From 9e6a4ba2577ffd9213f2d5373eded7c36b8b9a13 Mon Sep 17 00:00:00 2001 From: andreab1997 Date: Fri, 9 Dec 2022 12:44:31 +0100 Subject: [PATCH 1/4] Add mathematica notebook --- extras/Sv_generator.nb | 2805 ++++++++++++++++++++++++++++++++++++++++ 1 file changed, 2805 insertions(+) create mode 100644 extras/Sv_generator.nb diff --git a/extras/Sv_generator.nb b/extras/Sv_generator.nb new file mode 100644 index 00000000..e401f24d --- /dev/null +++ b/extras/Sv_generator.nb @@ -0,0 +1,2805 @@ +(* Content-type: application/vnd.wolfram.mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* CreatedBy='Mathematica 13.0' *) + +(*CacheID: 234*) +(* Internal cache information: +NotebookFileLineBreakTest +NotebookFileLineBreakTest +NotebookDataPosition[ 158, 7] +NotebookDataLength[ 110296, 2798] +NotebookOptionsPosition[ 94486, 2551] +NotebookOutlinePosition[ 94883, 2567] +CellTagsIndexPosition[ 94840, 2564] +WindowFrame->Normal*) + +(* Beginning of Notebook 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To do this we need to consider also the PDF because, while the \ +PDF to be used for scale variation is the one evaluated at the varied scale, \ +the PDF to be used for the central prediction is evaluated at the central \ +scale. Therefore, in this case it cannot be factored out and it does play a \ +role. +First of all we need to define the operator that transform the PDF evaluated \ +at the varied scale in the PDF evaluated at the central scale\ +\>", "Text", + CellChangeTimes->{{3.879506914797814*^9, + 3.8795071003097*^9}},ExpressionUUID->"2c370b6a-ea76-4196-9d9c-5031e32434d9"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"changepdf", "[", "a_", "]"}], " ", ":=", " ", + RowBox[{"1", " ", "+", " ", + RowBox[{"\[Kappa]F", "*", + RowBox[{"\[Gamma]", "[", "a", "]"}]}], "+", + RowBox[{ + RowBox[{"(", + RowBox[{"1", "/", "2"}], ")"}], "*", + SuperscriptBox["\[Kappa]F", "2"], "*", + RowBox[{"(", + RowBox[{ + SuperscriptBox[ + RowBox[{"\[Gamma]", "[", "a", "]"}], "2"], "+", " ", + RowBox[{ + RowBox[{"\[Beta]", "[", "a", "]"}], "*", + RowBox[{ + RowBox[{"\[Gamma]", "'"}], "[", "a", "]"}]}]}], ")"}]}], " ", "+", + " ", + SuperscriptBox[ + RowBox[{"O", "[", "a", "]"}], "3"]}]}]], "Input", + 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"+", + RowBox[{ + SuperscriptBox["\[Gamma]0", "2"], " ", + SuperscriptBox["\[Kappa]F", "2"]}]}], ")"}], " ", + SuperscriptBox["\[Alpha]F", "2"]}], "+", + InterpretationBox[ + SuperscriptBox[ + RowBox[{"O", "[", "\[Alpha]F", "]"}], "3"], + SeriesData[$CellContext`\[Alpha]F, 0, {}, 0, 3, 1], + Editable->False]}], + SeriesData[$CellContext`\[Alpha]F, 0, { + 1, $CellContext`\[Gamma]0 $CellContext`\[Kappa]F, + Rational[1, 2] ( + 2 $CellContext`\[Gamma]1 $CellContext`\[Kappa]F - $CellContext`\[Beta]0 \ +$CellContext`\[Gamma]0 $CellContext`\[Kappa]F^2 + $CellContext`\[Gamma]0^2 \ +$CellContext`\[Kappa]F^2)}, 0, 3, 1], + Editable->False]], "Output", + CellChangeTimes->{{3.879505347491722*^9, 