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vsht committed Aug 15, 2024
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2 changes: 1 addition & 1 deletion .gitignore
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@@ -1,6 +1,6 @@
FeynCalc/FeynArts
FeynCalc/FeynArts/*
FeynCalc/Database/*.db
FeynCalc/Database/*
FeynCalc/Examples/Temp
FeynCalc/Examples/*/WIP
FeynCalc/Examples/WIP
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627 changes: 66 additions & 561 deletions FeynCalc/Changelog.md

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577 changes: 577 additions & 0 deletions FeynCalc/ChangelogOld.md

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31 changes: 17 additions & 14 deletions FeynCalc/Dirac/DiracTrace.m
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Expand Up @@ -55,6 +55,7 @@
DiracTraceEvaluate -> False,
EpsContract -> False,
EpsExpand -> True,
EpsEvaluate -> True,
Expand -> True,
FCVerbose -> False,
Factoring -> Automatic,
Expand Down Expand Up @@ -217,18 +218,18 @@
unitMatrixTrace expr;

diracTraceEvaluate[expr_/;!FreeQ[expr,DiracGamma], opts:OptionsPattern[]] :=
Block[ { diractrres, tmp = expr, diractrfact,
diractrcoll,
dtmp,dWrap,wrapRule,prepSpur,time,time2,contract,spurHeadList,spurHeadListChiral,spurHeadListNonChiral,
gammaFree,gammaPart,
traceListChiral,traceListNonChiral,repRule,null1,null2,dummyIndexFreeQ},
Block[{ diractrres, tmp = expr, diractrfact, diractrcoll,
dtmp,dWrap,wrapRule,prepSpur,time,time2,contract,spurHeadList,
spurHeadListChiral,spurHeadListNonChiral,gammaFree,gammaPart,
traceListChiral,traceListNonChiral,repRule,null1,null2,dummyIndexFreeQ, epsEvaluate},

wrapRule = {dWrap[5]->0, dWrap[6]->1/2, dWrap[7]->1/2, dWrap[LorentzIndex[_,_:4],___]->0,
dWrap[_. Momentum[_,_:4]+_:0,___]->0};

diractrfact = OptionValue[DiracTrace,{opts},Factoring];
diractrcoll = OptionValue[DiracTrace,{opts},PairCollect];
contract = OptionValue[DiracTrace,{opts},Contract];
epsEvaluate = OptionValue[DiracTrace,{opts},EpsEvaluate];
west = OptionValue[DiracTrace,{opts},West];
larinMVV = OptionValue[DiracTrace,{opts},LarinMVV];

Expand Down Expand Up @@ -292,7 +293,6 @@
tmp = tmp/. spurHead[x__]/; !NonCommFreeQ[{x}/.DiracGamma->null1] :> noSpur[x];
FCPrint[3,"DiracTrace: diracTraceEvaluate: Trace contains unknown non-commutative objects: ", !FreeQ[tmp, noSpur], FCDoControl->diTrVerbose];


(* Split chiral projectors here *)
tmp = tmp /. {spurHead[x___,DiracGamma[6]] :> 1/2 spurHead[x] + 1/2 spurHead[x,DiracGamma[5]],
spurHead[x___,DiracGamma[7]] :> 1/2 spurHead[x] - 1/2 spurHead[x,DiracGamma[5]]} /. spurHead[] -> 1;
Expand Down Expand Up @@ -371,7 +371,7 @@

(* Evaluate the traces *)
time2=AbsoluteTime[];
FCPrint[1,"DiracTrace: diracTraceEvaluate: Calculating non-chiral traces.", FCDoControl->diTrVerbose];
FCPrint[1,"DiracTrace: diracTraceEvaluate: Calculating ", Length[spurHeadListChiral], " nonchiral traces.", FCDoControl->diTrVerbose];

traceListNonChiral = spurHeadListNonChiral/. spurHead-> spurNo5;
FCPrint[1,"DiracTrace: diracTraceEvaluate: Done calculating non-chiral traces, timing: ", N[AbsoluteTime[] - time2, 4], FCDoControl->diTrVerbose];
Expand All @@ -384,7 +384,7 @@
];

