(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 187662, 4992]*) (*NotebookOutlinePosition[ 216349, 5910]*) (* CellTagsIndexPosition[ 216305, 5906]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Recoil-Polarization Response Functions for Electroproduction of Pseudoscalar \ Mesons\ \>", "Title", TextAlignment->Center, TextJustification->0], Cell["James J. Kelly", "Author", TextAlignment->Center, TextJustification->0], Cell["\<\ Department of Physics University of Maryland College Park, MD 20742 jjkelly@physics.umd.edu\ \>", "Address", TextAlignment->Center, TextJustification->0], Cell["\<\ Recoil-polarization response functions for electroproduction of pseudoscalar \ mesons are derived in terms of CGLN and helicity amplitudes. Standard \ multipole expansions of the CGLN amplitudes are also used to express the \ response functions in terms of multipoles. These expansions are compared \ with those published by other authors.\ \>", "Abstract"], Cell[TextData[StyleBox["Created: July 31, 1998\nLast revised: October 12, \ 2000", FontSlant->"Italic"]], "Text", TextAlignment->Left, TextJustification->0], Cell[CellGroupData[{ Cell["Introduction", "SectionFirst"], Cell[TextData[{ "In this notebook I derive the recoil-polarization response functions for \ electroproduction of pseudoscalar mesons from nucleons. Although I and my \ former student, Thomas Payerle, have produced copious notes about the \ formalism, the algebra is quite complex and there are many sections of \ Payerle's work which I had not verified previously. Furthermore, his work \ was limited to recoil polarization and I would like to extend our code, ", StyleBox["epiprod", FontSlant->"Italic"], ", to include target polarization also. Finally, there appear to be \ discrepancies between the multipole expansions published by the Mainz group, \ namely Drechsel and Tiator, and those published by the MIT group, Raskin and \ Donnelly, and used by Lourie in several papers and proposals. Therefore, in \ this notebook I develop an independent derivation of the response functions \ in terms of both CGLN and helicity amplitudes. I also develop tools to \ expand these functions in terms of multipoles subject to specified \ constraints on maximum angular momentum and/or dominance of particular \ multipoles. For the most part I will try to keep the conventions and \ notation as close to Payerle's as possible so that his work and the \ expressions coded in ", StyleBox["epiprod", FontSlant->"Italic"], " can be checked easily. Some effort is made to simplify important \ expressions, but it is often difficult to compel ", StyleBox["Mathematica", FontSlant->"Italic"], " to format expressions in the same way one does with more traditional \ derivations." }], "Text"], Cell[CellGroupData[{ Cell["References", "Subsubsection"], Cell[TextData[{ "A.S. Raskin and T. W. Donnelly, ", StyleBox["Polarization in Coincidence Electron Scattering from Nuclei", FontSlant->"Italic"], ", Ann. Phys. (NY) ", StyleBox["191", FontWeight->"Bold"], ", 78 (1989)." }], "Text"], Cell[TextData[{ "R.W. Lourie ", StyleBox["et al.", FontSlant->"Italic"], ", ", StyleBox["Investigation of the ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`N \[Rule] \[CapitalDelta]\)]], " ", StyleBox["Transition via Polarization Observables in Hall A", FontSlant->"Italic"], ", TJNAF proposal 91-011." }], "Text"], Cell[TextData[{ "R.W. Lourie, ", StyleBox[ "Recoil Polarization Observables in Coincident Pion Electroproduction", FontSlant->"Italic"], ", Nucl. Phys. ", StyleBox["A509", FontWeight->"Bold"], ", 653 (1990)." }], "Text"], Cell[TextData[{ "R. W. Lourie, ", StyleBox[ "Quark Models and Polarized Electroproduction of the Roper Resonance", FontSlant->"Italic"], ", Z. Phys. C ", StyleBox["50", FontWeight->"Bold"], ", 345 (1991)." }], "Text"], Cell[TextData[{ "D. Drechsel and L. Tiator, ", StyleBox["Threshold Pion Photoproduction on Nucleons", FontSlant->"Italic"], ", J. Phys. G ", StyleBox["18", FontWeight->"Bold"], ", 449 (1992)." }], "Text"], Cell[TextData[{ "T.M. Payerle, ", StyleBox["User Manual for ", FontSlant->"Italic"], StyleBox["epiprod", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" Version 2.3.6", FontSlant->"Italic"], ", (UMd, 1993)", StyleBox[".", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ "P. Dennery, ", StyleBox["Theory of Electro- and Photoproduction of \[Pi] Mesons", FontSlant->"Italic"], ", Phys. Rev. ", StyleBox["124", FontWeight->"Bold"], ", 2000 (1961)." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Initialization", "Section"], Cell[CellGroupData[{ Cell["Defaults and packages", "Subsection"], Cell[BoxData[{ \(\(ClearAll["\"];\)\), "\n", \(\(Off[General::spell, \ General::spell1];\)\), "\n", \(\($TextStyle = {FontFamily \[Rule] "\", FontSize \[Rule] 12};\)\)}], "Input", CellLabel->"In[1]:="], Cell[BoxData[ \(Needs["\"]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[{ RowBox[{ RowBox[{"Symbolize", "[", TagBox[\(M\_+\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";", RowBox[{"Symbolize", "[", TagBox[\(M\_-\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Symbolize", "[", TagBox[\(E\_+\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";", RowBox[{"Symbolize", "[", TagBox[\(E\_-\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Symbolize", "[", TagBox[\(S\_+\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";", RowBox[{"Symbolize", "[", TagBox[\(S\_-\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";"}]}], "Input", CellLabel->"In[5]:="] }, Open ]], Cell[CellGroupData[{ Cell["Manipulation of complex quantities", "Subsection"], Cell["\<\ Throughout this notebook I assume that the only complex quantities are the \ electroproduction amplitudes themselves. Thus, it is useful to define rules \ and functions which simplify expressions in which the complex quantities are \ known to appear in specific patterns.\ \>", "Text"], Cell[BoxData[{ \(\(\(conjugate::usage\ = \n\t"\";\)\(\n\) \)\), "\n", \(\(conjugateRule\ = \ Complex[re_, im_] :> Complex[re, \(-im\)];\)\), "\n", \(\(conjugate[exp__]\ := \ exp /. conjugateRule;\)\)}], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \(HermitianConjugate[A_] := Transpose[conjugate[A]]\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(\(trigToExp[\[Phi]_] = {Cos[\[Phi]] \[RuleDelayed] \ \(\[ExponentialE]\^\(I\ \[Phi]\) + \[ExponentialE]\^\(\(-I\)\ \[Phi]\)\)\/2, Sin[\[Phi]] \[RuleDelayed] \(\[ExponentialE]\^\(I\ \[Phi]\) - \ \[ExponentialE]\^\(\(-I\)\ \[Phi]\)\)\/\(2 I\)};\)\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(\(expToTrig[\[Phi]_] = {E\^\(Complex[0, a_]\ \[Phi]\) \[Rule] Cos[a\ \[Phi]] + I\ Sin[a\ \[Phi]]};\)\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ \(\(\(ContractAmplitudeProducts = \n\t{\n\t c_. \(A_\_a_\) \(\((A_\_a_)\)\^*\) \[RuleDelayed] c\ Abs[A\_a]\^2, \n\t d_. + c_. \(A_\_a_\) \(\((B_\_b_)\)\^*\) \[RuleDelayed] d + c\ \((Re[\(A\_a\) \(\((B\_b)\)\^*\)] + I\ Im[\(A\_a\) \(\((B\_b)\)\^*\)])\) /; b > a, \n\t d_. + c_. \ \(A_\_a_\) \(\((B_\_b_)\)\^*\) \[RuleDelayed] d + c\ \((Re[\(B\_b\) \(\((A\_a)\)\^*\)] - I\ Im[\(B\_b\) \(\((A\_a)\)\^*\)])\) /; b < a, \n\t d_. \ Re[\(A_\_a_\) \(\((B_\_b_)\)\^*\)] + \ e_. Re[\(C_\_c_\) \(\((B_\_b_)\)\^*\)] \[RuleDelayed] d\ Re[\((A\_a + e\/d\ C\_c)\) \(\((B\_b)\)\^*\)], \n\t d_. \ Im[\(A_\_a_\) \(\((B_\_b_)\)\^*\)] + \ e_. Im[\(C_\_c_\) \(\((B_\_b_)\)\^*\)] \[RuleDelayed] d\ Im[\((A\_a + e\/d\ C\_c)\) \(\((B\_b)\)\^*\)]\n\t};\)\(\t\)\)\)], \ "Input", CellLabel->"In[14]:="], Cell[BoxData[{ \(MyRules = {\(0\^*\) \[Rule] 0}; MyAssumptions = {\[Alpha] \[Element] Reals, \[Beta] \[Element] Reals, \[Gamma] \[Element] Reals, \[Delta] \[Element] Reals, \[Theta] \[Element] Reals, \[Phi] \[Element] Reals, \[Omega] > 0, q > \[Omega], Q > 0, 0 < x < 1};\), "\[IndentingNewLine]", \(\(MySimplify = Simplify[# /. MyRules, MyAssumptions] &;\)\), "\[IndentingNewLine]", \(\(MyFullSimplify = FullSimplify[# /. MyRules, MyAssumptions] &;\)\)}], "Input", CellLabel->"In[15]:="] }, Open ]], Cell[CellGroupData[{ Cell["Pauli matrices", "Subsection"], Cell[BoxData[ \(\(\[Sigma]\&\[RightVector] = {{{0, 1}, {1, 0}}, \ {{0, \(-I\)}, {I, 0}}, {{1, 0}, {0, \(-1\)}}};\)\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \(PauliAmp[A : {{_, _}, {_, _}}, 0] := Module[{j}, \(1\/2\) Sum[A\[LeftDoubleBracket]j, j\[RightDoubleBracket], {j, 1, 2}]]; PauliAmp[A : {{_, _}, {_, _}}, i_] := Module[{j}, \(1\/2\) Sum[\((\[Sigma]\&\[RightVector]\[LeftDoubleBracket] i\[RightDoubleBracket] . A)\)\[LeftDoubleBracket]j, j\[RightDoubleBracket], {j, 1, 2}]] /; i > 0\)], "Input", CellLabel->"In[19]:="] }, Open ]], Cell[CellGroupData[{ Cell["Basis vectors", "Subsection"], Cell[TextData[ "The polar and azimuthal angles, \[Theta] and \[Phi], describe the pion cm \ angle relative to the momentum transfer vector and the scattering plane. "], "Text"], Cell[BoxData[{ \(\(p\&^\_\[Pi] = {Sin[\[Theta]] Cos[\[Phi]], Sin[\[Theta]] Sin[\[Phi]], Cos[\[Theta]]};\)\), "\n", \(\(q\&^ = {0, 0, 1};\)\)}], "Input", CellLabel->"In[20]:="], Cell["\<\ It is useful to formulate a very general basis for polarization vectors in \ terms of Euler angles.\ \>", "Text"], Cell[BoxData[{ \(\(rotz[\[Theta]_] := {{Cos[\[Theta]], Sin[\[Theta]], 0}, {\(-Sin[\[Theta]]\), Cos[\[Theta]], 0}, {0, 0, 1}};\)\), "\n", \(\(rotx[\[Theta]_] := {{1, 0, 0}, {0, Cos[\[Theta]], Sin[\[Theta]]}, {0, \(-Sin[\[Theta]]\), Cos[\[Theta]]}};\)\), "\n", \(\(roty[\[Theta]_] := {{Cos[\[Theta]], 0, \(-Sin[\[Theta]]\)}, {0, 1, 0}, {Sin[\[Theta]], 0, Cos[\[Theta]]}};\)\), "\n", \(\(euler[\[Alpha]_, \[Beta]_, \[Gamma]_] := rotz[\[Gamma]] . roty[\[Beta]] . rotz[\[Alpha]];\)\)}], "Input", CellLabel->"In[22]:="], Cell[TextData[{ "The most useful basis for recoil polarization is normally the ", StyleBox["ejectile basis", FontSlant->"Italic"], " defined with ", Cell[BoxData[ \(TraditionalForm\`L\&^\)]], " along the nucleon recoil momentum, ", Cell[BoxData[ \(TraditionalForm \`N\&^ = \ \(q\&^\[CircleTimes]L\&^ \)\/\(\(\[VerticalSeparator] q\)\&^\[CircleTimes]L\&^ \[VerticalSeparator] \)\)]], " normal to the reaction plane, and ", Cell[BoxData[ \(TraditionalForm\`S\&^ = N\&^\[CircleTimes]L\&^\)]], " within the reaction plane. