Issue with noninteger propagator in FCLoopFindTopologies/FCLoopTensorReduce/FCLoopApplyTopologyMappings

[mashhoudtt]

Hello,

So I'm doing a two-loop region expansion in which one of the loop momenta is collinear and the other is hard. I first perform the expansion and then use the mathematica functions successfully, leading to my form factor being written in GLI form. I then get the masters through Kira and solve to get some combination of gamma functions combined with a propagator to the power of epsilon. When trying to do this whole process again for the leftover collinear loop momentum, I get multiple errors. These include:
FCLoopIntegralToPropogators->Failed to extract the propagators.
Seeing this, I changed the masters result I plugged in from SFAD form to just a combination of SPDs in the denominator. This allowed for FCLoopFindTopologies to run and there was an integer form of the propagator from the masters result, so it was still found.
But then FCLoopTensorReduce gives an error as Uncontract cannot work with negative or symbolic powers of expressions.
And even when trying to bypass/skip that step, I get FCLoopApplyTopologyMappings failed to eliminate all loop momenta in the expression. I was just wondering whether or not these commands are compatible with noninteger powers of propagators or not. And if not, what would be the best workaround ?

Thanks !

[vsht]

Hi,

please provide a minimal working example that reproduces the problem. As a code that one copy-paste, not a screenshot.

Seeing this, I changed the masters result I plugged in from SFAD form to just a combination of SPDs in the denominator.
Inverse SPDs are not being treated as proper denominators, so this approach definitely won't work.

I then get the masters through Kira and solve to get some combination of gamma functions combined with a propagator to the power of epsilon.

On the FeynCalc side symbolic propagator powers should, at least in principle, work. But it's difficult to say without seeing the explicit expressions and error messages.

IBP reduction with symbolic powers is a totally different business, though. Might be doable with KIRA/FIRE, but not out of the box. LiteRed should be better for this, though.

[mashhoudtt]

Hi, so

First the diagram and amplitude:
diags = InsertFields[ CreateTopologies[2, 2 -> 1, ExcludeTopologies -> {Tadpoles}], {F[2, {1}], -F[2, {1}]} -> {V[1]}, InsertionLevel -> {Particles}, ExcludeParticles -> {V[2 | 3], S[_], U[_]}] ; Paint[DiagramExtract[diags, {84}], ColumnsXRows -> {1, 1}, SheetHeader -> False, Numbering -> None, ImageSize -> {768, 256}] ampRaw = FCFAConvert[ CreateFeynAmp[DiagramExtract[diags, {84}], Truncated -> True, GaugeRules -> {}, PreFactor -> 1], IncomingMomenta -> {p1, p2}, OutgoingMomenta -> {q}, LorentzIndexNames -> {\[Mu], \[Nu], \[Rho], \[Theta], \[Eta]}, LoopMomenta -> {k1, k2}, UndoChiralSplittings -> True, ChangeDimension -> D, List -> True, SMP -> True, DropSumOver -> True] /. {GaugeXi[A] -> 1} // SMPToSymbol // FCReplaceMomenta[#, {q -> p1 + p2, k1 -> -k1, k2 -> k2 - p2}] &

My kinematic rules (some are not needed in this case):
FCClearScalarProducts[]; SPD[p1] = m^2; SPD[p2] = m^2; SPD[p2m] = 0; SPD[p1, p2] = sh/2 + m^4/(2 sh); SPD[p1, p2m] = sh/2; SPD[p2, p2m] = m^2/2; SPD[p1, p2p] = m^4/(2 sh); SPD[p2, p2p] = m^2/2; SPD[p2m, p2p] = m^2/2; SPD[p1p, p2m] = sh/2; SPD[p1, p1p] = m^2/2; SPD[p1p, p2] = sh/2; SPD[p1p, p2p] = 0; SPD[p2p] = 0; SPD[p1p] = 0;

