @@ fctopology2,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc4*SPD[p1,p2],GLI[\
fctopology4,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc1*SPD[p1,Q],GLI[\
fctopology1,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc3*SPD[p2,p2],GLI[\
fctopology3,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc5*SPD[p2,Q],GLI[\
fctopology5,{1,1,1,1,1,1,1,1,1}]],{{FCTopology[fctopology3,{SFAD[{{p3,\
0},{0,1},1}],SFAD[{{p2,0},{0,1},1}],SFAD[{{p1,0},{0,1},1}],SFAD[{{p2+\
p3,0},{0,1},1}],SFAD[{{p1+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},1}],\
fctopology5,{1,1,1,1,1,1,1,1,1}]],{{{FCTopology[fctopology3,{SFAD[{{\
p3,0},{0,1},1}],SFAD[{{p2,0},{0,1},1}],SFAD[{{p1,0},{0,1},1}],SFAD[{{\
p2+p3,0},{0,1},1}],SFAD[{{p1+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},1}],\
SFAD[{{p2+p3-Q,0},{0,1},1}],SFAD[{{p1+p3-Q,0},{0,1},1}],SFAD[{{p1+p2+\
p3-Q,0},{0,1},1}]}],{p1->-p1-p3+Q,p2->-p2-p3+Q,p3->p3},GLI[\
fctopology3,{n1_,n7_,n8_,n5_,n6_,n4_,n2_,n3_,n9_}]:>GLI[fctopology1,{\
Expand All @@ -40,18 +40,140 @@ n9}]},{FCTopology[fctopology5,{SFAD[{{p3,0},{0,1},1}],SFAD[{{p2,0},{0,\
,{0,1},1}],SFAD[{{p1-Q,0},{0,1},1}],SFAD[{{p1+p3-Q,0},{0,1},1}],SFAD[{\
{p1+p2-Q,0},{0,1},1}],SFAD[{{p1+p2+p3-Q,0},{0,1},1}]}],{p1->p2,p2->p1,\
p3->p3},GLI[fctopology5,{n1_,n3_,n2_,n4_,n6_,n5_,n7_,n8_,n9_}]:>GLI[\
fctopology2,{n1,n2,n3,n4,n5,n6,n7,n8,n9}]}},Head->gliProduct]",
"gliProduct[cc6*Pair[Momentum[p1, D], Momentum[p1, D]],
GLI[fctopology1, {1, 1, 2, 1, 1, 1, 1, 1, 1}]] +
gliProduct[cc2*Pair[Momentum[p1, D], Momentum[p2, D]],
GLI[fctopology2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] +
gliProduct[cc1*Pair[Momentum[p1, D], Momentum[Q, D]],
GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] +
gliProduct[cc5*Pair[Momentum[p1, D], Momentum[Q, D]],
GLI[fctopology2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] +
gliProduct[cc4*Pair[Momentum[-p1 + Q, D], Momentum[-p2 + Q, D]],
GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] +
gliProduct[
cc3*Pair[Momentum[-p2 - p3 + Q, D], Momentum[-p2 - p3 + Q, D]],
fctopology2,{n1,n2,n3,n4,n5,n6,n7,n8,n9}]}},{
FCTopology[fctopology1,{SFAD[{{p3,0},{0,1},1}],SFAD[{{p2,0},{0,1},1}],\
SFAD[{{p1,0},{0,1},1}],SFAD[{{p2+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},\
1}],
SFAD[{{p1-Q,0},{0,1},1}],SFAD[{{p2+p3-Q,0},{0,1},1}],SFAD[{{p1+p3-Q,0}\
,{0,1},1}],SFAD[{{p1+p2+p3-Q,0},{0,1},1}]},{p1,p2,p3},{Q},{},{}],
FCTopology[fctopology2,{SFAD[{{p3,0},{0,1},1}],SFAD[{{p2,0},{0,1},1}],\
SFAD[{{p1,0},{0,1},1}],SFAD[{{p2+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},\
1}],
SFAD[{{p1-Q,0},{0,1},1}],SFAD[{{p2+p3-Q,0},{0,1},1}],SFAD[{{p1+p2-Q,0}\
,{0,1},1}],SFAD[{{p1+p2+p3-Q,0},{0,1},1}]},{p1,p2,p3},{Q},{},{}]}},\
Head->gliProduct]",
"(cc4*GLI[fctopology1, {0, 1, 1, 1, 1, 1, 1, 1, 1}])/2 + ((cc1 - \