3.8795053727741957`*^9}, { + 3.879505459812097*^9, 3.879505465511239*^9}, 3.879568932655694*^9}, + CellLabel->"Out[32]=",ExpressionUUID->"f1a50d83-5b05-4d57-93ed-c1efd2a90958"] +}, Open ]], + +Cell["\<\ +This is the definition of scheme C written in terms of the central \[Alpha] \ +and the central PDF\ 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RowBox[{"Simplify", "[", + RowBox[{ + RowBox[{"FschemeCa", "[", + RowBox[{"m", ",", "ord"}], "]"}], "-", " ", + RowBox[{ + RowBox[{"Series", "[", + RowBox[{ + RowBox[{"c", "[", + RowBox[{"\[Alpha]", ",", "m"}], "]"}], ",", " ", + RowBox[{"{", + RowBox[{"\[Alpha]", ",", " ", "0", ",", " ", + RowBox[{"m", "+", "ord"}]}], "}"}]}], "]"}], "*", "f"}]}], " ", + "]"}]}], "\[IndentingNewLine]", + RowBox[{"DiffschemeCcentral", "[", + RowBox[{"2", ",", "1"}], "]"}]}], "Input", + CellChangeTimes->{{3.879505567929284*^9, 3.8795055719671392`*^9}, { + 3.879505611168491*^9, 3.879505619022008*^9}, {3.879505652108387*^9, + 3.879505652192531*^9}, {3.879505751976781*^9, 3.879505756550097*^9}, { + 3.879505990790626*^9, 3.879506010799488*^9}, {3.879507149071569*^9, + 3.879507241264182*^9}}, + CellLabel->"In[36]:=",ExpressionUUID->"9b4d00e7-450b-4bd0-add9-53d1aa68bf3e"], + +Cell[BoxData[ + InterpretationBox[ + SuperscriptBox[ + RowBox[{"O", "[", "\[Alpha]", "]"}], "4"], + SeriesData[$CellContext`\[Alpha], 0, {}, 4, 4, 1], + Editable->False]], "Output", + CellChangeTimes->{{3.8795072255048227`*^9, 3.8795072416900177`*^9}, { + 3.8795689393969717`*^9, 3.879568947627707*^9}}, + CellLabel->"Out[37]=",ExpressionUUID->"ff65344b-91ba-4038-95ea-9752fc848c41"] +}, Open ]], + +Cell["\<\ +As expected the difference with the central prediction is always higher order +This is the definition of the scheme B prediction in terms of the central \ +\[Alpha] and the central PDF\ +\>", "Text", + CellChangeTimes->{{3.8795072578694353`*^9, + 3.879507297503524*^9}},ExpressionUUID->"88c2acb9-343d-4fb3-9908-\ +1b23e27e95ac"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"FschemeBa", "[", + RowBox[{"m_", ",", " ", "ord_"}], "]"}], " ", ":=", " ", + RowBox[{ + RowBox[{"Series", "[", + RowBox[{ + RowBox[{"c", "[", + RowBox[{"\[Alpha]", ",", "m"}], "]"}], ",", " ", + RowBox[{"{", + RowBox[{"\[Alpha]", ",", " ", "0", ",", " ", + RowBox[{"m", "+", "ord"}]}], "}"}]}], "]"}], "*", + RowBox[{"seriesaschemeBev", "[", "ord", "]"}], "*", + RowBox[{"(", + RowBox[{"changepdf", "[", "\[Alpha]", "]"}], ")"}], "*", "f"}]}]], "Input",\ + + CellChangeTimes->{{3.8795058017838697`*^9, 3.8795058486868057`*^9}, { + 3.879505935960575*^9, 3.879505940822781*^9}, 3.879505976698153*^9}, + CellLabel->"In[38]:=",ExpressionUUID->"eb0830bb-b146-483b-a8f8-a7e24299b0ca"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"DiffschemeBcentral", "[", + RowBox[{"m_", ",", "ord_"}], "]"}], ":=", + RowBox[{"Simplify", "[", + RowBox[{ + RowBox[{"FschemeBa", "[", + RowBox[{"m", ",", "ord"}], "]"}], "-", " ", + RowBox[{ + RowBox[{"Series", "[", + RowBox[{ + RowBox[{"c", "[", + RowBox[{"\[Alpha]", ",", "m"}], "]"}], ",", " ", + RowBox[{"{", + RowBox[{"\[Alpha]", ",", " ", "0", ",", " ", + RowBox[{"m", "+", "ord"}]}], "}"}]}], "]"}], "*", "f"}]}], " ", + "]"}]}]], "Input", + CellChangeTimes->{{3.879507306695356*^9, 3.879507311926631*^9}}, + CellLabel->"In[39]:=",ExpressionUUID->"820f3b96-60ff-4a4e-aa8f-66ea2aab1afa"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"DiffschemeBcentral", "[", + RowBox[{"1", ",", "1"}], "]"}]], "Input", + CellChangeTimes->{{3.879505861560878*^9, 3.879505910421152*^9}, { + 3.879505949264722*^9, 3.879505954883952*^9}, {3.8795059853767014`*^9, + 3.8795060238672657`*^9}, {3.879507326356173*^9, 3.879507336276525*^9}}, + CellLabel->"In[40]:=",ExpressionUUID->"eddb4ffa-baf7-41db-bfe8-d05e284b8c55"], + +Cell[BoxData[ + InterpretationBox[ + SuperscriptBox[ + RowBox[{"O", "[", "\[Alpha]", "]"}], "3"], + SeriesData[$CellContext`\[Alpha], 0, {}, 3, 3, 1], + Editable->False]], "Output", + CellChangeTimes->{{3.879505862546219*^9, 3.879505910844171*^9}, { + 3.879505944292346*^9, 3.879505988355105*^9}, {3.87950601874118*^9, + 3.879506024286263*^9}, {3.879507331419197*^9, 3.879507336683976*^9}, + 3.8795689549266653`*^9}, + CellLabel->"Out[40]=",ExpressionUUID->"a82b7a29-bf48-454d-841e-cab0ec15437a"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell["Wrong starting point leads to wrong results", "Subsubsection", + CellChangeTimes->{{3.8795723096731377`*^9, + 3.8795723217005653`*^9}},ExpressionUUID->"b3288821-db7b-4258-ab4b-\ +9d1e6b710783"], + +Cell["\<\ +As expected the difference with the central prediction is always higher order\ +\[LineSeparator]Now we want to show the even little modifications of the \ +expression give wrong answers, i.e. the scale variations terms are not higher \ +orders anymore. +As a first example, let\[CloseCurlyQuote]s take scheme B as a function of the \ +central \[Alpha]\ +\>", "Text", + CellChangeTimes->{{3.8795073461274223`*^9, 3.8795074051029673`*^9}, { + 3.8795074610113792`*^9, 3.879507476919169*^9}, {3.8795722925979967`*^9, + 3.8795722949939117`*^9}},ExpressionUUID->"f75db1db-0759-4d6a-9c07-\ +c35755aea26d"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"seriesaschemeBevWrong", "[", "ord_", "]"}], " ", ":=", " ", + RowBox[{"Series", "[", + RowBox[{ + RowBox[{"schemeBev", "[", "\[Alpha]", "]"}], ",", " ", + RowBox[{"{", + RowBox[{"\[Alpha]", ",", " ", "0", ",", " ", "ord"}], "}"}]}], + "]"}]}]], "Input", + CellChangeTimes->{{3.8795074409174767`*^9, 3.879507455526186*^9}}, + CellLabel->"In[41]:=",ExpressionUUID->"fc27f870-45da-42b3-9b93-dc5170d5e759"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"Simplify", "[", + RowBox[{ + RowBox[{"seriesaschemeBevWrong", "[", "2", "]"}], " ", "-", " ", + RowBox[{"seriesaschemeBev", "[", "2", "]"}]}], "]"}]], "Input", + CellChangeTimes->{{3.879507498087685*^9, 3.879507537784192*^9}}, + CellLabel->"In[42]:=",ExpressionUUID->"1bf93260-2bc0-4d90-9627-9c0a4885fc21"], + +Cell[BoxData[ + InterpretationBox[ + RowBox[{ + RowBox[{"\[Beta]0", " ", "\[Gamma]0", " ", + SuperscriptBox["\[Kappa]F", "2"], " ", + SuperscriptBox["\[Alpha]", "2"]}], "+", + InterpretationBox[ + SuperscriptBox[ + RowBox[{"O", "[", "\[Alpha]", "]"}], "3"], + SeriesData[$CellContext`\[Alpha], 0, {}, 2, 3, 1], + Editable->False]}], + SeriesData[$CellContext`\[Alpha], + 0, {$CellContext`\[Beta]0 $CellContext`\[Gamma]0 $CellContext`\[Kappa]F^2}, + 2, 3, 1], + Editable->False]], "Output", + CellChangeTimes->{{3.8795075016699963`*^9, 3.8795075382064457`*^9}, + 3.8795689586938953`*^9}, + CellLabel->"Out[42]=",ExpressionUUID->"1f56629d-7d19-4359-b54d-97d29e1c8d76"] +}, Open ]], + +Cell["\<\ +Of course the two are different at NNLO, but let\[CloseCurlyQuote]s see the \ +impact on the prediction and on the difference between schemes\ +\>", "Text", + CellChangeTimes->{{3.879507545082168*^9, + 3.879507566363866*^9}},ExpressionUUID->"bb0648e8-bc2a-4e5a-9da9-\ +3928e31a0e85"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"diffBminusCaWrong", "[", + RowBox[{"m_", ",", " ", "ord_"}], "]"}], " ", ":=", " ", + RowBox[{"Series", "[", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"seriesaschemeBevWrong", "[", "ord", "]"}], "*", + RowBox[{"Series", "[", + RowBox[{ + RowBox[{"c", "[", + RowBox[{"\[Alpha]", ",", "m"}], "]"}], ",", " ", + RowBox[{"{", + RowBox[{"\[Alpha]", ",", " ", "0", ",", " ", + RowBox[{"m", "+", "ord"}]}], "}"}]}], "]"}]}], " ", "-", " ", + RowBox[{"seriesschemeCacoeff", "[", + RowBox[{"m", ",", " ", "ord"}], "]"}]}], ",", " ", + RowBox[{"{", + RowBox[{"\[Alpha]", ",", " ", "0", ",", " ", + RowBox[{"m", "+", "ord"}]}], "}"}]}], "]"}]}]], "Input", + CellChangeTimes->{{3.879507583726852*^9, 3.879507594993668*^9}}, + CellLabel->"In[43]:=",ExpressionUUID->"c1f68153-e3b0-4fa4-ac14-6b014877ea45"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"diffBminusCaWrong", "[", + RowBox[{"0", ",", "1"}], "]"}]], "Input", + CellChangeTimes->{{3.879507607376096*^9, 3.8795076130735397`*^9}}, + CellLabel->"In[44]:=",ExpressionUUID->"3d0d93cb-42bf-4afb-b36f-dd5406a9088a"], + +Cell[BoxData[ + InterpretationBox[ + SuperscriptBox[ + RowBox[{"O", "[", "\[Alpha]", "]"}], "2"], + SeriesData[$CellContext`\[Alpha], 0, {}, 2, 2, 1], + Editable->False]], "Output", + CellChangeTimes->{3.87950761358458*^9, 3.879568962296973*^9}, + CellLabel->"Out[44]=",ExpressionUUID->"a0605048-cac7-42f7-8b02-e3318d266ec0"] +}, Open ]], + +Cell["\<\ +As expected, at NLO there is no difference and the difference between schemes \ +still works\ +\>", "Text", + CellChangeTimes->{{3.879507618274199*^9, + 3.87950763750298*^9}},ExpressionUUID->"3783e889-2a8f-4d7e-8d3c-\ +4e1310bc354b"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"diffBminusCaWrong", "[", + RowBox[{"0", ",", "2"}], "]"}]], "Input", + CellChangeTimes->{{3.879507644645513*^9, 3.879507644677538*^9}}, + CellLabel->"In[45]:=",ExpressionUUID->"27c7883c-9920-4bb9-a347-6d378eb20890"], + +Cell[BoxData[ + InterpretationBox[ + RowBox[{ + RowBox[{"C0", " ", "\[Beta]0", " ", "\[Gamma]0", " ", + SuperscriptBox["\[Kappa]F", "2"], " ", + SuperscriptBox["\[Alpha]", "2"]}], "+", + InterpretationBox[ + SuperscriptBox[ + RowBox[{"O", "[", "\[Alpha]", "]"}], "3"], + SeriesData[$CellContext`\[Alpha], 0, {}, 2, 3, 1], + Editable->False]}], + SeriesData[$CellContext`\[Alpha], + 0, {$CellContext`C0 $CellContext`\[Beta]0 $CellContext`\[Gamma]0 \ +$CellContext`\[Kappa]F^2}, 2, 3, 1], + Editable->False]], "Output", + CellChangeTimes->{3.879507645316019*^9, 3.879568964388379*^9}, + CellLabel->"Out[45]=",ExpressionUUID->"cde93bbe-c8df-4c80-b193-05b0802badf7"] +}, Open ]], + +Cell["At NNLO instead the difference is no more higher order! ", "Text", + CellChangeTimes->{{3.879507650855632*^9, + 3.8795076602628183`*^9}},ExpressionUUID->"7bc8ef07-4774-4408-b21f-\ +914197f2445d"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"FschemeBaWrong", "[", + RowBox[{"m_", ",", " ", "ord_"}], "]"}], " ", ":=", " ", + RowBox[{ + RowBox[{"Series", "[", + RowBox[{ + RowBox[{"c", "[", + RowBox[{"\[Alpha]", ",", "m"}], "]"}], ",", " ", + RowBox[{"{", + RowBox[{"\[Alpha]", ",", " ", "0", ",", " ", + RowBox[{"m", "+", "ord"}]}], "}"}]}], "]"}], "*", + RowBox[{"seriesaschemeBevWrong", "[", "ord", "]"}], "*", + RowBox[{"(", + RowBox[{"changepdf", "[", "\[Alpha]", "]"}], ")"}], "*", "f"}]}]], "Input",\ + + CellChangeTimes->{{3.879507768843651*^9, 3.879507774316888*^9}}, + CellLabel->"In[46]:=",ExpressionUUID->"c67d40db-b34c-4956-bd8b-2afdce382aeb"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"DiffschemeBcentralWrong", "[", + RowBox[{"m_", ",", "ord_"}], "]"}], ":=", + RowBox[{"Simplify", "[", + RowBox[{ + RowBox[{"FschemeBaWrong", "[", + RowBox[{"m", ",", "ord"}], "]"}], "-", " ", + RowBox[{ + RowBox[{"Series", "[", + RowBox[{ + RowBox[{"c", "[", + RowBox[{"\[Alpha]", ",", "m"}], "]"}], ",", " ", + RowBox[{"{", + RowBox[{"\[Alpha]", ",", " ", "0", ",", " ", + RowBox[{"m", "+", "ord"}]}], "}"}]}], "]"}], "*", "f"}]}], " ", + "]"}]}]], "Input", + CellChangeTimes->{{3.879507793614092*^9, 3.879507798037092*^9}}, + CellLabel->"In[47]:=",ExpressionUUID->"788a72b0-92f2-46a4-901a-18597120fedc"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"DiffschemeBcentralWrong", "[", + RowBox[{"0", ",", "1"}], "]"}]], "Input", + CellChangeTimes->{{3.879507800870521*^9, 3.879507805942073*^9}}, + CellLabel->"In[48]:=",ExpressionUUID->"071bc1e7-1196-48b1-bb09-a8b165e196a8"], + +Cell[BoxData[ + InterpretationBox[ + SuperscriptBox[ + RowBox[{"O", "[", "\[Alpha]", "]"}], "2"], + SeriesData[$CellContext`\[Alpha], 0, {}, 2, 2, 1], + Editable->False]], "Output", + CellChangeTimes->{3.879507806834816*^9, 3.879568969938611*^9}, + CellLabel->"Out[48]=",ExpressionUUID->"822de04e-5a69-46c0-abd6-29f0da8221be"] +}, Open ]], + +Cell["Again, at NLO everything seems to work. But at NNLO...", "Text", + CellChangeTimes->{{3.879507812127903*^9, + 