Expanding the relevant propagators:
ampRawSplit = FeynAmpDenominatorSplit[ampRaw]; FADrepk1p2 = FeynAmpDenominator[ PropagatorDenominator[Momentum[p2, D] - Momentum[k1, D], me]] -> -SFAD[{{0, 2 (SPD[k1, p2m])}, 0}] + ( SPD[k1]) SFAD[{{0, 2 (SPD[k1, p2m])}, 0}]^2 FADrepk2p2 = FeynAmpDenominator[ PropagatorDenominator[Momentum[k2, D] - Momentum[p2, D], me]] -> SFAD[{{k2, -2 (SPD[k2, p2m])}, 0}] + (m^2/ sh) (2 SPD[k2, p2m]) SFAD[{{k2, -2 (SPD[k2, p2m])}, 0}]^2 ampRawRep = ampRawSplit /. {FADrepk2p2, FADrepk1p2} ampRawtr = ampRawRep[[1]] /. {me -> m}

Applying projection operator:
g1[1] := -1/(e * Nc) * 1/(4 (1 - \[Epsilon]))* 1/(s/m^2 - 4); g1[2] := 1/(e * Nc) * 1/(1 - \[Epsilon])* 1/((s/m^2) (s/m^2 - 4)); g2[1] := 1/(e * Nc) * (3 - 2 \[Epsilon])/(1 - \[Epsilon])* 1/(s/m^2 - 4)^2; g2[2] := -1/(e * Nc) * 1/(1 - \[Epsilon])* 1/(s/m^2 - 4)^2 * (4 m^2/s + 2 - 2 \[Epsilon]); P[j_, \[Mu]_] := ((GSD[p1] + m)/ m ) . (I*g1[j]* GAD[\[Mu]] + (I/(2 m))* g2[j] (FVD[p2, \[Mu]] - FVD[p1, \[Mu]])) . ((GSD[p2] - m)/m); AbsoluteTiming[ ampForms = Table[TR[ P[j, \[Mu]] . ampRawtr /. {s -> 2 m^2 + sh + m^4/sh, Nc -> 1}], {j, 1, 2}];] ampFormsp2m = ampForms /. {Momentum[p2, D] -> Momentum[p2m, D]};

Topology and mappings:
{amp1, topos1} = FCLoopFindTopologies[ampFormsp2m[[1]], {k2}, FCParallelize -> True]; {amp2, topos2} = FCLoopFindTopologies[ampFormsp2m[[2]], {k2}, FCParallelize -> True]; TopoAch = {FCTopology[ topoAch, {SFAD[k2], SFAD[{k1 - k2}], SFAD[{{k2, -2 SPD[k2, p2m]}, 0 }] }, {k2}, {k1, p2m}, {}, {}]} mappings1 = FCLoopFindTopologyMappings[topos1, PreferredTopologies -> TopoAch, FCParallelize -> True]; completedToposc = FCLoopBasisFindCompletion[TopoAch]; basisCompletionRules1 = FCLoopCreateRuleGLIToGLI[completedToposc, List /@ topos1] // Flatten; basisCompletionRules2 = FCLoopCreateRuleGLIToGLI[completedToposc, List /@ topos2] // Flatten; mappings2 = {mappings1[[1]] /. basisCompletionRules1, completedToposc}; SPGLIrules = FCLoopCreateRulesToGLI[TopoAch];

AbsoluteTiming[ ampReduced1 = FCLoopTensorReduce[amp1, topos1, FCParallelize -> True];] AbsoluteTiming[ ampReduced2 = FCLoopTensorReduce[amp2, topos2, FCParallelize -> True];] AbsoluteTiming[ ampFinal1 = FCLoopApplyTopologyMappings[ ampReduced1 /. basisCompletionRules1 /. SPGLIrules, mappings2, FCParallelize -> True];] AbsoluteTiming[ ampFinal2 = FCLoopApplyTopologyMappings[ ampReduced2 /. basisCompletionRules2 /. SPGLIrules, mappings2, FCParallelize -> True];] ampCollected1 = Collect[ampFinal1, GLI[___]]; ampCollected2 = Collect[ampFinal2, GLI[___]];