time2=AbsoluteTime[];
FCPrint[1,"DiracTrace: diracTraceEvaluate: Calculating chiral traces.", FCDoControl->diTrVerbose];
FCPrint[1,"DiracTrace: diracTraceEvaluate: Calculating ",Length[spurHeadListChiral]," chiral traces.", FCDoControl->diTrVerbose];
(* Purely 4 dimensional traces are always computed in the same way, regardless of the chosen scheme:
Eq 2.18 of R. Mertig, M. Boehm, A. Denner. Comp. Phys. Commun., 64 (1991)) *)
traceListChiral = spurHeadListChiral/. spurHead[x__]/;(FCGetDimensions[{x},ChangeDimension->True]==={4}) :> spur5In4Dim[x];
Expand All @@ -395,22 +395,25 @@

(* NDR *)
"NDR",
FCPrint[3,"DiracTrace: diracTraceEvaluate: Chiral traces will be left untouched (NDR scheme).", FCDoControl->diTrVerbose];
FCPrint[2,"DiracTrace: diracTraceEvaluate: Chiral traces will be left untouched (NDR scheme).", FCDoControl->diTrVerbose];
traceListChiral = traceListChiral/. spurHead -> noSpur,

(* NDR-Discard *)
"NDR-Discard",
FCPrint[3,"DiracTrace: diracTraceEvaluate: Chiral traces are set to zero (NDR-Discard scheme).", FCDoControl->diTrVerbose];
FCPrint[2,"DiracTrace: diracTraceEvaluate: Chiral traces are set to zero (NDR-Discard scheme).", FCDoControl->diTrVerbose];
traceListChiral = ConstantArray[0, Length[traceListChiral]],

(* Larin *)
"Larin",
FCPrint[3,"DiracTrace: diracTraceEvaluate: Chiral traces will be computed using Larin scheme", FCDoControl->diTrVerbose];
FCPrint[2,"DiracTrace: diracTraceEvaluate: Chiral traces will be computed using Larin scheme.", FCDoControl->diTrVerbose];

If[ larinMVV,
(* Larin, MVV trace formula *)
FCPrint[2,"DiracTrace: diracTraceEvaluate: Using the MVV trace formula.", FCDoControl->diTrVerbose];

traceListChiral = traceListChiral/. spurHead -> spur5LarinMVV,
(* Laring, standard (slow!) trace formula *)

FCPrint[2,"DiracTrace: diracTraceEvaluate: Using the standard slow trace formula.", FCDoControl->diTrVerbose];

traceListChiral = traceListChiral/. spurHead -> spur5Larin
],

Expand Down Expand Up @@ -496,7 +499,7 @@
];

(* Special expansion for expressions that contain Levi-Civita tensors*)
If[ !FreeQ[tmp, Eps],
If[ !FreeQ[tmp, Eps] && epsEvaluate,
time=AbsoluteTime[];
FCPrint[1,"DiracTrace: diracTraceEvaluate: Treating Eps tensors.", FCDoControl->diTrVerbose];
tmp = EpsEvaluate[tmp,FCI->True, EpsExpand->OptionValue[DiracTrace,{opts},EpsExpand]]//Expand;
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12 changes: 10 additions & 2 deletions FeynCalc/Dirac/DiracTrick.m
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Expand Up @@ -349,12 +349,17 @@ create replacement rules (standard) and substitute the results back.
diracTrickEvalFast[DiracGamma[6|7]]:=
1/2/; insideDiracTrace;

(* Reordering of Dirac matrices using cyclicity of the trace *)

diracTrickEvalFast[DOT[DiracGamma[(a:6|7)],b___,DiracGamma[(a:6|7)]]] :=
diracTrickEvalFast[DOT[b,DiracGamma[a]]]/; insideDiracTrace;

diracTrickEvalFast[DOT[DiracGamma[(h1:5|6|7)],b___,DiracGamma[(h2:5|6|7)]]] :=
ga67MatSign[h1,h2] diracTrickEvalFast[DOT[b,ga67Mat[h1,h2]]]/; h1=!=h2 && insideDiracTrace;

diracTrickEvalFast[DOT[DiracGamma[(h:5|6|7)],b__]] :=
diracTrickEvalFast[DOT[b,DiracGamma[h]]]/; insideDiracTrace;