cc4)*GLI[fctopology1, {1, 1, 0, 1, 1, 1, 1, 1, 1}])/2 - \
(cc4*GLI[fctopology1, {1, 1, 1, 0, 1, 1, 1, 1, 1}])/2 - ((cc1 - \
cc4)*GLI[fctopology1, {1, 1, 1, 1, 1, 0, 1, 1, 1}])/2 + \
cc3*GLI[fctopology1, {1, 1, 1, 1, 1, 1, 0, 1, 1}] - \
(cc4*GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 0, 1}])/2 + \
(cc4*GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 1, 0}])/2 + \
(cc5*GLI[fctopology2, {1, 1, 0, 1, 1, 1, 1, 1, 1}])/2 - \
(cc2*GLI[fctopology2, {1, 1, 1, 1, 0, 1, 1, 1, 1}])/2 - ((cc2 + \
cc5)*GLI[fctopology2, {1, 1, 1, 1, 1, 0, 1, 1, 1}])/2 + \
(cc2*GLI[fctopology2, {1, 1, 1, 1, 1, 1, 1, 0, 1}])/2 + ((cc2 + \
cc5)*GLI[fctopology2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]*Pair[Momentum[Q, \
D], Momentum[Q, D]])/2 + (GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 1, \
1}]*(2*cc6 + cc1*Pair[Momentum[Q, D], Momentum[Q, D]] + \
cc4*Pair[Momentum[Q, D], Momentum[Q, D]]))/2"},
{"fcstFCLoopApplyTopologyMappings-ID2",
"FCLoopApplyTopologyMappings[gliProduct[cc6*SPD[p1,p1],GLI[\
fctopology1,{1,1,2,1,1,1,1,1,1}]]+gliProduct[cc2*SPD[p1,p2],GLI[\
fctopology2,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc4*SPD[p1,p2],GLI[\
fctopology4,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc1*SPD[p1,Q],GLI[\
fctopology1,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc3*SPD[p2,p2],GLI[\
fctopology3,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc5*SPD[p2,Q],GLI[\
fctopology5,{1,1,1,1,1,1,1,1,1}]],{{{FCTopology[fctopology3,{SFAD[{{\
p3,0},{0,1},1}],SFAD[{{p2,0},{0,1},1}],SFAD[{{p1,0},{0,1},1}],SFAD[{{\
p2+p3,0},{0,1},1}],SFAD[{{p1+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},1}],\
SFAD[{{p2+p3-Q,0},{0,1},1}],SFAD[{{p1+p3-Q,0},{0,1},1}],SFAD[{{p1+p2+\
p3-Q,0},{0,1},1}]}],{p1->-p1-p3+Q,p2->-p2-p3+Q,p3->p3},GLI[\
fctopology3,{n1_,n7_,n8_,n5_,n6_,n4_,n2_,n3_,n9_}]:>GLI[fctopology1,{\
n1,n2,n3,n4,n5,n6,n7,n8,n9}]},{FCTopology[fctopology4,{SFAD[{{p3,0},{\
0,1},1}],SFAD[{{p2,0},{0,1},1}],SFAD[{{p1,0},{0,1},1}],SFAD[{{p2+p3,0}\
,{0,1},1}],SFAD[{{p1+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},1}],SFAD[{{\
p1-Q,0},{0,1},1}],SFAD[{{p1+p3-Q,0},{0,1},1}],SFAD[{{p1+p2+p3-Q,0},{0,\
1},1}]}],{p1->-p2+Q,p2->-p1+Q,p3->-p3},GLI[fctopology4,{n1_,n6_,n5_,\
n8_,n7_,n3_,n2_,n4_,n9_}]:>GLI[fctopology1,{n1,n2,n3,n4,n5,n6,n7,n8,\
n9}]},{FCTopology[fctopology5,{SFAD[{{p3,0},{0,1},1}],SFAD[{{p2,0},{0,\
1},1}],SFAD[{{p1,0},{0,1},1}],SFAD[{{p1+p3,0},{0,1},1}],SFAD[{{p2-Q,0}\
,{0,1},1}],SFAD[{{p1-Q,0},{0,1},1}],SFAD[{{p1+p3-Q,0},{0,1},1}],SFAD[{\
{p1+p2-Q,0},{0,1},1}],SFAD[{{p1+p2+p3-Q,0},{0,1},1}]}],{p1->p2,p2->p1,\
p3->p3},GLI[fctopology5,{n1_,n3_,n2_,n4_,n6_,n5_,n7_,n8_,n9_}]:>GLI[\
fctopology2,{n1,n2,n3,n4,n5,n6,n7,n8,n9}]}},{
FCTopology[fctopology1,{SFAD[{{p3,0},{0,1},1}],SFAD[{{p2,0},{0,1},1}],\