Extracting GLIs from amplitude to create necessary Kira files cleanly:
onlyGLIs1 = Cases[ampCollected1, GLI[___], Infinity]; neededGLIs1 = DeleteDuplicates@Cases[ampCollected1, _GLI, Infinity]; onlyGLIs2 = Cases[ampCollected2, GLI[___], Infinity]; neededGLIs2 = DeleteDuplicates@Cases[ampCollected2, _GLI, Infinity]; neededKeys1 = DeleteDuplicates@(neededGLIs1 /. GLI[topoAch, idx_List] :> topoAch @@ idx); neededKeys2 = DeleteDuplicates@(neededGLIs2 /. GLI[topoAch, idx_List] :> topoAch @@ idx); onlyGLIs1 === onlyGLIs2 onlyGLIs = DeleteDuplicates@(Join[onlyGLIs1, onlyGLIs2] /. GLI[topoAch, idx_List] :> topoAch @@ idx);

Kira Files:
KiraCreateJobFile[TopoAch, onlyGLIs, "job_Ach"] KiraCreateIntegralFile[ampCollected1, TopoAch, "job_Ach"] KiraCreateConfigFiles[TopoAch, onlyGLIs, "job_Ach", KiraMassDimensions -> {m -> 1, sh -> 2}]

Replacing the GLIs with the masters
<< C:\Users\duohh\OneDrive\Desktop\TopoAccCOMPLETE\topoAch\t\ opoAch\kira_KiraLoopIntegrals.m; AchRep = % /. {d -> D, x -> SPD[k1], y -> SPD[k1, p2m]}; AchAssoc = Association@AchRep; neededAssoc1 = KeyTake[AchAssoc, neededKeys1]; neededAssoc2 = KeyTake[AchAssoc, neededKeys2]; ampReplaced1 = ampCollected1 /. GLI[topoAch, idx_List] :> Lookup[neededAssoc1, topoAch @@ idx, GLI[topoAch, idx]]; ampReplaced2 = ampCollected2 /. GLI[topoAch, idx_List] :> Lookup[neededAssoc2, topoAch @@ idx, GLI[topoAch, idx]]; ampMI1 = ampReplaced1 /. HoldPattern[topoAch[inds__]] :> GLI[topoAch, {inds}]; ampMI2 = ampReplaced2 /. HoldPattern[topoAch[inds__]] :> GLI[topoAch, {inds}];

Solved Masters integral for hard loop:
solvedMI1 = (Gamma[1 - \[Epsilon]]^2 Gamma[\[Epsilon]])/((4 \[Pi])^( 2 - \[Epsilon]) Gamma[2 - 2 \[Epsilon]]) (-SFAD[(SPD[k1] - 2 SPD[k1, p2m])])^\[Epsilon]

Plugging Master result in:
ampExpanded = {ampMI1, ampMI2}; ampMIrep = ampExpanded /. {GLI[topoAch, {0, 1, 1}] -> solvedMI1};

Here is where I try to run the FCLoopFindTopologies again:
{amp11, topos11} = FCLoopFindTopologies[ampMIrep[[1]], {k1}, FCParallelize -> True]; {amp22, topos22} = FCLoopFindTopologies[ampMIrep[[2]], {k1}, FCParallelize -> True];

So using the master integral result in SFAD form just gives me the error that FCLoopIntegralToPropagators encountered a fatal problem and must abort the computation. The problem reads: Failed to extract the propagators. And from then on it continues when I do
TopoAch2 = {FCTopology[ topoAch2, {SFAD[k1], SFAD[{k1 + p1, m^2}], SFAD[{{0, 2 (SPD[k1, p2m])}, 0}] }, {k1}, {p1, p2m}, {}, {}]} mappings11 = FCLoopFindTopologyMappings[topos11, PreferredTopologies -> TopoAch2, FCParallelize -> True]; completedToposc2 = FCLoopBasisFindCompletion[TopoAch2]; basisCompletionRules11 = FCLoopCreateRuleGLIToGLI[completedToposc2, List /@ topos11] // Flatten; basisCompletionRules22 = FCLoopCreateRuleGLIToGLI[completedToposc2, List /@ topos22] // Flatten; mappings22 = {mappings11[[1]] /. basisCompletionRules11, completedToposc2}; SPGLIrules2 = FCLoopCreateRulesToGLI[TopoAch2];

I know Kira can handle non-integer propagators, however this is the issue I face before even reaching that point. My only other option I guess would be to not include the epsilon propagator at all and then include it at the end once I get the master for the collinear loop.