(* Generic simplifications *)

diracTrickEvalFast[DOT[b___,DiracGamma[l_LorentzIndex], DiracGamma[l_LorentzIndex], d___]] :=
Expand All @@ -375,6 +380,8 @@ create replacement rules (standard) and substitute the results back.
diracTrickEvalFast[DOT[b___,DiracGamma[l_LorentzIndex, dim_:4], DiracGamma[c_Momentum, dim_:4], DiracGamma[l_LorentzIndex, dim_:4], d___]] :=
(2 - dim) diracTrickEvalFast[DOT[ b, DiracGamma[c, dim], d ]];

(* Neighboring g^5 and chiral projectors *)

diracTrickEvalFast[DOT[b___,DiracGamma[5], DiracGamma[5], d___]] :=
diracTrickEvalFast[DOT[ b,d ]];

Expand All @@ -387,6 +394,7 @@ create replacement rules (standard) and substitute the results back.
diracTrickEvalFast[DOT[b___, DiracGamma[(hh1:6|7)],DiracGamma[(hh2:6|7)], c___]] :=
ga67Val2[hh1,hh2] diracTrickEvalFast[DOT[b, DiracGamma[hh2], c]];

(* D-dimensional formulas for anticommuting g^5 (NDR only) *)

diracTrickEvalFast[DOT[b___,DiracGamma[5], c:DiracGamma[_[_,__],_].. , d___]] :=
(-1)^Length[{c}] diracTrickEvalFast[DOT[ b,c,DiracGamma[5],d]]/; MemberQ[{"NDR","NDR-Discard"},FeynCalc`Package`DiracGammaScheme] &&
Expand Down Expand Up @@ -424,7 +432,7 @@ create replacement rules (standard) and substitute the results back.
diracTrickEvalFast[DOT[b, dg, xy, DiracGamma[h], c]]/; OddQ[Length[{xy}]] && MemberQ[{"NDR","NDR-Discard"},FeynCalc`Package`DiracGammaScheme] &&
MatchQ[FCGetDimensions[{dg,xy}],{_Symbol}];


(* 4-dimensional formulas for anticommuting g^5 *)
diracTrickEvalFast[DOT[b___,DiracGamma[5], c:DiracGamma[_[__]].. , d___]] :=
(-1)^Length[{c}] diracTrickEvalFast[DOT[ b,c,DiracGamma[5],d]];

Expand Down Expand Up @@ -1562,7 +1570,7 @@ create replacement rules (standard) and substitute the results back.
Block[{li1,li2,li3},
{li1,li2,li3} = LorentzIndex[#,dim]& /@ Unique[{"dtlarLia","dtlarLib","dtlarLic"}];
mass chiralTrickLarin[b,DiracGamma[7],d] +
1/2 chiralTrickLarin[b,dg,d] +
1/2 chiralTrickLarin[b,dg,d] -
I/12 $LeviCivitaSign Eps[dg[[1]], li1, li2, li3] chiralTrickLarin[b,DiracGamma[li1,dim],DiracGamma[li2,dim],DiracGamma[li3,dim],d]
]/; !FreeQ2[{d},{DiracGamma[5],DiracGamma[6],DiracGamma[7]}] && NonCommFreeQ[mass];

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4 changes: 0 additions & 4 deletions FeynCalc/Documentation/Markdown/$FCDefaultLightconeVectorN.md
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@@ -1,7 +1,3 @@
```mathematica
```

## $FCDefaultLightconeVectorN

`$FCDefaultLightconeVectorN` is a global variable which is set to FCGV["n"]. It denotes the default name for the vector $n$ in the lightcone decomposition for Lorentz tensors.
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@@ -1,7 +1,3 @@
```mathematica
```

## $FCDefaultLightconeVectorNB

`$FCDefaultLightconeVectorNB` is a global variable which is set to FCGV["nb"]. It denotes the default name for the vector $\bar{n}$ in the lightcone decomposition for Lorentz tensors.
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2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$FeynArtsDirectory.md
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Expand Up @@ -4,7 +4,7 @@

### See also

[Overview](Extra/FeynCalc.md), [$FeynCalcDirectory]($FeynCalcDirectory.md).
[Overview](Extra/FeynCalc.md), [\$FeynCalcDirectory](\$FeynCalcDirectory.md).