3.8795078256213408`*^9}},ExpressionUUID->"a2b8b98a-8779-43b0-9bee-\ +e3e438f3215a"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"DiffschemeBcentralWrong", "[", + RowBox[{"0", ",", "2"}], "]"}]], "Input", + CellChangeTimes->{{3.8795078342079363`*^9, 3.879507834297463*^9}}, + CellLabel->"In[49]:=",ExpressionUUID->"78b2bf6d-4ecb-4e2f-bcad-e620fe9e7a6a"], + +Cell[BoxData[ + InterpretationBox[ + RowBox[{ + RowBox[{"C0", " ", "f", " ", "\[Beta]0", " ", "\[Gamma]0", " ", + SuperscriptBox["\[Kappa]F", "2"], " ", + SuperscriptBox["\[Alpha]", "2"]}], "+", + InterpretationBox[ + SuperscriptBox[ + RowBox[{"O", "[", "\[Alpha]", "]"}], "3"], + SeriesData[$CellContext`\[Alpha], 0, {}, 2, 3, 1], + Editable->False]}], + SeriesData[$CellContext`\[Alpha], + 0, {$CellContext`C0 $CellContext`f $CellContext`\[Beta]0 $CellContext`\ +\[Gamma]0 $CellContext`\[Kappa]F^2}, 2, 3, 1], + Editable->False]], "Output", + CellChangeTimes->{3.879507837344734*^9, 3.879568971594101*^9}, + CellLabel->"Out[49]=",ExpressionUUID->"8977bdd8-ba8d-40d6-a004-8c4f1f6f5e80"] +}, Open ]], + +Cell["\<\ +we get that the scale variation terms are no more of higher orders. \ +\>", "Text", + CellChangeTimes->{{3.879507843456785*^9, + 3.87950785920944*^9}},ExpressionUUID->"0329460f-8abc-42c3-ac20-\ +5bb53d785e87"], + +Cell["\<\ +The same can be shown for scheme C +\ +\>", "Text", + CellChangeTimes->{{3.879507915594734*^9, + 3.879507922600642*^9}},ExpressionUUID->"92da9939-aca0-49b2-8cad-\ +7a12702a7822"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"schemeCacoeffWrong", "[", " ", "m_", "]"}], " ", ":=", " ", + RowBox[{ + RowBox[{"c", "[", + RowBox[{"\[Alpha]", ",", " ", "m"}], "]"}], "*", + RowBox[{"(", + RowBox[{"schemeBev", "[", "\[Alpha]", "]"}], ")"}]}]}]], "Input", + CellChangeTimes->{{3.8795079501979847`*^9, 3.879507962634523*^9}}, + CellLabel->"In[50]:=",ExpressionUUID->"1de8a6e7-2abf-4f57-98b6-64719c53de7a"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"seriesschemeCacoeffWrong", "[", + RowBox[{"m_", ",", " ", "ord_"}], "]"}], " ", ":=", " ", + RowBox[{"Series", "[", + RowBox[{ + RowBox[{"schemeCacoeffWrong", "[", "m", "]"}], ",", " ", + RowBox[{"{", + RowBox[{"\[Alpha]", ",", " ", "0", ",", + RowBox[{"m", "+", "ord"}]}], " ", "}"}]}], "]"}]}]], "Input", + CellChangeTimes->{{3.87950799021835*^9, 3.87950799452682*^9}}, + CellLabel->"In[51]:=",ExpressionUUID->"899fbe7e-34fc-4486-8bb2-5e627bae6d8f"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"FschemeCaWrong", "[", + RowBox[{"m_", ",", " ", "ord_"}], "]"}], " ", ":=", " ", + RowBox[{ + RowBox[{"seriesschemeCacoeffWrong", "[", + RowBox[{"m", ",", "ord"}], "]"}], "*", + RowBox[{"(", + RowBox[{"changepdf", "[", "\[Alpha]", "]"}], ")"}], "*", "f"}]}]], "Input",\ + + CellChangeTimes->{{3.879508029778556*^9, 3.87950803446424*^9}}, + CellLabel->"In[52]:=",ExpressionUUID->"c530757d-5402-4fff-836c-d80423b91089"], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{ + RowBox[{"DiffschemeCcentralWrong", "[", + RowBox[{"m_", ",", "ord_"}], "]"}], ":=", + RowBox[{"Simplify", "[", + RowBox[{ + RowBox[{"FschemeCaWrong", "[", + RowBox[{"m", ",", "ord"}], "]"}], "-", " ", + RowBox[{ + RowBox[{"Series", "[", + RowBox[{ + RowBox[{"c", "[", + RowBox[{"\[Alpha]", ",", "m"}], "]"}], ",", " ", + RowBox[{"{", + RowBox[{"\[Alpha]", ",", " ", "0", ",", " ", + RowBox[{"m", "+", "ord"}]}], "}"}]}], "]"}], "*", "f"}]}], " ", + "]"}]}], "\[IndentingNewLine]", + RowBox[{"DiffschemeCcentralWrong", "[", + RowBox[{"0", ",", "2"}], "]"}]}], "Input", + CellChangeTimes->{{3.879508047140024*^9, 3.879508069456108*^9}}, + CellLabel->"In[55]:=",ExpressionUUID->"85f4b64b-9450-40d0-8d2b-b4cd4540fab7"], + +Cell[BoxData[ + InterpretationBox[ + RowBox[{ + RowBox[{"C0", " ", "f", " ", "\[Beta]0", " ", "\[Gamma]0", " ", + SuperscriptBox["\[Kappa]F", "2"], " ", + SuperscriptBox["\[Alpha]", "2"]}], "+", + InterpretationBox[ + SuperscriptBox[ + RowBox[{"O", "[", "\[Alpha]", "]"}], "3"], + SeriesData[$CellContext`\[Alpha], 0, {}, 2, 3, 1], + Editable->False]}], + SeriesData[$CellContext`\[Alpha], + 0, {$CellContext`C0 $CellContext`f $CellContext`\[Beta]0 $CellContext`\ +\[Gamma]0 $CellContext`\[Kappa]F^2}, 2, 3, 1], + Editable->False]], "Output", + CellChangeTimes->{{3.879508057890173*^9, 3.879508069840438*^9}, { + 3.879568978337335*^9, 3.87956898032793*^9}}, + CellLabel->"Out[56]=",ExpressionUUID->"2fe479b6-b298-460a-9087-7c873a187a02"] +}, Open ]], + +Cell["Again at NNLO we get the wrong result", "Text", + CellChangeTimes->{{3.8795080757411118`*^9, + 3.879508081422022*^9}},ExpressionUUID->"714487f6-1b64-44a5-8ba0-\ +5e68c94d17c4"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell["Joint SV", "Subsection", + CellChangeTimes->{{3.879572333912809*^9, + 3.879572344686664*^9}},ExpressionUUID->"8740f4a2-8e2a-4217-89ab-\ +1fb3630a0e0a"], + +Cell["\<\ +In this case we want to do joint factorization and renormalization sv. In the \ +scheme B case this is trivial because the two are completely separated. \ +Instead I can compute the schemeC coeff.\ +\>", "Text", + CellChangeTimes->{{3.879572348078421*^9, 3.879572382889886*^9}, { + 3.879573897982828*^9, + 3.8795739087972603`*^9}},ExpressionUUID->"61affe21-362d-4c55-985b-\ +1badd5549198"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"schemeCjointaR", "[", + RowBox[{"m_", ",", "ord_"}], "]"}], " ", ":=", " ", + RowBox[{ + RowBox[{"seriesaRcrenbar", "[", + RowBox[{"m", ",", "ord"}], "]"}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{"seriesaschemeBev", "[", "ord", "]"}], "/.", + RowBox[{"\[Alpha]", "->", " ", + RowBox[{"\[Alpha]R", " ", "-", " ", + RowBox[{"\[Kappa]R", "*", "\[Beta]0", "*", + SuperscriptBox["\[Alpha]R", "2"]}], " ", "-", + RowBox[{"\[Kappa]R", "*", + SuperscriptBox["\[Alpha]R", "3"], "*", + RowBox[{"(", + RowBox[{"\[Beta]1", " ", "-", " ", + RowBox[{"\[Kappa]R", "*", "\[Beta]0", "*", "\[Beta]0"}]}], ")"}]}], + "+", " ", + SuperscriptBox[ + 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