[vsht]

I cannot completely rerun your code, because you seem to use customized Kira config files. In my case already
KiraCreateJobFile fails because you give it integrals with the topology name as head, instead of GLIs. Also the value of k1
is not fixed, so the reduction won't run out of the box.

But actually I don't really need that part if it works for you this or the other way. It seems that it gets interesting after the reduction, so I'd like to have the expressions ampMI1 and ampMI2.

The way you write SFAD in solvedMI1 is obviously incorrect, you can see it immediately by checking the output of

-SFAD[(SPD[k1] - 2 SPD[k1, p2m])] // FeynAmpDenominatorExplicit

This should be the correct form
SFAD[{{k1, -2 k1 . p2m}, 0, ep}]

But I'm still not 100% sure that it can go through this way until the end.

[mashhoudtt]

I'm not sure I understand what you mean for the KiraCreateJobFile failing, TopoAch for me is
TopoAch = {FCTopology[ topoAch, {SFAD[k2], SFAD[{k1 - k2}], SFAD[{{k2, -2 SPD[k2, p2m]}, 0 }] }, {k2}, {k1, p2m}, {}, {}]}
and my sectors are given by:
onlyGLIs1 = Cases[ampCollected1, GLI[___], Infinity];
which are GLIs, no? At least, KiraCreateJobFile works for me still, as far as I know.
For the Kira reduction side, my config and job files worked smoothly, I just had to implement the product rules and did [[k1,k1], x] for example and then when recovering the replacement for the master I performed x->SPD[k1] in mathematica.

What a silly SFAD mistake on my part ! I did fix it but when I run FCLoopFindTopologies now I get "The input expression contains integrals that require partial fractioning." When I try to use ApartFF I then get the "Failed to extract the propagators" error once again.

AmpMI1 and AmpMI2 are quite long, I wish I could copy paste it to save you from the hassle, but you can just go through the process to get to ampCollected, and the neededAssoc1 and 2, i.e. the master replacements are given by

NeededAssoc1 = <|topoAch[-1, 0, 2] -> 0, topoAch[-2, 1, 2] -> (-2 SPD[k1, k1] SPD[k1, p2m] + D SPD[k1, p2m]^2) topoAch[0, 1, 1] (-SFAD[{{k1, -2 SPD[k1, p2m]}}]), topoAch[0, 0, 1] -> 0, topoAch[-1, 1, 1] -> SPD[k1, p2m] topoAch[0, 1, 1], topoAch[0, 1, 0] -> 0, topoAch[-1, 1, 2] -> (-SPD[k1, k1] + (-2 + D) SPD[k1, p2m]) topoAch[0, 1, 1] (-SFAD[{{k1, -2 SPD[k1, p2m]}}])|>

as well as:
neededAssoc2 = <|topoAch[-1, 0, 2] -> 0, topoAch[-2, 1, 2] -> (-2 SPD[k1, k1] SPD[k1, p2m] + D SPD[k1, p2m]^2) topoAch[0, 1, 1] ((-SFAD[{{k1, -2 SPD[k1, p2m]}}])), topoAch[0, 1, 0] -> 0, topoAch[0, 0, 1] -> 0, topoAch[-1, 1, 1] -> SPD[k1, p2m] topoAch[0, 1, 1], topoAch[-1, 1, 2] -> (-SPD[k1, k1] + (-2 + D) SPD[k1, p2m]) topoAch[0, 1, 1] ((-SFAD[{{k1, -2 SPD[k1, p2m]}}]))|>

I think you need to do Normal[neededAssoc1] and Normal[neededAssoc2] for it work with my code!
when going through this I realized I had made another mistake, which is that I blindly did the AchRep = % /. {d -> D, x -> SPD[k1], y -> SPD[k1, p2m]}; replacement after running the kira reduction instead of going through and replacing the denominator products to be in SFAD form, so I had to just manually go and do that right now. However, this did not affect the errors I run into.