### Examples

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2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$FeynCalcDirectory.md
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Expand Up @@ -4,7 +4,7 @@

### See also

[Overview](Extra/FeynCalc.md), [$FeynArtsDirectory]($FeynArtsDirectory.md).
[Overview](Extra/FeynCalc.md), [\$FeynArtsDirectory](\$FeynArtsDirectory.md).

### Examples

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```mathematica
```

## $FeynCalcLastCommitDateHash

The setting of `$FeynCalcLastCommitDateHash` provides the date and the hash of the last commit in the branch from which the current FeynCalc version originates.
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4 changes: 0 additions & 4 deletions FeynCalc/Documentation/Markdown/$LeviCivitaSign.md
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@@ -1,7 +1,3 @@
```mathematica
```

## $LeviCivitaSign

`$LeviCivitaSign` is a global variable that determines the sign in the result of a Dirac trace of four gamma matrices and $\gamma^5$. `$LeviCivitaSign` is by default set to `-1` which corresponds to the convention `TR[LC[a,b,c,d,5]] = -4*I*Eps[a,b,c,d]`. Setting `$LeviCivitaSign=-I` will switch to the FORM-convention.
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2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$LimitTo4.md
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Expand Up @@ -17,7 +17,7 @@ The modern and more flexible way to simplify amplitudes involving IR-finite `PaV

### See also

[Overview](Extra/FeynCalc.md), [PaVe](PaVe.md), [PaVeReduce](PaVeReduce.md), [OneLoop](OneLoop.md), [$LimitTo4IRUnsafe]($LimitTo4IRUnsafe.md), [PaVeLimitTo4](PaVeLimitTo4.md).
[Overview](Extra/FeynCalc.md), [PaVe](PaVe.md), [PaVeReduce](PaVeReduce.md), [OneLoop](OneLoop.md), [\$LimitTo4IRUnsafe](\$LimitTo4IRUnsafe.md), [PaVeLimitTo4](PaVeLimitTo4.md).

### Examples

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2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$LimitTo4IRUnsafe.md
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Expand Up @@ -4,7 +4,7 @@

### See also

[Overview](Extra/FeynCalc.md), [PaVe](PaVe.md), [PaVeReduce](PaVeReduce.md), [OneLoop](OneLoop.md), [$LimitTo4]($LimitTo4.md), [PaVeLimitTo4](PaVeLimitTo4.md).
[Overview](Extra/FeynCalc.md), [PaVe](PaVe.md), [PaVeReduce](PaVeReduce.md), [OneLoop](OneLoop.md), [\$LimitTo4](\$LimitTo4.md), [PaVeLimitTo4](PaVeLimitTo4.md).

### Examples

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2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$LoadAddOns.md
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Expand Up @@ -4,6 +4,6 @@

### See also

[Overview](Extra/FeynCalc.md), [$RenameFeynCalcObjects]($RenameFeynCalcObjects.md).
[Overview](Extra/FeynCalc.md), [\$RenameFeynCalcObjects](\$RenameFeynCalcObjects.md).

### Examples
2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$LoadFeynArts.md
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Expand Up @@ -4,7 +4,7 @@

### See also

[Overview](Extra/FeynCalc.md), [$LoadAddOns]($LoadAddOns.md).
[Overview](Extra/FeynCalc.md), [\$LoadAddOns](\$LoadAddOns.md).

### Examples

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2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$LoadPhi.md
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Expand Up @@ -4,7 +4,7 @@

### See also

[Overview](Extra/FeynCalc.md), [$LoadAddOns]($LoadAddOns.md).
[Overview](Extra/FeynCalc.md), [\$LoadAddOns](\$LoadAddOns.md).

### Examples

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2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$LoadTARCER.md
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Expand Up @@ -4,7 +4,7 @@

### See also

[Overview](Extra/FeynCalc.md), [$LoadAddOns]($LoadAddOns.md).
[Overview](Extra/FeynCalc.md), [\$LoadAddOns](\$LoadAddOns.md).

### Examples

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15 changes: 15 additions & 0 deletions FeynCalc/Documentation/Markdown/$ParallelizeFeynCalc.md
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@@ -0,0 +1,15 @@
## $ParallelizeFeynCalc

`$ParallelizeFeynCalc` is a global switch that enables FeynCalc to evaluate some subroutines on using parallel kernels. It should be explicitly activated by setting `$ParallelizeFeynCalc` to `True`. However, before that one should evaluate `LaunchKernels[n]` with `n` being the number of parallel kernels to launch. The default value is `False`.

### See also

[Overview](Extra/FeynCalc.md)

### Examples