SFAD[{{p1,0},{0,1},1}],SFAD[{{p2+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},\
1}],
SFAD[{{p1-Q,0},{0,1},1}],SFAD[{{p2+p3-Q,0},{0,1},1}],SFAD[{{p1+p3-Q,0}\
,{0,1},1}],SFAD[{{p1+p2+p3-Q,0},{0,1},1}]},{p1,p2,p3},{Q},{},{}],
FCTopology[fctopology2,{SFAD[{{p3,0},{0,1},1}],SFAD[{{p2,0},{0,1},1}],\
SFAD[{{p1,0},{0,1},1}],SFAD[{{p2+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},\
1}],
SFAD[{{p1-Q,0},{0,1},1}],SFAD[{{p2+p3-Q,0},{0,1},1}],SFAD[{{p1+p2-Q,0}\
,{0,1},1}],SFAD[{{p1+p2+p3-Q,0},{0,1},1}]},{p1,p2,p3},{Q},{},{}]}},\
Head->gliProduct,FCLoopCreateRulesToGLI->False]",
"gliProduct[cc6*Pair[Momentum[p1, D], Momentum[p1, D]], \
GLI[fctopology1, {1, 1, 2, 1, 1, 1, 1, 1, 1}]] + \
gliProduct[cc2*Pair[Momentum[p1, D], Momentum[p2, D]], \
GLI[fctopology2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] + \
gliProduct[cc1*Pair[Momentum[p1, D], Momentum[Q, D]], \
GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] + \
gliProduct[cc5*Pair[Momentum[p1, D], Momentum[Q, D]], \
GLI[fctopology2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] + \
gliProduct[cc4*Pair[Momentum[-p1 + Q, D], Momentum[-p2 + Q, D]], \
GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] + \
gliProduct[cc3*Pair[Momentum[-p2 - p3 + Q, D], Momentum[-p2 - p3 + Q, \
D]], GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 1, 1}]]"},
{"fcstFCLoopApplyTopologyMappings-ID3",
"FCLoopApplyTopologyMappings[gliProduct[cc6*SPD[p1,p1],GLI[\
fctopology1,{1,1,2,1,1,1,1,1,1}]]+gliProduct[cc2*SPD[p1,p2],GLI[\
fctopology2,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc4*SPD[p1,p2],GLI[\
fctopology4,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc1*SPD[p1,Q],GLI[\
fctopology1,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc3*SPD[p2,p2],GLI[\
fctopology3,{1,1,1,1,1,1,1,1,1}]]+gliProduct[cc5*SPD[p2,Q],GLI[\
fctopology5,{1,1,1,1,1,1,1,1,1}]],{{{FCTopology[fctopology3,{SFAD[{{\
p3,0},{0,1},1}],SFAD[{{p2,0},{0,1},1}],SFAD[{{p1,0},{0,1},1}],SFAD[{{\
p2+p3,0},{0,1},1}],SFAD[{{p1+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},1}],\
SFAD[{{p2+p3-Q,0},{0,1},1}],SFAD[{{p1+p3-Q,0},{0,1},1}],SFAD[{{p1+p2+\
p3-Q,0},{0,1},1}]}],{p1->-p1-p3+Q,p2->-p2-p3+Q,p3->p3},GLI[\
fctopology3,{n1_,n7_,n8_,n5_,n6_,n4_,n2_,n3_,n9_}]:>GLI[fctopology1,{\
n1,n2,n3,n4,n5,n6,n7,n8,n9}]},{FCTopology[fctopology4,{SFAD[{{p3,0},{\
0,1},1}],SFAD[{{p2,0},{0,1},1}],SFAD[{{p1,0},{0,1},1}],SFAD[{{p2+p3,0}\
,{0,1},1}],SFAD[{{p1+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},1}],SFAD[{{\
p1-Q,0},{0,1},1}],SFAD[{{p1+p3-Q,0},{0,1},1}],SFAD[{{p1+p2+p3-Q,0},{0,\
1},1}]}],{p1->-p2+Q,p2->-p1+Q,p3->-p3},GLI[fctopology4,{n1_,n6_,n5_,\
n8_,n7_,n3_,n2_,n4_,n9_}]:>GLI[fctopology1,{n1,n2,n3,n4,n5,n6,n7,n8,\
n9}]},{FCTopology[fctopology5,{SFAD[{{p3,0},{0,1},1}],SFAD[{{p2,0},{0,\