[vsht]

I'm not sure I understand what you mean for the KiraCreateJobFile failing, TopoAch for me is
TopoAch = {FCTopology[ topoAch, {SFAD[k2], SFAD[{k1 - k2}], SFAD[{{k2, -2 SPD[k2, p2m]}, 0 }] }, {k2}, {k1, p2m}, {}, {}]}
and my sectors are given by:
onlyGLIs1 = Cases[ampCollected1, GLI[___], Infinity];
which are GLIs, no? At least, KiraCreateJobFile works for me still, as far as I know.

In your code you have

onlyGLIs = 
  DeleteDuplicates@(Join[onlyGLIs1, onlyGLIs2] /.  
     GLI[topoAch, idx_List] :> topoAch @@ idx);

which replaces GLIs with heads made of their IDs. This is why I wrote that you must be using a more customized setup.

Anyway, using

ruIBPs = Normal[NeededAssoc1] /. topoAch[ids__] :> GLI[topoAch, {ids}]
tmp = Collect2[{ampCollected1, ampCollected2} /. ruIBPs, GLI];
solvedMI1 = (Gamma[
       1 - \[Epsilon]]^2 Gamma[\[Epsilon]])/((4 \[Pi])^(2 - \
\[Epsilon]) Gamma[2 - 2 \[Epsilon]]) SFAD[{{k1, -2 k1 . p2m}, 
    0, \[Epsilon]}]
ampMIrep = tmp /. {GLI[topoAch, {0, 1, 1}] -> solvedMI1};

I can get to ampMIrep. From there

{amp11, topos11} = 
  FCLoopFindTopologies[ampMIrep[[1]], {k1}, FCParallelize -> True];
{amp22, topos22} = 
  FCLoopFindTopologies[ampMIrep[[2]], {k1}, FCParallelize -> True];

works fine for me:

(*topos11 // FCE // InputForm*)
{FCTopology["fctopology1", {SFAD[{{k1, 0}, {0, 1}, 1}], SFAD[{{0, 2*k1 . p2m}, {0, 1}, 1}], SFAD[{{k1 + p1, 0}, {m^2, 1}, 1}], 
   SFAD[{{k1, -2*k1 . p2m}, {0, 1}, 1}]}, {k1}, {p1, p2m}, {Hold[Pair][Momentum[p1, D], Momentum[p1, D]] -> m^2, 
   Hold[Pair][Momentum[p2m, D], Momentum[p2m, D]] -> 0, Hold[Pair][Momentum[p1, D], Momentum[p2, D]] -> m^4/(2*sh) + sh/2, 
   Hold[Pair][Momentum[p1, D], Momentum[p2m, D]] -> sh/2, Hold[Pair][Momentum[p2, D], Momentum[p2m, D]] -> m^2/2, 
   Hold[Pair][Momentum[p1, D], Momentum[p2p, D]] -> m^4/(2*sh), Hold[Pair][Momentum[p2m, D], Momentum[p2p, D]] -> m^2/2, 
   Hold[Pair][Momentum[p1p, D], Momentum[p2m, D]] -> sh/2, Hold[Pair][Momentum[p1, D], Momentum[p1p, D]] -> m^2/2}, {}]}

(*amp11 /. 
 FCGV["GLIProduct"][a_, b_] :> FCGV["GLIProduct"][Isolate[a], b]*)