```mathematica
$ParallelizeFeynCalc
```

$$\text{False}$$
2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$RenameFeynCalcObjects.md
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Expand Up @@ -4,7 +4,7 @@

### See also

[Overview](Extra/FeynCalc.md), [$LoadAddOns]($LoadAddOns.md).
[Overview](Extra/FeynCalc.md), [\$LoadAddOns](\$LoadAddOns.md).

### Examples

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2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$TypesettingDim4.md
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Expand Up @@ -4,7 +4,7 @@ The string value of $TypesettingDim4 determines which symbols will be displayed

### See also

[Overview](Extra/FeynCalc.md), [$TypesettingDimD]($TypesettingDimD.md), [$TypesettingDimE]($TypesettingDimE.md).
[Overview](Extra/FeynCalc.md), [\$TypesettingDimD](\$TypesettingDimD.md), [\$TypesettingDimE](\$TypesettingDimE.md).

### Examples

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2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$TypesettingDimD.md
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Expand Up @@ -4,7 +4,7 @@ The string value of `$TypesettingDimD` determines which symbols will be displaye

### See also

[Overview](Extra/FeynCalc.md), [$TypesettingDim4]($TypesettingDim4.md), [$TypesettingDimE]($TypesettingDimE.md).
[Overview](Extra/FeynCalc.md), [\$TypesettingDim4](\$TypesettingDim4.md), [\$TypesettingDimE](\$TypesettingDimE.md).

### Examples

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2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/$TypesettingDimE.md
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Expand Up @@ -4,7 +4,7 @@ The string value of `$TypesettingDimE` determines which symbols will be displaye

### See also

[Overview](Extra/FeynCalc.md), [$TypesettingDim4]($TypesettingDim4.md), [$TypesettingDimD]($TypesettingDimD.md).
[Overview](Extra/FeynCalc.md), [\$TypesettingDim4](\$TypesettingDim4.md), [\$TypesettingDimD](\$TypesettingDimD.md).

### Examples

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1 change: 0 additions & 1 deletion FeynCalc/Documentation/Markdown/A0.md
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Expand Up @@ -26,7 +26,6 @@ $$-\frac{2 m^2 \;\text{B}_0\left(0,m^2,m^2\right)}{2-D}$$

