(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 9.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 56560, 1513] NotebookOptionsPosition[ 53740, 1414] NotebookOutlinePosition[ 54138, 1432] CellTagsIndexPosition[ 54095, 1429] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ "This notebook is an auxiliary file to the paper:\n\n\ \[OpenCurlyDoubleQuote]Higher dimension operators in the SM EFT\ \[CloseCurlyDoubleQuote], \nAuthors: Brian Henning, Xiaochuan Lu, Tom Melia, \ and Hitoshi Murayama. \n\nAll equation numbers in the below relate to this \ paper.\n\nWe stress that the implementation of our method contained in this \ notebook is geared towards the pedagogical, rather than optimized for speed. \ The calculation of the dimension 6 Hilbert series takes ~7 min; for dimension \ 7 ~30min; dimension 8 ~2hr. ", StyleBox["This unoptimized implementation is thus prohibitaviely slow for \ higher dimensions than this.", FontSlant->"Italic"], " In a separate notebook, also given as an auxilliary file to the above \ paper, we provide the Hilbert series up to dimension 12 in the SM EFT, with \ full Nf dependence, that we produce using an optimized version of the below \ code. \n\nThe below code generates Hilbert series for the SM (with one \ fermion generation) for a specified mass dimension (dimension five and above):" }], "Text", CellChangeTimes->{{3.656270300905652*^9, 3.656270582461783*^9}, { 3.6562706705899143`*^9, 3.656270695381528*^9}, {3.656270726772388*^9, 3.6562707539075747`*^9}, {3.656270831934865*^9, 3.6562708581399927`*^9}, { 3.6562711381117268`*^9, 3.656271138268882*^9}, {3.6562716680482407`*^9, 3.6562717120305223`*^9}, {3.656279651854205*^9, 3.656279671581016*^9}, { 3.65627980364716*^9, 3.656279928856657*^9}, {3.657380364243438*^9, 3.6573803957948647`*^9}, {3.657380749580721*^9, 3.657380754347816*^9}, { 3.6573815704755087`*^9, 3.6573815791119223`*^9}, 3.658764387077196*^9}], Cell[BoxData[ RowBox[{"Quit", "[", "]"}]], "Input", CellChangeTimes->{{3.6562706320116253`*^9, 3.656270635334462*^9}}], Cell[CellGroupData[{ Cell["Haar measures", "Subtitle", CellChangeTimes->{{3.656270610152011*^9, 3.656270616982602*^9}}], Cell["The Haar measures for SO(4), as per eq. (3.10).", "Text", CellChangeTimes->{{3.656270644615546*^9, 3.656270666909636*^9}, { 3.656270700660892*^9, 3.656270714980976*^9}, 3.6562708784115877`*^9, { 3.6562709280229187`*^9, 3.656270928279564*^9}, {3.656271092614208*^9, 3.656271098582245*^9}, {3.658764445870736*^9, 3.658764447141282*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"hrso4", "[", RowBox[{"\[Alpha]_", ",", "\[Beta]_"}], "]"}], ":=", RowBox[{ FractionBox["1", RowBox[{"4", "\[Alpha]", " ", "\[Beta]"}]], RowBox[{"(", RowBox[{"1", "-", SuperscriptBox["\[Alpha]", "2"]}], ")"}], RowBox[{"(", RowBox[{"1", "-", FractionBox["1", SuperscriptBox["\[Alpha]", "2"]]}], ")"}], RowBox[{"(", RowBox[{"1", "-", SuperscriptBox["\[Beta]", "2"]}], ")"}], RowBox[{"(", RowBox[{"1", "-", FractionBox["1", SuperscriptBox["\[Beta]", "2"]]}], ")"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.6545397506714697`*^9, 3.654539750718873*^9}, 3.6545400602395353`*^9, {3.654541476693202*^9, 3.65454156332617*^9}, 3.654541684625875*^9, {3.654541741051137*^9, 3.654541968532803*^9}, { 3.6562709183012857`*^9, 3.656270919250659*^9}}], Cell["\<\ The Haar measures for U(1), SU(2) and SU(3), as per eqs. (3.12a,b,c)\ \>", "Text", CellChangeTimes->{{3.656270644615546*^9, 3.656270666909636*^9}, { 3.656270700660892*^9, 3.656270714980976*^9}, 3.6562708784115877`*^9, { 3.6562709280229187`*^9, 3.656270928279564*^9}, {3.6562710875993233`*^9, 3.656271091005937*^9}, {3.6587644568851852`*^9, 3.65876446822053*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"hru1", "[", "x_", "]"}], ":=", FractionBox["1", "x"]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"hrsu2", "[", "y_", "]"}], ":=", RowBox[{ FractionBox["1", RowBox[{"2", "y"}]], RowBox[{"(", RowBox[{"1", "-", SuperscriptBox["y", "2"]}], ")"}], RowBox[{"(", RowBox[{"1", "-", FractionBox["1", SuperscriptBox["y", "2"]]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"hrsu3", "[", RowBox[{"z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ FractionBox["1", RowBox[{"6", "z1", " ", "z2"}]], RowBox[{"(", RowBox[{"1", "-", RowBox[{"z1", " ", "z2"}]}], ")"}], RowBox[{"(", RowBox[{"1", "-", FractionBox[ SuperscriptBox["z1", "2"], "z2"]}], ")"}], RowBox[{"(", RowBox[{"1", "-", FractionBox[ SuperscriptBox["z2", "2"], "z1"]}], ")"}], RowBox[{"(", RowBox[{"1", "-", FractionBox["1", RowBox[{"z1", " ", "z2"}]]}], ")"}], RowBox[{"(", RowBox[{"1", "-", FractionBox["z1", SuperscriptBox["z2", "2"]]}], ")"}], RowBox[{"(", RowBox[{"1", "-", FractionBox["z2", SuperscriptBox["z1", "2"]]}], ")"}]}]}], ";"}]}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Momentum generating function", "Subtitle", CellChangeTimes->{{3.656270610152011*^9, 3.656270616982602*^9}, { 3.656271012918799*^9, 3.656271020573649*^9}, {3.656271128119276*^9, 3.656271130905558*^9}}], Cell["Eq. (3.4)", "Text", CellChangeTimes->{{3.6562711573122883`*^9, 3.656271159648291*^9}, { 3.658764478372033*^9, 3.658764480588212*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"P", "[", RowBox[{"t_", ",", "\[Alpha]_", ",", "\[Beta]_"}], "]"}], ":=", FractionBox["1", RowBox[{ RowBox[{"(", RowBox[{"1", "-", RowBox[{"t", " ", "\[Alpha]", " ", "\[Beta]"}]}], ")"}], RowBox[{"(", RowBox[{"1", "-", FractionBox["t", RowBox[{"\[Alpha]", " ", "\[Beta]"}]]}], ")"}], RowBox[{"(", RowBox[{"1", "-", RowBox[{"t", " ", FractionBox["\[Alpha]", "\[Beta]"]}]}], ")"}], RowBox[{"(", RowBox[{"1", "-", RowBox[{"t", " ", FractionBox["\[Beta]", "\[Alpha]"]}]}], ")"}]}]]}], ";"}]], "Input", CellChangeTimes->{{3.654603742567218*^9, 3.6546037759222383`*^9}, { 3.65460403529431*^9, 3.6546040595632877`*^9}, {3.654604145498789*^9, 3.6546041491683702`*^9}, 3.6547205139509974`*^9, {3.6562711137202997`*^9, 3.656271113831664*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Characters of represenations", "Subtitle", CellChangeTimes->{{3.656270610152011*^9, 3.656270616982602*^9}, { 3.656271012918799*^9, 3.656271020573649*^9}}], Cell["\<\ Characters of conformal representations (with the suprion scaling in (2.7)), \ eqs (3.3a-e)\ \>", "Text", CellChangeTimes->{{3.656271030321748*^9, 3.656271072830447*^9}, { 3.6562711181355467`*^9, 3.656271118382443*^9}, {3.656271165560074*^9, 3.6562711681840982`*^9}, {3.656271751925622*^9, 3.656271753564726*^9}, { 3.658764533827952*^9, 3.658764579520208*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"\[Chi]scal", "[", RowBox[{"t_", ",", "\[Alpha]_", ",", "\[Beta]_"}], "]"}], ":=", RowBox[{ RowBox[{"P", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], RowBox[{"(", RowBox[{"1", "-", SuperscriptBox["t", "2"]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]fermL", "[", RowBox[{"t_", ",", "\[Alpha]_", ",", "\[Beta]_"}], "]"}], ":=", RowBox[{ RowBox[{"P", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Alpha]", "+", FractionBox["1", "\[Alpha]"]}], ")"}], "-", RowBox[{"t", RowBox[{"(", RowBox[{"\[Beta]", "+", FractionBox["1", "\[Beta]"]}], ")"}]}]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]fermR", "[", RowBox[{"t_", ",", "\[Alpha]_", ",", "\[Beta]_"}], "]"}], ":=", RowBox[{ RowBox[{"P", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"\[Beta]", "+", FractionBox["1", "\[Beta]"]}], ")"}], "-", RowBox[{"t", RowBox[{"(", RowBox[{"\[Alpha]", "+", FractionBox["1", "\[Alpha]"]}], ")"}]}]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]fsL", "[", RowBox[{"t_", ",", "\[Alpha]_", ",", "\[Beta]_"}], "]"}], ":=", RowBox[{ RowBox[{"P", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], RowBox[{"(", RowBox[{ SuperscriptBox["\[Alpha]", "2"], "+", "1", "+", FractionBox["1", SuperscriptBox["\[Alpha]", "2"]], "-", RowBox[{"t", RowBox[{"(", RowBox[{"\[Alpha]", "+", FractionBox["1", "\[Alpha]"]}], ")"}], RowBox[{"(", RowBox[{"\[Beta]", "+", FractionBox["1", "\[Beta]"]}], ")"}]}], "+", SuperscriptBox["t", "2"]}], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]fsR", "[", RowBox[{"t_", ",", "\[Alpha]_", ",", "\[Beta]_"}], "]"}], ":=", RowBox[{ RowBox[{"P", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], RowBox[{"(", RowBox[{ SuperscriptBox["\[Beta]", "2"], "+", "1", "+", FractionBox["1", SuperscriptBox["\[Beta]", "2"]], "-", RowBox[{"t", RowBox[{"(", RowBox[{"\[Alpha]", "+", FractionBox["1", "\[Alpha]"]}], ")"}], RowBox[{"(", RowBox[{"\[Beta]", "+", FractionBox["1", "\[Beta]"]}], ")"}]}], "+", SuperscriptBox["t", "2"]}], ")"}]}]}], ";"}]}], "Input", CellChangeTimes->{{3.6545395410555058`*^9, 3.654539548364653*^9}, { 3.654539597272024*^9, 3.654539729929957*^9}, {3.6545398410599203`*^9, 3.654539847686799*^9}, {3.654540043745878*^9, 3.6545400520930643`*^9}, { 3.6545437609008904`*^9, 3.65454377686987*^9}, {3.654604064324539*^9, 3.6546041926080923`*^9}, {3.656271471216955*^9, 3.656271475945755*^9}}], Cell["\<\ Characters needed for SM representations under U(1)xSU(2)xSU(3), eqs \ (3.5)-(3.8)\ \>", "Text", CellChangeTimes->{{3.656271030321748*^9, 3.656271072830447*^9}, { 3.6562711181355467`*^9, 3.656271118382443*^9}, {3.656271165560074*^9, 3.656271204487073*^9}, {3.658764587959903*^9, 3.658764594927217*^9}}], Cell["U(1), charge q", "Text", CellChangeTimes->{{3.656271230592223*^9, 3.656271256534485*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"\[Chi]u1", "[", RowBox[{"q_", ",", "x_"}], "]"}], ":=", SuperscriptBox["x", "q"]}], ";"}]], "Input", CellChangeTimes->{{3.654543291491376*^9, 3.65454331352217*^9}}], Cell["SU(2) fundamental, anti-fundamental, adjoint", "Text", CellChangeTimes->{{3.6562712381745653`*^9, 3.656271262638506*^9}, { 3.656271315604828*^9, 3.6562713208131733`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"\[Chi]su2f", "[", "y_", "]"}], ":=", RowBox[{"y", "+", FractionBox["1", "y"]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]su2fb", "[", "y_", "]"}], ":=", RowBox[{"y", "+", FractionBox["1", "y"]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]su2ad", "[", "y_", "]"}], ":=", RowBox[{ SuperscriptBox["y", "2"], " ", "+", " ", "1", " ", "+", FractionBox["1", SuperscriptBox["y", "2"]]}]}], ";"}]}], "Input", CellChangeTimes->{{3.654542064979409*^9, 