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(EjectileBasis\ = \ Thread[{S\&^, N\&^, L\&^} \[Rule] Transpose[euler[0, \[Pi] + \[Theta], \[Pi] - \[Phi]]]]\)], "Input", CellLabel->"In[26]:="], Cell[BoxData[ \({\(S\&^\) \[Rule] {Cos[\[Theta]]\ Cos[\[Phi]], Cos[\[Theta]]\ Sin[\[Phi]], \(-Sin[\[Theta]]\)}, \(N\&^\) \[Rule] \ {Sin[\[Phi]], \(-Cos[\[Phi]]\), 0}, \(L\&^\) \[Rule] {\(-Cos[\[Phi]]\)\ Sin[\[Theta]], \(-Sin[\ \[Theta]]\)\ Sin[\[Phi]], \(-Cos[\[Theta]]\)}}\)], "Output", CellLabel->"Out[26]="] }, Open ]], Cell[TextData[{ "Similarly, the most useful basis for target polarization is the ", StyleBox["target basis", FontSlant->"Italic"], " defined with ", Cell[BoxData[ \(TraditionalForm\`L\&^ = q\&^\)]], " along the momentum transfer, ", Cell[BoxData[ \(TraditionalForm \`N\&^ = \ \(q\&^\[CircleTimes]p\&^ \)\/\(\(\[VerticalSeparator] q\)\&^\[CircleTimes]p\&^ \[VerticalSeparator] \)\)]], " normal to the reaction plane, and ", Cell[BoxData[ \(TraditionalForm\`S\&^ = N\&^\[CircleTimes]L\&^\)]], " within the reaction plane. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TargetBasis\ = \ Thread[{S\&^, N\&^, L\&^} \[Rule] Transpose[euler[0, 0, \[Phi]]]]\)], "Input", CellLabel->"In[27]:="], Cell[BoxData[ \({\(S\&^\) \[Rule] {Cos[\[Phi]], \(-Sin[\[Phi]]\), 0}, \(N\&^\) \[Rule] {Sin[\[Phi]], Cos[\[Phi]], 0}, \(L\&^\) \[Rule] {0, 0, 1}}\)], "Output", CellLabel->"Out[27]="] }, Open ]], Cell["\<\ Polarizations are determined here in the barycentric frame and can be \ transformed to the lab frame using a Wigner rotation later.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Helicity state vectors", "Subsection"], Cell["\<\ The following notation for the phases is based upon Payerle's notes, although \ permitting 4 independent phases is clearly extravagant. Most results will be \ based upon the Jacob and Wick (JW) conventions for these phases.\ \>", "Text"], Cell[BoxData[{ \(\(\[Chi]\_\(i, 1\) = \(\[ExponentialE]\^\(I\ \[Gamma]\)\) {0, \ \(-1\)};\)\), "\n", \(\(\[Chi]\_\(i, 2\) = \(\[ExponentialE]\^\(I\ \((\[Gamma] + \ \[Delta])\)\)\) {1, 0};\)\), "\n", \(\(\[Chi]\_i = Transpose[{\[Chi]\_\(i, 1\), \[Chi]\_\(i, 2\)}];\)\)}], "Input", CellLabel->"In[28]:="], Cell[BoxData[{ \(\(\[Chi]\_\(f, 1\) = \(\[ExponentialE]\^\(I\ \[Alpha]\)\) {Sin[\[Theta]\ \/2] \[ExponentialE]\^\(\(-I\)\ \[Phi]\/2\), \(-Cos[\[Theta]\/2]\) \ \[ExponentialE]\^\(I\ \[Phi]\/2\)};\)\), "\n", \(\(\[Chi]\_\(f, 2\) = \(\[ExponentialE]\^\(I \((\ \[Alpha] + \ \[Beta])\)\)\) {Cos[\[Theta]\/2] \[ExponentialE]\^\(\(-I\)\ \[Phi]\/2\), Sin[\[Theta]\/2] \[ExponentialE]\^\(I\ \[Phi]\/2\)};\)\), "\n", \(\(\[Chi]\_f = Transpose[{\[Chi]\_\(f, 1\), \[Chi]\_\(f, 2\)}];\)\)}], "Input", CellLabel->"In[31]:="], Cell[BoxData[ \(\(phases[ JW]\ = \ {\[Alpha] \[Rule] \[Pi] - \[Phi]\/2, \[Beta] \[Rule] \ \[Pi] + \[Phi], \[Gamma] \[Rule] \[Pi], \[Delta] \[Rule] \[Pi]};\)\)], "Input",\ CellLabel->"In[34]:="] }, Open ]], Cell[CellGroupData[{ Cell["Virtual photon polarization vectors", "Subsection"], Cell[BoxData[ \(\(a\&\[RightVector] = {a\_x, a\_y, a\_z};\)\)], "Input", CellLabel->"In[35]:="], Cell[BoxData[{ \(\(rule[a\_0] = {a\_x \[Rule] 0, a\_y \[Rule] 0, a\_z \[Rule] \(-\(Q\/\[Omega]\)\)};\)\), "\n", \(\(rule[a\_1] = {a\_x \[Rule] \(-\ 1\)\/\@2, a\_y \[Rule] \(-\ I\)\/\@2, a\_z \[Rule] 0};\)\), "\n", \(\(rule[a\_\(-1\)] = {a\_x \[Rule] 1\/\@2, a\_y \[Rule] \(-\ I\)\/\@2, a\_z \[Rule] 0};\)\)}], "Input", CellLabel->"In[36]:="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Current operator in helicity representation", "Section"], Cell[CellGroupData[{ Cell["General form", "Subsection"], Cell["\<\ The most general form of the current for pion electroproduction operator has \ been given by CGLN as follows. \ \>", "Text"], Cell[BoxData[ \(J\&\[RightVector] = \[ImaginaryI]\ \[Sigma]\&\[RightVector]\ F\_1 + \(F\ \_2\) \((p\&^\_\[Pi] . \[Sigma]\&\[RightVector])\) \[Sigma]\&\[RightVector]\ \[CircleTimes]q\&^ + \(p\&^\_\[Pi]\) \((\[ImaginaryI] q\&^\ . \[Sigma]\&\[RightVector] F\_3)\) + \(p\&^\_\[Pi]\) \((\[ImaginaryI] p\&^\_\[Pi] . \[Sigma]\&\[RightVector] F\_4)\) + \(q\&^\) \((\[ImaginaryI] q\&^\ . \[Sigma]\&\[RightVector]\ F\_5)\) + \(q\&^\) \((\ \[ImaginaryI] p\&^\_\[Pi] . \[Sigma]\&\[RightVector] F\_6)\)\)], "DisplayFormula", Evaluatable->False, FontFamily->"Times New Roman"], Cell[TextData[{ "In order to formulate this expression