{FCGV["GLIProduct"][HoldForm[KK[911]], GLI["fctopology1", {1, 2, 1, \[Epsilon]}]] + FCGV["GLIProduct"][HoldForm[KK[917]], GLI["fctopology1", {1, 3, 1, \[Epsilon]}]] + 
  FCGV["GLIProduct"][HoldForm[KK[918]], GLI["fctopology1", {1, 4, 1, \[Epsilon]}]] + FCGV["GLIProduct"][HoldForm[KK[921]], GLI["fctopology1", {1, 2, 1, 1 + \[Epsilon]}]] + 
  FCGV["GLIProduct"][HoldForm[KK[923]], GLI["fctopology1", {1, 3, 1, 1 + \[Epsilon]}]] + FCGV["GLIProduct"][HoldForm[KK[924]], GLI["fctopology1", {1, 4, 1, 1 + \[Epsilon]}]]}

But this is because recently I disabled the overdetermined basis check in FCLoopFindTopologies. At that stage it is not really necessary.

This does not change the fact that your propagator basis is overdetermined.

FCLoopBasisOverdeterminedQ[topos11]
(*True*)

The problem is that to arrive at a proper basis one has to remove your propagator with symbolic index in one of the resulting topologies. Obviously, if the index is a finite integer, you can just apply partial fractioning until it is gone. But for a symbolic
index this procedure never terminates

FCLoopCreatePartialFractioningRules[{GLI[
    "fctopology1", {1, 2, 1, \[Epsilon]}]} /. \[Epsilon] -> 
   1, topos11]

FCLoopCreatePartialFractioningRules[{GLI[
    "fctopology1", {1, 2, 1, \[Epsilon]}]} /. \[Epsilon] -> 
  2, topos11]

FCLoopCreatePartialFractioningRules[{GLI[
    "fctopology1", {1, 2, 1, \[Epsilon]}]} , topos11]

It wouldn't be an issue if the linear dependence would affect other propagators. The one could just keep the symbolic propagators as it is. But in your case this propagator is precisely the reason for the propagator basis being overdetermined

FCLoopBasisOverdeterminedQ[FCLoopRemovePropagator[topos11, {{4}}]]

So it is not obvious for the code how to resolve this.

[mashhoudtt]

Sure, I'll just attach my notebook.

For the loop-by-loop, I guess I don't even need to do that anymore considering my mixed k1 and k2 propagator is being expanded after anyway. The paper I'm basing my work off suggests the loop-by-loop approach, so I was just trying to streamline as efficiently as possible using FeynCalc/Helpers as opposed to something like FORM.
ABCH.zip
Hopefully you can make sense of my disorganized work, and thank you for your patience !

[vsht]

Hopefully you can make sense of my disorganized work, and thank you for your patience !

Sorry, I'm afraid that I can't, because the notebook cannot be evaluated right from the start without running into errors right after the << C : \Users\duohh\OneDrive\Desktop\TopoAccCOMPLETE\topoAch\topoAch\kira_KiraLoopIntegrals.m;-part (which I can't run anyway).

What I initially meant is that you include just the expressions that cause problems with FCLoopTensorReduce and KiraCreateJobFile so that one can isolate the issue. The rest of the calculation is irrelevant for that anyway. And please check beforehand that one can run it without having all of your other code parts at hand.

[vsht]

I can imagine, though, that KiraCreateJobFileA create job file cannot deal with GLIs containing symbolic propagator powers: Those are needed to determine the sectors for the reduction, which works only for integer powers. You can specify the sectors and r,s-variables directly, though.

[mashhoudtt]

without running into errors right after the << C : \Users\duohh\OneDrive\Desktop\TopoAccCOMPLETE\topoAch\topoAch\kira_KiraLoopIntegrals.m;-part (which I can't run anyway)

Right, my mistake, sorry ! I do think you can ignore that segment, though and use the neededAssoc terms that are written below. In any case, thank you very much for your help. I will create the Kira files by hand; hopefully that's not too much of an ask of myself :p

[vsht]

I think that it would take some work to add support for symbolic powers to the KIRA interface. I never needed it personally, so it was never a top priority. Let's see if I can find time for it.