3.6545421963493233`*^9}, { 3.654542330815556*^9, 3.6545423389645557`*^9}, {3.654543266694426*^9, 3.654543266844611*^9}, {3.654720549664468*^9, 3.6547205500141573`*^9}, { 3.6562713245982*^9, 3.656271335117722*^9}, {3.656271419968123*^9, 3.656271444787531*^9}, {3.656281240196089*^9, 3.6562812693867397`*^9}}], Cell["SU(3) fundamental, antifundamental, adjoint", "Text", CellChangeTimes->{{3.656271305198346*^9, 3.656271313829035*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"\[Chi]su3f", "[", RowBox[{"z1_", ",", "z2_"}], "]"}], ":=", RowBox[{"z1", "+", FractionBox["z2", "z1"], "+", FractionBox["1", "z2"]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]su3fb", "[", RowBox[{"z1_", ",", "z2_"}], "]"}], ":=", RowBox[{"z2", "+", FractionBox["z1", "z2"], "+", FractionBox["1", "z1"]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]su3ad", "[", RowBox[{"z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"z1", " ", "z2"}], "+", FractionBox[ SuperscriptBox["z2", "2"], "z1"], "+", FractionBox[ SuperscriptBox["z1", "2"], "z2"], "+", "2", "+", FractionBox["z1", SuperscriptBox["z2", "2"]], "+", FractionBox["z2", SuperscriptBox["z1", "2"]], "+", FractionBox["1", RowBox[{"z1", " ", "z2"}]]}]}], ";"}]}], "Input", CellChangeTimes->{{3.654542221876007*^9, 3.6545422597188187`*^9}, { 3.654542368788877*^9, 3.654542494699017*^9}, {3.6545974027023077`*^9, 3.654597448473558*^9}, 3.656271282471035*^9, {3.656271450851338*^9, 3.656271463451845*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["SM field characters", "Subtitle", CellChangeTimes->{{3.656270610152011*^9, 3.656270616982602*^9}, { 3.656271012918799*^9, 3.656271020573649*^9}, {3.656271533002306*^9, 3.656271535521137*^9}}], Cell["\<\ The label d stands for \[OpenCurlyQuote]dagger\[CloseCurlyQuote], i.e. Hd is \ the hermitian conjugate of H. We have the SM fields (H, Hd, Q, Qd, u, ud, d, dd, L, Ld, e, ed, Bl, Br, Wl, Wr, Gl, Gr) The hypercharges have been scaled so as to take integer values.\ \>", "Text", CellChangeTimes->{{3.656271865488701*^9, 3.65627194727914*^9}, { 3.6562720474340878`*^9, 3.656272064393836*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"\[Chi]H", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]scal", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{"3", ",", "x"}], "]"}], "*", RowBox[{"su2f", "[", "y", "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]Hd", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]scal", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{ RowBox[{"-", "3"}], ",", "x"}], "]"}], "*", RowBox[{"su2fb", "[", "y", "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]Q", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fermL", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{"1", ",", "x"}], "]"}], "*", RowBox[{"su2f", "[", "y", "]"}], "*", RowBox[{"su3f", "[", RowBox[{"z1", ",", "z2"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]Qd", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fermR", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{ RowBox[{"-", "1"}], ",", "x"}], "]"}], "*", RowBox[{"su2fb", "[", "y", "]"}], "*", RowBox[{"su3fb", "[", RowBox[{"z1", ",", "z2"}], "]"}]}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"\[Chi]u", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fermL", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{ RowBox[{"-", "4"}], ",", "x"}], "]"}], "*", RowBox[{"su3fb", "[", RowBox[{"z1", ",", "z2"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]ud", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fermR", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{"4", ",", "x"}], "]"}], "*", RowBox[{"su3f", "[", RowBox[{"z1", ",", "z2"}], "]"}]}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"\[Chi]d", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fermL", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{"2", ",", "x"}], "]"}], "*", RowBox[{"su3fb", "[", RowBox[{"z1", ",", "z2"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]dd", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fermR", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{ RowBox[{"-", "2"}], ",", "x"}], "]"}], "*", RowBox[{"su3f", "[", RowBox[{"z1", ",", "z2"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]L", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fermL", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{ RowBox[{"-", "3"}], ",", "x"}], "]"}], "*", RowBox[{"su2f", "[", "y", "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]Ld", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fermR", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{"3", ",", "x"}], "]"}], "*", RowBox[{"su2fb", "[", "y", "]"}]}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"\[Chi]e", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fermL", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{"6", ",", "x"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]ed", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fermR", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"u1", "[", RowBox[{ RowBox[{"-", "6"}], ",", "x"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]Bl", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{"\[Chi]fsL", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]Br", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{"\[Chi]fsR", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]Wl", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fsL", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"su2ad", "[", "y", "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]Wr", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fsR", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"su2ad", "[", "y", "]"}]}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"\[Chi]Gl", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fsL", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"su3ad", "[", RowBox[{"z1", ",", "z2"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Chi]Gr", "[", RowBox[{ "t_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "x_", ",", "y_", ",", "z1_", ",", "z2_"}], "]"}], ":=", RowBox[{ RowBox[{"\[Chi]fsR", "[", RowBox[{"t", ",", "\[Alpha]", ",", "\[Beta]"}], "]"}], "*", RowBox[{"su3ad", "[", RowBox[{"z1", ",", "z2"}], "]"}]}]}], ";"}]}], "Input", CellChangeTimes->{{3.654543352888482*^9, 3.654543387264287*^9}, { 3.654543463309079*^9, 3.654543599810466*^9}, {3.654543698886324*^9, 3.654543707058783*^9}, {3.654543998164711*^9, 3.6545439985076103`*^9}, { 3.654544142542554*^9, 3.654544142856732*^9}, {3.6548122461129417`*^9, 3.654812283264229*^9}, {3.6555587501301947`*^9, 3.6555587596727858`*^9}, { 3.655561041693078*^9, 3.655561044353457*^9}, {3.6555610820772038`*^9, 3.65556108421859*^9}, {3.656271771471933*^9, 3.656271858456442*^9}, { 3.656272005410849*^9, 3.656272036002406*^9}, {3.656272071702858*^9, 3.656272187762096*^9}, {3.6562725330425463`*^9, 3.656272591351241*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Integrand of eq.