properly using ", StyleBox["Mathematica", FontSlant->"Italic"], ", it is useful to employ a generic vector ", Cell[BoxData[ \(TraditionalForm\`s\&\[RightVector]\)]], " in place of ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\&\[RightVector]\)]], " first and to substitute after the vector operations have been evaluated. \ [Otherwise, the ", Cell[BoxData[ \(TraditionalForm\`F\_2\)]], " term in particular becomes troublesome.] The form below is expressed in \ the spin basis." }], "Text"], Cell[BoxData[ \(\(s\&\[RightVector] = {s\_x, s\_y, s\_z};\)\)], "Input", CellLabel->"In[39]:="], Cell[CellGroupData[{ Cell[BoxData[ \(T[spin]\ = \ \(\(-I\) \((I\ s\&\[RightVector] . a\&\[RightVector]\ \ F\_1 + \(F\_2\) \((p\&^\_\[Pi] . s\&\[RightVector])\) . \((a\&\[RightVector]\ . Cross[s\&\[RightVector], q\&^])\) + p\&^\_\[Pi] . a\&\[RightVector]\ \ \((I q\&^\ . s\&\[RightVector] F\_3)\) + p\&^\_\[Pi] . a\&\[RightVector]\ \ \((I p\&^\_\[Pi] . s\&\[RightVector] F\_4)\) + q\&^ . a\&\[RightVector]\ \ \((I q\&^\ . s\&\[RightVector]\ F\_5)\) + q\&^ . a\&\[RightVector]\ \ \((I p\&^\_\[Pi] . s\&\[RightVector] F\_6)\))\) /. {s\_x \[Rule] \[Sigma]\&\[RightVector]\ \[LeftDoubleBracket]1\[RightDoubleBracket], s\_y \[Rule] \[Sigma]\&\[RightVector]\[LeftDoubleBracket]2\ \[RightDoubleBracket], s\_z \[Rule] \[Sigma]\&\[RightVector]\[LeftDoubleBracket]3\ \[RightDoubleBracket]}\) /. trigToExp[\[Phi]] // MySimplify\)], "Input", CellLabel->"In[40]:="], Cell[BoxData[ \({{1\/2\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Phi]\)\ \ \((Sin[\[Theta]]\ \((\[ImaginaryI]\ a\_y\ \((2\ F\_2 - \((\(-1\) + \ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ \((F\_3 + Cos[\[Theta]]\ F\_4)\))\) + a\_x\ \((2\ F\_2 + \((1 + \[ExponentialE]\^\(2\ \ \[ImaginaryI]\ \[Phi]\))\)\ \((F\_3 + Cos[\[Theta]]\ F\_4)\))\))\) + 2\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ a\_z\ \((F\_1 + Cos[\[Theta]]\ F\_3 + Cos[\[Theta]]\^2\ F\_4 + F\_5 + Cos[\[Theta]]\ F\_6)\))\), 1\/2\ \[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ \[Phi]\)\ \((\(-\ \[ImaginaryI]\)\ a\_y\ \((2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ F\ \_1 - 2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ Cos[\[Theta]]\ F\_2 + \ \((\(-1\) + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ \ Sin[\[Theta]]\^2\ F\_4)\) + a\_x\ \((2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ F\_1 \ - 2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ Cos[\[Theta]]\ F\_2 + \ \((1 + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ Sin[\[Theta]]\^2\ \ F\_4)\) + 2\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ Sin[\[Theta]]\ a\ \_z\ \((Cos[\[Theta]]\ F\_4 + F\_6)\))\)}, {1\/2\ \((\[ImaginaryI]\ a\_y\ \((2\ F\_1 - 2\ Cos[\[Theta]]\ F\_2 - \((\(-1\) + \[ExponentialE]\^\(2\ \ \[ImaginaryI]\ \[Phi]\))\)\ Sin[\[Theta]]\^2\ F\_4)\) + a\_x\ \((2\ F\_1 - 2\ Cos[\[Theta]]\ F\_2 + \((1 + \[ExponentialE]\^\(2\ \ \[ImaginaryI]\ \[Phi]\))\)\ Sin[\[Theta]]\^2\ F\_4)\) + 2\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ Sin[\[Theta]]\ a\ \_z\ \((Cos[\[Theta]]\ F\_4 + F\_6)\))\), 1\/2\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Phi]\)\ \((\(-Sin[\ \[Theta]]\)\ \((\(-\[ImaginaryI]\)\ a\_y\ \((2\ \[ExponentialE]\^\(2\ \ \[ImaginaryI]\ \[Phi]\)\ F\_2 + \((\(-1\) + \[ExponentialE]\^\(2\ \ \[ImaginaryI]\ \[Phi]\))\)\ \((F\_3 + Cos[\[Theta]]\ F\_4)\))\) + a\_x\ \((2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ \ F\_2 + \((1 + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ \((F\_3 + Cos[\[Theta]]\ F\_4)\))\))\) - 2\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ a\_z\ \((F\_1 + Cos[\[Theta]]\ F\_3 + Cos[\[Theta]]\^2\ F\_4 + F\_5 + Cos[\[Theta]]\ F\_6)\))\)}}\)], "Output", CellLabel->"Out[40]="] }, Open ]], Cell["\<\ The transition operator can now be transformed into the helicity basis.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(T[helicity] = HermitianConjugate[\[Chi]\_f] . T[spin] . \[Chi]\_i // MySimplify\)], "Input", CellLabel->"In[41]:="], Cell[BoxData[ \({{1\/4\ \[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \[Alpha] \ - 2\ \[Gamma] + 3\ \[Phi])\)\)\ \((\(-2\)\ Sin[\[Theta]\/2]\ a\_x\ \((2\ \ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ F\_1 + 2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ F\_2 + \ \((1 + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ \((1 + Cos[\[Theta]])\)\ \((F\_3 + F\_4)\))\) + \[ImaginaryI]\ \((2\ Sin[\[Theta]\/2]\ \ a\_y\ \((2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ F\_1 + 2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ F\ \_2 + \((\(-1\) + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ \((1 + Cos[\[Theta]])\)\ \((F\_3 + F\_4)\))\) + 4\ \[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ \ \[Phi]\)\ Cos[\[Theta]\/2]\ a\_z\ \((F\_1 + Cos[\[Theta]]\ F\_3 + Cos[\[Theta]]\ F\_4 + F\_5 + F\_6)\))\))\), 