1},1}],SFAD[{{p1,0},{0,1},1}],SFAD[{{p1+p3,0},{0,1},1}],SFAD[{{p2-Q,0}\
,{0,1},1}],SFAD[{{p1-Q,0},{0,1},1}],SFAD[{{p1+p3-Q,0},{0,1},1}],SFAD[{\
{p1+p2-Q,0},{0,1},1}],SFAD[{{p1+p2+p3-Q,0},{0,1},1}]}],{p1->p2,p2->p1,\
p3->p3},GLI[fctopology5,{n1_,n3_,n2_,n4_,n6_,n5_,n7_,n8_,n9_}]:>GLI[\
fctopology2,{n1,n2,n3,n4,n5,n6,n7,n8,n9}]}},{
FCTopology[fctopology1,{SFAD[{{p3,0},{0,1},1}],SFAD[{{p2,0},{0,1},1}],\
SFAD[{{p1,0},{0,1},1}],SFAD[{{p2+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},\
1}],
SFAD[{{p1-Q,0},{0,1},1}],SFAD[{{p2+p3-Q,0},{0,1},1}],SFAD[{{p1+p3-Q,0}\
,{0,1},1}],SFAD[{{p1+p2+p3-Q,0},{0,1},1}]},{p1,p2,p3},{Q},{},{}],
FCTopology[fctopology2,{SFAD[{{p3,0},{0,1},1}],SFAD[{{p2,0},{0,1},1}],\
SFAD[{{p1,0},{0,1},1}],SFAD[{{p2+p3,0},{0,1},1}],SFAD[{{p2-Q,0},{0,1},\
1}],
SFAD[{{p1-Q,0},{0,1},1}],SFAD[{{p2+p3-Q,0},{0,1},1}],SFAD[{{p1+p2-Q,0}\
,{0,1},1}],SFAD[{{p1+p2+p3-Q,0},{0,1},1}]},{p1,p2,p3},{Q},{},{}]}},\
Head->gliProduct,GLIMultiply->False]",
"gliProduct[cc6*GLI[fctopology1, {0, 0, -1, 0, 0, 0, 0, 0, 0}], \
GLI[fctopology1, {1, 1, 2, 1, 1, 1, 1, 1, 1}]] + \
gliProduct[cc3*GLI[fctopology1, {0, 0, 0, 0, 0, 0, -1, 0, 0}], \
GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] + \
gliProduct[(cc1*(GLI[fctopology1, {0, 0, -1, 0, 0, 0, 0, 0, 0}] - \
GLI[fctopology1, {0, 0, 0, 0, 0, -1, 0, 0, 0}] + Pair[Momentum[Q, D], \
Momentum[Q, D]]))/2, GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] + \
gliProduct[(cc5*(GLI[fctopology2, {0, 0, -1, 0, 0, 0, 0, 0, 0}] - \
GLI[fctopology2, {0, 0, 0, 0, 0, -1, 0, 0, 0}] + Pair[Momentum[Q, D], \
Momentum[Q, D]]))/2, GLI[fctopology2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] + \
gliProduct[(cc2*(-GLI[fctopology2, {0, 0, 0, 0, -1, 0, 0, 0, 0}] - \
GLI[fctopology2, {0, 0, 0, 0, 0, -1, 0, 0, 0}] + GLI[fctopology2, {0, \
0, 0, 0, 0, 0, 0, -1, 0}] + Pair[Momentum[Q, D], Momentum[Q, D]]))/2, \
GLI[fctopology2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] + \
gliProduct[cc4*((-GLI[fctopology1, {0, -1, 0, 0, 0, 0, 0, 0, 0}] + \
GLI[fctopology1, {0, 0, 0, 0, -1, 0, 0, 0, 0}] - Pair[Momentum[Q, D], \
Momentum[Q, D]])/2 + (-GLI[fctopology1, {0, 0, -1, 0, 0, 0, 0, 0, 0}] \
+ GLI[fctopology1, {0, 0, 0, 0, 0, -1, 0, 0, 0}] - Pair[Momentum[Q, \
D], Momentum[Q, D]])/2 + Pair[Momentum[Q, D], Momentum[Q, D]] + \
(GLI[fctopology1, {-1, 0, 0, 0, 0, 0, 0, 0, 0}] + GLI[fctopology1, \
{0, -1, 0, 0, 0, 0, 0, 0, 0}] - GLI[fctopology1, {0, 0, 0, -1, 0, 0, \
0, 0, 0}] - GLI[fctopology1, {0, 0, 0, 0, -1, 0, 0, 0, 0}] - \
GLI[fctopology1, {0, 0, 0, 0, 0, 0, 0, -1, 0}] + GLI[fctopology1, {0, \
0, 0, 0, 0, 0, 0, 0, -1}] + Pair[Momentum[Q, D], Momentum[Q, D]])/2), \
GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 1, 1}]]"}
});

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