(3.14): 1/P * (Plethystic exponential), at given order in \ mass dimension\ \>", "Subtitle", CellChangeTimes->{{3.6562726020602617`*^9, 3.656272607618363*^9}, { 3.6562733370941763`*^9, 3.656273364723043*^9}, {3.658764612574771*^9, 3.658764614422879*^9}}], Cell["Input mass dimension order", "Text", CellChangeTimes->{{3.6562726378308153`*^9, 3.65627267075735*^9}, { 3.656272826799415*^9, 3.656272827277956*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"integrand", "[", "massord_", "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ "arg", ",", "scal", ",", "integrand", ",", "i", ",", "Pinvineps", ",", "expineps"}], "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"argument", " ", "of", " ", "the", " ", "PE"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"arg", "=", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "r"], RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["H", "r"], " ", RowBox[{"\[Chi]H", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["Hd", "r"], " ", RowBox[{"\[Chi]Hd", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["Bl", "r"], " ", RowBox[{"\[Chi]Bl", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["Br", "r"], " ", RowBox[{"\[Chi]Br", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", "\[IndentingNewLine]", RowBox[{ SuperscriptBox["Wl", "r"], " ", RowBox[{"\[Chi]Wl", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["Wr", "r"], RowBox[{"\[Chi]Wr", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["Gl", "r"], " ", RowBox[{"\[Chi]Gl", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["Gr", "r"], " ", RowBox[{"\[Chi]Gr", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}]}], ")"}]}], "+", RowBox[{ FractionBox["1", "r"], SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], RowBox[{"r", "+", "1"}]], RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["Q", "r"], " ", RowBox[{"\[Chi]Q", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["Qd", "r"], " ", RowBox[{"\[Chi]Qd", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["u", "r"], " ", RowBox[{"\[Chi]u", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["ud", "r"], " ", RowBox[{"\[Chi]ud", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["d", "r"], " ", RowBox[{"\[Chi]d", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["dd", "r"], " ", RowBox[{"\[Chi]dd", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["L", "r"], " ", RowBox[{"\[Chi]L", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["Ld", "r"], " ", RowBox[{"\[Chi]Ld", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["e", "r"], " ", RowBox[{"\[Chi]e", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}], "+", RowBox[{ SuperscriptBox["ed", "r"], " ", RowBox[{"\[Chi]ed", "[", RowBox[{ SuperscriptBox["t", "r"], ",", SuperscriptBox["\[Alpha]", "r"], ",", SuperscriptBox["\[Beta]", "r"], ",", SuperscriptBox["x", "r"], ",", SuperscriptBox["y", "r"], ",", SuperscriptBox["z1", "r"], ",", SuperscriptBox["z2", "r"]}], "]"}]}]}], ")"}]}]}], ",", RowBox[{"{", RowBox[{"r", ",", "1", ",", "massord"}], "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{ "scale", " ", "according", " ", "to", " ", "mass", " ", "dimension"}], "*)"}], "\[IndentingNewLine]", RowBox[{"scal", "=", RowBox[{"{", RowBox[{ RowBox[{"t", "\[Rule]", RowBox[{"\[Epsilon]", " ", "t"}]}], ",", RowBox[{"H", "->", RowBox[{"\[Epsilon]", " ", "H"}]}], ",", RowBox[{"Hd", "->", RowBox[{"\[Epsilon]", " ", "Hd"}]}], ",", RowBox[{"Q", "->", RowBox[{ SuperscriptBox["\[Epsilon]", RowBox[{"3", "/", "2"}]], "Q"}]}], ",", RowBox[{"Qd", "->", RowBox[{ SuperscriptBox["\[Epsilon]", RowBox[{"3", "/", "2"}]], "Qd"}]}], ",", RowBox[{"u", "->", RowBox[{ SuperscriptBox["\[Epsilon]", RowBox[{"3", "/", "2"}]], "u"}]}], ",", RowBox[{"ud", "->", RowBox[{ SuperscriptBox["\[Epsilon]", RowBox[{"3", "/", "2"}]], "ud"}]}], ",", RowBox[{"d", "->", RowBox[{ SuperscriptBox["\[Epsilon]", RowBox[{"3", "/", "2"}]], "d"}]}], ",", RowBox[{"dd", "->", RowBox[{ SuperscriptBox["\[Epsilon]", RowBox[{"3", "/", "2"}]], "dd"}]}], ",", RowBox[{"L", "->", RowBox[{ SuperscriptBox["\[Epsilon]", RowBox[{"3", "/", "2"}]], "L"}]}], ",", RowBox[{"Ld", "->", RowBox[{ SuperscriptBox["\[Epsilon]", RowBox[{"3", "/", "2"}]], "Ld"}]}], ",", RowBox[{"e", "->", RowBox[{ SuperscriptBox["\[Epsilon]", RowBox[{"3", "/", "2"}]], "e"}]}], ",", RowBox[{"ed", "->", RowBox[{ SuperscriptBox["\[Epsilon]", RowBox[{"3", "/", "2"}]], "ed"}]}], ",", RowBox[{"Bl", "->", RowBox[{ SuperscriptBox["\[Epsilon]", "2"], "Bl"}]}], ",", RowBox[{"Br", "->", RowBox[{ SuperscriptBox["\[Epsilon]", "2"], "Br"}]}], ",", RowBox[{"Wl", "->", RowBox[{ SuperscriptBox["\[Epsilon]", "2"], "Wl"}]}], ",", RowBox[{"Wr", "->", RowBox[{ SuperscriptBox["\[Epsilon]", "2"], "Wr"}]}], ",", RowBox[{"Gl", "->", RowBox[{ SuperscriptBox["\[Epsilon]", "2"], "Gl"}]}], ",", RowBox[{"Gr", "->", RowBox[{ SuperscriptBox["\[Epsilon]", "2"], "Gr"}]}]}], "}"}]}], ";", "\[IndentingNewLine]", RowBox[{"arg", "=", RowBox[{"arg", "/.", "scal"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{ "take", " ", "coefficient", " ", "at", " ", "order", " ", "massord"}], "*)"}], "\[IndentingNewLine]", RowBox[{"integrand", "=", RowBox[{"SeriesCoefficient", "[", RowBox[{ RowBox[{ FractionBox["1", RowBox[{"P", "[", RowBox[{ RowBox[{"\[Epsilon]", " ", "*", " ", "t"}], ",", "\[Alpha]", ",", "\[Beta]"}], "]"}]], "*", RowBox[{"Exp", "[", "arg", "]"}]}], ",", RowBox[{"{", RowBox[{"\[Epsilon]", ",", "0", ",", "massord"}], "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"integrand", "//", "Expand"}]}]}], "\[IndentingNewLine]", "]"}]}]], "Input", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmViYGAQA2IQ3SJwIOGk7WvH7um7ikD0oh3LG0C0XHvFUhD9TSJ/ FYiu+WO8AUT/WOzDfQpIFySWg+mDT7xlQfRZ6y/yIPrV9BhrED1BtN8WROtz dDuD6AgpuXAQvdnkbiyIbjm2OxFE252PzAfRDWazCkC0xwGvShB93TGpFkS/ abZsBtFM+sktINqslKMHrK7CDkxH2BZPAtGXPLvAtM6cc9NB9Brbplkg+rMs yykQ/alw7xkQrf7dM+EMkO6RLgTTv8Lk5oDoa+kr5oNomY0TQ8+C/Ln/Gph2 ydicDaJnGXjmgOjH3ZFFIFprdU4piJ5TY14FopUzC8H0OucWlRN2rx25fC1V QTQAdRe0KA== "]] }, Open ]], Cell[CellGroupData[{ Cell["Obtain Hilbert series: performing the Haar integrals", "Subtitle", CellChangeTimes->{{3.6562726020602617`*^9, 3.656272607618363*^9}, { 3.6562733370941763`*^9, 3.656273409770512*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"getHS", "[", "massord_", "]"}], ":=", RowBox[{"DynamicModule", "[", RowBox[{ RowBox[{"{", RowBox[{"int", ",", "i", ",", "res", ",", "tot", ",", "alltoone"}], "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{"massord", "<", "5"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ "Print", "[", "\"\5\>\"", "]"}], ";", "\[IndentingNewLine]", RowBox[{ "tot", "=", "\"\