1\/2\ \[ExponentialE]\^\(\[ImaginaryI]\ \((\[Gamma] + \[Delta])\) - 1\ \/2\ \[ImaginaryI]\ \((2\ \[Alpha] + \[Phi])\)\)\ \((\(-Cos[\[Theta]\/2]\)\ \ \((\[ImaginaryI]\ a\_y\ \((2\ F\_1 - 2\ Cos[\[Theta]]\ F\_2 - \((\(-1\) + \ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ Sin[\[Theta]]\^2\ F\_4)\) + a\_x\ \((2\ F\_1 - 2\ Cos[\[Theta]]\ F\_2 + \((1 + \ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ Sin[\[Theta]]\^2\ F\_4)\) + 2\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ Sin[\ \[Theta]]\ a\_z\ \((Cos[\[Theta]]\ F\_4 + F\_6)\))\) + Sin[\[Theta]\/2]\ \((Sin[\[Theta]]\ \((\[ImaginaryI]\ a\_y\ \ \((2\ F\_2 - \((\(-1\) + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ \ \((F\_3 + Cos[\[Theta]]\ F\_4)\))\) + a\_x\ \((2\ F\_2 + \((1 + \[ExponentialE]\^\(2\ \ \[ImaginaryI]\ \[Phi]\))\)\ \((F\_3 + Cos[\[Theta]]\ F\_4)\))\))\) + 2\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ a\_z\ \((F\ \_1 + Cos[\[Theta]]\ F\_3 + Cos[\[Theta]]\^2\ F\_4 + F\_5 + Cos[\[Theta]]\ F\_6)\))\))\)}, {1\/4\ \ \[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \[Alpha] + 2\ \[Beta] \ - 2\ \[Gamma] + 3\ \[Phi])\)\)\ \((\(-2\)\ Cos[\[Theta]\/2]\ a\_x\ \((2\ \ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ F\_1 - 2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ F\_2 + \ \((1 + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ \((\(-1\) + Cos[\[Theta]])\)\ \((F\_3 - F\_4)\))\) + \[ImaginaryI]\ \((2\ Cos[\[Theta]\/2]\ \ a\_y\ \((2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ F\_1 - 2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ F\ \_2 + \((\(-1\) + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ \((\(-1\) \ + Cos[\[Theta]])\)\ \((F\_3 - F\_4)\))\) - 4\ \[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ \ \[Phi]\)\ Sin[\[Theta]\/2]\ a\_z\ \((F\_1 + Cos[\[Theta]]\ F\_3 - Cos[\[Theta]]\ F\_4 + F\_5 - F\_6)\))\))\), 1\/4\ \[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \[Alpha] \ + 2\ \[Beta] - 2\ \[Gamma] - 2\ \[Delta] + \[Phi])\)\)\ \((2\ \[ImaginaryI]\ \ Sin[\[Theta]\/2]\ a\_y\ \((2\ F\_1 + 2\ F\_2 - \((\(-1\) + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \ \[Phi]\))\)\ \((1 + Cos[\[Theta]])\)\ \((F\_3 + F\_4)\))\) + 2\ Sin[\[Theta]\/2]\ a\_x\ \((2\ F\_1 + 2\ F\_2 + \((1 + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \ \[Phi]\))\)\ \((1 + Cos[\[Theta]])\)\ \((F\_3 + F\_4)\))\) + 4\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ Cos[\[Theta]\/2]\ \ a\_z\ \((F\_1 + Cos[\[Theta]]\ F\_3 + Cos[\[Theta]]\ F\_4 + F\_5 + F\_6)\))\)}}\)], "Output", CellLabel->"Out[41]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Linear combination of CGLN amplitudes", "Subsection"], Cell[BoxData[ \(\(Fto\[ScriptCapitalF] = {F\_1 \[Rule] \[ScriptCapitalF]\_1, F\_2 \[Rule] \[ScriptCapitalF]\_2, F\_3 \[Rule] \[ScriptCapitalF]\_3, F\_4 \[Rule] \[ScriptCapitalF]\_4, F\_5 \[Rule] \[ScriptCapitalF]\_5 - \[ScriptCapitalF]\_1 - Cos[\[Theta]] \[ScriptCapitalF]\_3, F\_6 \[Rule] \[ScriptCapitalF]\_6 - \(\[ScriptCapitalF]\_4\) Cos[\[Theta]]};\)\)], "Input", CellLabel->"In[42]:="], Cell[BoxData[ \(\(\[ScriptCapitalF]toF = {\[ScriptCapitalF]\_1 \[Rule] F\_1, \[ScriptCapitalF]\_2 \[Rule] F\_2, \[ScriptCapitalF]\_3 \[Rule] F\_3, \[ScriptCapitalF]\_4 \[Rule] F\_4, \[ScriptCapitalF]\_5 \[Rule] F\_1 + Cos[\[Theta]] F\_3 + F\_5, \[ScriptCapitalF]\_6 \[Rule] \(F\_4\) Cos[\[Theta]]\ + \ F\_6};\)\)], "Input", CellLabel->"In[43]:="], Cell[CellGroupData[{ Cell[BoxData[ \(T[helicity] /. Fto\[ScriptCapitalF] // MySimplify\)], "Input", CellLabel->"In[44]:="], Cell[BoxData[ \({{1\/4\ \[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \[Alpha] \ - 2\ \[Gamma] + 3\ \[Phi])\)\)\ \((\(-2\)\ Sin[\[Theta]\/2]\ a\_x\ \((2\ \ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ \[ScriptCapitalF]\_1 + 2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ \ \[ScriptCapitalF]\_2 + \((1 + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \ \[Phi]\))\)\ \((1 + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 + \ \[ScriptCapitalF]\_4)\))\) + \[ImaginaryI]\ \((2\ Sin[\[Theta]\/2]\ a\_y\ \ \((2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ \[ScriptCapitalF]\_1 + 2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ \ \[ScriptCapitalF]\_2 + \((\(-1\) + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \ \[Phi]\))\)\ \((1 + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 + \ \[ScriptCapitalF]\_4)\))\) + 4\ \[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ \ \[Phi]\)\ Cos[\[Theta]\/2]\ a\_z\ \((\[ScriptCapitalF]\_5 + \[ScriptCapitalF]\ \_6)\))\))\), 1\/4\ \[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \[Alpha] \ - 2\ \[Gamma] - 2\ \[Delta] + \[Phi])\)\)\ \((\(-2\)\ Cos[\[Theta]\/2]\ a\_x\ \ \((2\ \[ScriptCapitalF]\_1 - 2\ \[ScriptCapitalF]\_2 + \((1 + \[ExponentialE]\^\(2\ \ \[ImaginaryI]\ \[Phi]\))\)\ \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \ \[ScriptCapitalF]\_4)\))\) + \[ImaginaryI]\ \((2\ Cos[\[Theta]\/2]\ a\_y\ \((\ \(-2\)\ \[ScriptCapitalF]\_1 + 2\ \[ScriptCapitalF]\_2 + \((\(-1\) + \ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \ \[ScriptCapitalF]\_4)\))\) - 4\ \[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ \ \[Phi]\)\ Sin[\[Theta]\/2]\ a\_z\ \((\[ScriptCapitalF]\_5 - \[ScriptCapitalF]\ \_6)\))\))\)}, {1\/4\ \[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \ \[Alpha] + 2\ \[Beta] - 2\ \[Gamma] + 3\ \[Phi])\)\)\ \((\(-2\)\ Cos[\[Theta]\ \/2]\ a\_x\ \((2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ \ \[ScriptCapitalF]\_1 - 2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ \ \[ScriptCapitalF]\_2 + \((1 + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \ \[Phi]\))\)\ \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \ \[ScriptCapitalF]\_4)\))\) + \[ImaginaryI]\ \((2\ Cos[\[Theta]\/2]\ a\_y\ \ \((2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ \[ScriptCapitalF]\_1 - 2\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ \ \[ScriptCapitalF]\_2 + \((\(-1\) + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \ \[Phi]\))\)\ \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \ \[ScriptCapitalF]\_4)\))\) - 4\ \[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ \ \[Phi]\)\ Sin[\[Theta]\/2]\ a\_z\ \((\[ScriptCapitalF]\_5 - \[ScriptCapitalF]\ \_6)\))\))\), 1\/4\ \[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \[Alpha] \ + 2\ \[Beta] - 2\ \[Gamma] - 2\ \[Delta] + \[Phi])\)\)\ \((2\ \[ImaginaryI]\ \ Sin[\[Theta]\/2]\ a\_y\ \((2\ \[ScriptCapitalF]\_1 + 2\ \[ScriptCapitalF]\_2 - \((\(-1\) + \ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\))\)\ \((1 + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 + \ \[ScriptCapitalF]\_4)\))\) + 2\ Sin[\[Theta]\/2]\ a\_x\ \((2\ \[ScriptCapitalF]\_1 + 2\ \[ScriptCapitalF]\_2 + \((1 + \[ExponentialE]\^\(2\ \ \[ImaginaryI]\ \[Phi]\))\)\ \((1 + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 + \ \[ScriptCapitalF]\_4)\))\) + 4\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ Cos[\[Theta]\/2]\ \ a\_z\ \((\[ScriptCapitalF]\_5 + \[ScriptCapitalF]\_6)\))\)}}\)], "Output", CellLabel->"Out[44]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Transition matrices with general phases", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(T\_0 = \((\(\(T[helicity] /. rule[a\_0]\) /. Fto\[ScriptCapitalF]\) /. trigToExp[\[Phi]])\) // Simplify\)], "Input", CellLabel->"In[45]:="], Cell[BoxData[ \({{\(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \[Alpha] - 2\ \ \[Gamma] + \[Phi])\)\)\ Q\ Cos[\[Theta]\/2]\ \((\[ScriptCapitalF]\_5 + \ \[ScriptCapitalF]\_6)\)\)\/\[Omega], \(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \ \[ImaginaryI]\ \((2\ \[Alpha] - 2\ \[Gamma] - 2\ \[Delta] - \[Phi])\)\)\ Q\ \ Sin[\[Theta]\/2]\ \((\(-\[ScriptCapitalF]\_5\) + \[ScriptCapitalF]\_6)\)\)\/\ \[Omega]}, {\(-\(\(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \ \[Alpha] + 2\ \[Beta] - 2\ \[Gamma] + \[Phi])\)\)\ Q\ Sin[\[Theta]\/2]\ \((\ \[ScriptCapitalF]\_5 - \[ScriptCapitalF]\_6)\)\)\/\[Omega]\)\), \(-\(\(\ \[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \[Alpha] + 2\ \[Beta] \ - 2\ \[Gamma] - 2\ \[Delta] - \[Phi])\)\)\ Q\ Cos[\[Theta]\/2]\ \((\ \[ScriptCapitalF]\_5 + \[ScriptCapitalF]\_6)\)\)\/\[Omega]\)\)}}\)], "Output",\ CellLabel->"Out[45]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(T\_1 = \((\(\(T[helicity] /. rule[a\_1]\) /. Fto\[ScriptCapitalF]\) /. trigToExp[\[Phi]])\) // Simplify\)], "Input", CellLabel->"In[46]:="], Cell[BoxData[ \({{\(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \[Alpha] - 2\ \ \[Gamma] - \[Phi])\)\)\ Sin[\[Theta]\/2]\ \((2\ \[ScriptCapitalF]\_1 + 2\ \ \[ScriptCapitalF]\_2 + \((1 + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 + \ \[ScriptCapitalF]\_4)\))\)\)\/\@2, \(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \ \[ImaginaryI]\ \((2\ \[Alpha] - 2\ \[Gamma] - 2\ \[Delta] - 3\ \[Phi])\)\)\ \ Sin[\[Theta]\/2]\ Sin[\[Theta]]\ \((\(-\[ScriptCapitalF]\_3\) + \ \[ScriptCapitalF]\_4)\)\)\/\@2}, {\(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \ \[ImaginaryI]\ \((2\ \[Alpha] + 2\ \[Beta] - 2\ \[Gamma] - \[Phi])\)\)\ Cos[\ \[Theta]\/2]\ \((2\ \[ScriptCapitalF]\_1 - 2\ \[ScriptCapitalF]\_2 + \ \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \[ScriptCapitalF]\_4)\ \))\)\)\/\@2, \(-\(\(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \ \[Alpha] + 2\ \[Beta] - 2\ \[Gamma] - 2\ \[Delta] - 3\ \[Phi])\)\)\ \((1 + Cos[\[Theta]])\)\ Sin[\[Theta]\/2]\ \ \((\[ScriptCapitalF]\_3 + \[ScriptCapitalF]\_4)\)\)\/\@2\)\)}}\)], "Output", CellLabel->"Out[46]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(T\_\(-1\) = \((\(\(T[helicity] /. rule[a\_\(-1\)]\) /. Fto\[ScriptCapitalF]\) /. trigToExp[\[Phi]])\) // Simplify\)], "Input", CellLabel->"In[47]:="], Cell[BoxData[ \({{\(-\(\(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \ \[Alpha] - 2\ \[Gamma] + 3\ \[Phi])\)\)\ \((1 + Cos[\[Theta]])\)\ Sin[\[Theta]\/2]\ \ \((\[ScriptCapitalF]\_3 + \[ScriptCapitalF]\_4)\)\)\/\@2\)\), \(-\(\(\ \[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \[Alpha] - 2\ \[Gamma] \ - 2\ \[Delta] + \[Phi])\)\)\ Cos[\[Theta]\/2]\ \((2\ \[ScriptCapitalF]\_1 - 2\ \[ScriptCapitalF]\_2 + \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \ \[ScriptCapitalF]\_4)\))\)\)\/\@2\)\)}, \ {\(-\(\(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \[Alpha] + 2\ \ \[Beta] - 2\ \[Gamma] + 3\ \[Phi])\)\)\ Cos[\[Theta]\/2]\ \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \ \[ScriptCapitalF]\_4)\)\)\/\@2\)\), \(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \ \[ImaginaryI]\ \((2\ \[Alpha] + 2\ \[Beta] - 2\ \[Gamma] - 2\ \[Delta] + \ \[Phi])\)\)\ Sin[\[Theta]\/2]\ \((2\ \[ScriptCapitalF]\_1 + 2\ \ \[ScriptCapitalF]\_2 + \((1 + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 + \ \[ScriptCapitalF]\_4)\))\)\)\/\@2}}\)], "Output", CellLabel->"Out[47]="] }, Open ]], Cell["Phase relationships.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({T\_0\[LeftDoubleBracket]2, 2\[RightDoubleBracket]/ T\_0\[LeftDoubleBracket]1, 1\[RightDoubleBracket], \ T\_0\[LeftDoubleBracket]2, 1\[RightDoubleBracket]/ T\_0\[LeftDoubleBracket]1, 2\[RightDoubleBracket]} // Simplify\)], "Input", CellLabel->"In[48]:="], Cell[BoxData[ \({\(-\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \((\[Beta] - \[Delta] - \ \[Phi])\)\)\), \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \((\[Beta] + \[Delta] + \ \[Phi])\)\)}\)], "Output", CellLabel->"Out[48]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({T\_\(-1\)\[LeftDoubleBracket]1, 1\[RightDoubleBracket]/ T\_1\[LeftDoubleBracket]2, 2\[RightDoubleBracket], T\_\(-1\)\[LeftDoubleBracket]1, 2\[RightDoubleBracket]/ T\_1\[LeftDoubleBracket]2, 1\[RightDoubleBracket], T\_\(-1\)\[LeftDoubleBracket]2, 1\[RightDoubleBracket]/ T\_1\[LeftDoubleBracket]1, 2\[RightDoubleBracket], T\_\(-1\)\[LeftDoubleBracket]2, 2\[RightDoubleBracket]/ T\_1\[LeftDoubleBracket]1, 1\[RightDoubleBracket]} // Simplify\)], "Input", CellLabel->"In[49]:="], Cell[BoxData[ \({\[ExponentialE]\^\(\[ImaginaryI]\ \((\[Beta] - \[Delta] - 3\ \[Phi])\)\ \), \(-\[ExponentialE]\^\(\[ImaginaryI]\ \((\[Beta] + \[Delta] - \ \[Phi])\)\)\), \(-\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \((\[Beta] + \ \[Delta] + 3\ \[Phi])\)\)\), \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \((\[Beta] - \[Delta] + \[Phi])\)\)}\)], "Output", CellLabel->"Out[49]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Transition matrices using Jacob-Wick phases", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(T\_0 /. phases[JW] // Simplify\)], "Input", CellLabel->"In[50]:="], Cell[BoxData[ \({{\(Q\ Cos[\[Theta]\/2]\ \((\[ScriptCapitalF]\_5 + \ \[ScriptCapitalF]\_6)\)\)\/\[Omega], \(\[ExponentialE]\^\(\[ImaginaryI]\ \ \[Phi]\)\ Q\ Sin[\[Theta]\/2]\ \((\[ScriptCapitalF]\_5 - \ \[ScriptCapitalF]\_6)\)\)\/\[Omega]}, \ {\(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Phi]\)\ Q\ Sin[\[Theta]\/2]\ \((\ \[ScriptCapitalF]\_5 - \[ScriptCapitalF]\_6)\)\)\/\[Omega], \(-\(\(Q\ Cos[\ \[Theta]\/2]\ \((\[ScriptCapitalF]\_5 + \[ScriptCapitalF]\_6)\)\)\/\[Omega]\)\ \)}}\)], "Output", CellLabel->"Out[50]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(T\_1 /. phases[JW] // Simplify\)], "Input", CellLabel->"In[51]:="], Cell[BoxData[ \({{\(\[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ Sin[\[Theta]\/2]\ \((2\ \ \[ScriptCapitalF]\_1 + 2\ \[ScriptCapitalF]\_2 + \((1 + Cos[\[Theta]])\)\ \((\ \[ScriptCapitalF]\_3 + \[ScriptCapitalF]\_4)\))\)\)\/\@2, \(\[ExponentialE]\^\ \(2\ \[ImaginaryI]\ \[Phi]\)\ Sin[\[Theta]\/2]\ Sin[\[Theta]]\ \((\ \[ScriptCapitalF]\_3 - \[ScriptCapitalF]\_4)\)\)\/\@2}, \ {\(-\(\(Cos[\[Theta]\/2]\ \((2\ \[ScriptCapitalF]\_1 - 2\ \[ScriptCapitalF]\_2 + \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \ \[ScriptCapitalF]\_4)\))\)\)\/\@2\)\), \ \(-\(\(\[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ \((1 + Cos[\[Theta]])\)\ Sin[\[Theta]\/2]\ \ \((\[ScriptCapitalF]\_3 + \[ScriptCapitalF]\_4)\)\)\/\@2\)\)}}\)], "Output", CellLabel->"Out[51]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(T\_\(-1\) /. phases[JW] // Simplify\)], "Input", CellLabel->"In[52]:="], Cell[BoxData[ \({{\(-\(\(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Phi]\)\ \((1 + Cos[\[Theta]])\)\ Sin[\[Theta]\/2]\ \ \((\[ScriptCapitalF]\_3 + \[ScriptCapitalF]\_4)\)\)\/\@2\)\), \ \(Cos[\[Theta]\/2]\ \((2\ \[ScriptCapitalF]\_1 - 2\ \[ScriptCapitalF]\_2 + \ \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \[ScriptCapitalF]\_4)\ \))\)\)\/\@2}, {\(\[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ \[Phi]\)\ Cos[\ \[Theta]\/2]\ \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \ \[ScriptCapitalF]\_4)\)\)\/\@2, \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \[Phi]\)\ Sin[\[Theta]\/2]\ \((2\ \[ScriptCapitalF]\_1 + 2\ \[ScriptCapitalF]\ \_2 + \((1 + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 + \ \[ScriptCapitalF]\_4)\))\)\)\/\@2}}\)], "Output", CellLabel->"Out[52]="] }, Open ]], Cell["Phase relationships.