```mathematica
SetOptions[A0, A0ToB0 -> False];
```

According to the rules of dimensional regularization $A_0(0)$ is set to 0.
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2 changes: 1 addition & 1 deletion FeynCalc/Documentation/Markdown/Abbreviation.md
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Expand Up @@ -4,7 +4,7 @@

### See also

[Overview](Extra/FeynCalc.md), [$Abbreviations]($Abbreviations.md), [OneLoop](OneLoop.md), [PaVeReduce](PaVeReduce.md), [WriteOut](WriteOut.md), [WriteOutPaVe](WriteOutPaVe.md), [GluonPropagator](GluonPropagator.md), [GluonVertex](GluonVertex.md), [QuarkPropagator](QuarkPropagator.md).
[Overview](Extra/FeynCalc.md), [\$Abbreviations](\$Abbreviations.md), [OneLoop](OneLoop.md), [PaVeReduce](PaVeReduce.md), [WriteOut](WriteOut.md), [WriteOutPaVe](WriteOutPaVe.md), [GluonPropagator](GluonPropagator.md), [GluonVertex](GluonVertex.md), [QuarkPropagator](QuarkPropagator.md).

### Examples

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5 changes: 1 addition & 4 deletions FeynCalc/Documentation/Markdown/Amplitude.md
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Expand Up @@ -23,10 +23,7 @@ This is the amplitude of a gluon self-energy diagram:
```mathematica
Amplitude["se1g1"]
Explicit[%]
Explicit[%]
```

$$\text{SUNDeltaContract}\left(f^{\text{FCGV}(\text{a})\text{FCGV}(\text{c})\text{FCGV}(\text{e})} f^{\text{FCGV}(\text{b})\text{FCGV}(\text{d})\text{FCGV}(\text{f})} \Pi _{\text{FCGV}(\text{e})\text{FCGV}(\text{f})}^{\text{FCGV}(\beta )\text{FCGV}(\sigma )}(\text{FCGV}(\text{q})) V^{\text{FCGV}(\mu )\text{FCGV}(\alpha )\text{FCGV}(\beta )}(\text{FCGV}(\text{p})\text{, }\;\text{FCGV}(\text{q})-\text{FCGV}(\text{p})\text{, }-\text{FCGV}(\text{q})) V^{\text{FCGV}(\nu )\text{FCGV}(\rho )\text{FCGV}(\sigma )}(-\text{FCGV}(\text{p})\text{, }\;\text{FCGV}(\text{p})-\text{FCGV}(\text{q})\text{, }\;\text{FCGV}(\text{q})) \Pi _{\text{FCGV}(\text{c})\text{FCGV}(\text{d})}^{\text{FCGV}(\alpha )\text{FCGV}(\rho )}(\text{FCGV}(\text{p})-\text{FCGV}(\text{q}))\right)$$
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4 changes: 0 additions & 4 deletions FeynCalc/Documentation/Markdown/ApartFF.md
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@@ -1,7 +1,3 @@
```mathematica
```

## ApartFF

`ApartFF[amp, {q1, q2, ...}]` partial fractions loop integrals by decomposing them into simpler integrals that contain only linearly independent propagators. It uses `FCApart` as a backend and is equally suitable for 1-loop and multi-loop integrals.
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4 changes: 0 additions & 4 deletions FeynCalc/Documentation/Markdown/CFAD.md
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@@ -1,7 +1,3 @@
```mathematica
```

## CFAD

`CFAD[{{q1 +..., p1 . q2 +...,} {m^2, s}, n}, ...]` denotes a Cartesian propagator given by $\frac{1}{[(q_1+\ldots)^2 + p_1 \cdot q_2 ... + m^2 + s i \eta]^n}$, where $q_1^2$ and $p_1 \cdot q_2$ are Cartesian scalar products in $D-1$ dimensions.
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5 changes: 1 addition & 4 deletions FeynCalc/Documentation/Markdown/CLC.md
Original file line number Diff line number Diff line change
Expand Up @@ -45,10 +45,7 @@ $$\bar{\delta }^{jl} \bar{\delta }^{km}-\bar{\delta }^{jm} \bar{\delta }^{kl}$$
```mathematica
CLC[i, j, k] CV[Subscript[p, 1], i] CV[Subscript[p, 2], j] CV[Subscript[p, 3], k]
Contract[%]
Contract[%]
```

$$\overline{p}_1{}^i \overline{p}_2{}^j \overline{p}_3{}^k \bar{\epsilon }^{ijk}$$
Expand Down
5 changes: 1 addition & 4 deletions FeynCalc/Documentation/Markdown/CLCD.md
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Expand Up @@ -45,10 +45,7 @@ $$(2-D) (3-D) (p\cdot q)$$
```mathematica
CLCD[i, j, k] CVD[Subscript[p, 1], i] CVD[Subscript[p, 2], j] CVD[Subscript[p, 3], k]
Contract[%]
Contract[%]
```

$$p_1{}^i p_2{}^j p_3{}^k \overset{\text{}}{\epsilon }^{ijk}$$
Expand Down
5 changes: 1 addition & 4 deletions FeynCalc/Documentation/Markdown/CartesianPairContract.md
Original file line number Diff line number Diff line change
Expand Up @@ -43,10 +43,7 @@ CartesianPair[CartesianIndex[i], CartesianMomentum[p + q]] CartesianPair[Cartesi
% /. CartesianPair -> CartesianPairContract
% /. CartesianPairContract -> CartesianPair
% /. CartesianPairContract -> CartesianPair
```

$$\left(\overline{p}+\overline{q}\right)^i \left(\overline{r}+\overline{s}\right)^i$$
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