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(T\_0 /. phases[JW]\) /. \[Phi] \[Rule] 0 // Simplify\)], "Input", CellLabel->"In[53]:="], Cell[BoxData[ \({{\(Q\ Cos[\[Theta]\/2]\ \((\[ScriptCapitalF]\_5 + \ \[ScriptCapitalF]\_6)\)\)\/\[Omega], \(Q\ Sin[\[Theta]\/2]\ \((\ \[ScriptCapitalF]\_5 - \[ScriptCapitalF]\_6)\)\)\/\[Omega]}, {\(Q\ Sin[\ \[Theta]\/2]\ \((\[ScriptCapitalF]\_5 - \[ScriptCapitalF]\_6)\)\)\/\[Omega], \ \(-\(\(Q\ Cos[\[Theta]\/2]\ \((\[ScriptCapitalF]\_5 + \[ScriptCapitalF]\_6)\)\ \)\/\[Omega]\)\)}}\)], "Output", CellLabel->"Out[53]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(T\_1 /. phases[JW]\) /. \[Phi] \[Rule] 0 // Simplify\)], "Input", CellLabel->"In[54]:="], Cell[BoxData[ \({{\(Sin[\[Theta]\/2]\ \((2\ \[ScriptCapitalF]\_1 + 2\ \[ScriptCapitalF]\ \_2 + \((1 + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 + \ \[ScriptCapitalF]\_4)\))\)\)\/\@2, \(Sin[\[Theta]\/2]\ Sin[\[Theta]]\ \((\ \[ScriptCapitalF]\_3 - \[ScriptCapitalF]\_4)\)\)\/\@2}, \ {\(-\(\(Cos[\[Theta]\/2]\ \((2\ \[ScriptCapitalF]\_1 - 2\ \[ScriptCapitalF]\_2 + \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \ \[ScriptCapitalF]\_4)\))\)\)\/\@2\)\), \(-\(\(\((1 + Cos[\[Theta]])\)\ Sin[\[Theta]\/2]\ \ \((\[ScriptCapitalF]\_3 + \[ScriptCapitalF]\_4)\)\)\/\@2\)\)}}\)], "Output", CellLabel->"Out[54]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(T\_\(-1\) /. phases[JW]\) /. \[Phi] \[Rule] 0 // Simplify\)], "Input",\ CellLabel->"In[55]:="], Cell[BoxData[ \({{\(-\(\(\((1 + Cos[\[Theta]])\)\ Sin[\[Theta]\/2]\ \ \((\[ScriptCapitalF]\_3 + \[ScriptCapitalF]\_4)\)\)\/\@2\)\), \ \(Cos[\[Theta]\/2]\ \((2\ \[ScriptCapitalF]\_1 - 2\ \[ScriptCapitalF]\_2 + \ \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \[ScriptCapitalF]\_4)\ \))\)\)\/\@2}, {\(Cos[\[Theta]\/2]\ \((\(-1\) + Cos[\[Theta]])\)\ \((\ \[ScriptCapitalF]\_3 - \[ScriptCapitalF]\_4)\)\)\/\@2, \(Sin[\[Theta]\/2]\ \ \((2\ \[ScriptCapitalF]\_1 + 2\ \[ScriptCapitalF]\_2 + \((1 + \ Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 + \ \[ScriptCapitalF]\_4)\))\)\)\/\@2}}\)], "Output", CellLabel->"Out[55]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\({T\_0\[LeftDoubleBracket]2, 2\[RightDoubleBracket]/ T\_0\[LeftDoubleBracket]1, 1\[RightDoubleBracket], \ T\_0\[LeftDoubleBracket]2, 1\[RightDoubleBracket]/ T\_0\[LeftDoubleBracket]1, 2\[RightDoubleBracket]} /. phases[JW]\) /. \[Phi] \[Rule] 0 // Simplify\)], "Input", CellLabel->"In[56]:="], Cell[BoxData[ \({\(-1\), 1}\)], "Output", CellLabel->"Out[56]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\({T\_\(-1\)\[LeftDoubleBracket]1, 1\[RightDoubleBracket]/ T\_1\[LeftDoubleBracket]2, 2\[RightDoubleBracket], T\_\(-1\)\[LeftDoubleBracket]1, 2\[RightDoubleBracket]/ T\_1\[LeftDoubleBracket]2, 1\[RightDoubleBracket], T\_\(-1\)\[LeftDoubleBracket]2, 1\[RightDoubleBracket]/ T\_1\[LeftDoubleBracket]1, 2\[RightDoubleBracket], T\_\(-1\)\[LeftDoubleBracket]2, 2\[RightDoubleBracket]/ T\_1\[LeftDoubleBracket]1, 1\[RightDoubleBracket]} /. phases[JW]\) /. \[Phi] \[Rule] 0 // Simplify\)], "Input", CellLabel->"In[57]:="], Cell[BoxData[ \({1, \(-1\), \(-1\), 1}\)], "Output", CellLabel->"Out[57]="] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Relationship between helicity and CGLN amplitudes", "Section"], Cell["\<\ The six independent helicity amplitudes are historically numbered as follows.\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(HelicityToCGLN\ = \n{H\_1 \[Rule] T\_1\[LeftDoubleBracket]2, 2\[RightDoubleBracket], H\_2 \[Rule] T\_1\[LeftDoubleBracket]2, 1\[RightDoubleBracket], H\_3 \[Rule] T\_1\[LeftDoubleBracket]1, 2\[RightDoubleBracket], \n\t H\_4 \[Rule] T\_1\[LeftDoubleBracket]1, 1\[RightDoubleBracket], H\_5 \[Rule] T\_0\[LeftDoubleBracket]1, 1\[RightDoubleBracket], H\_6 \[Rule] T\_0\[LeftDoubleBracket]1, 2\[RightDoubleBracket]} // Simplify\)], "Input", CellLabel->"In[58]:="], Cell[BoxData[ \({H\_1 \[Rule] \(-\(\(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \ \((2\ \[Alpha] + 2\ \[Beta] - 2\ \[Gamma] - 2\ \[Delta] - 3\ \[Phi])\)\)\ \ \((1 + Cos[\[Theta]])\)\ Sin[\[Theta]\/2]\ \((\[ScriptCapitalF]\_3 + \ \[ScriptCapitalF]\_4)\)\)\/\@2\)\), H\_2 \[Rule] \(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \ \[Alpha] + 2\ \[Beta] - 2\ \[Gamma] - \[Phi])\)\)\ Cos[\[Theta]\/2]\ \((2\ \ \[ScriptCapitalF]\_1 - 2\ \[ScriptCapitalF]\_2 + \((\(-1\) + Cos[\[Theta]])\)\ \ \((\[ScriptCapitalF]\_3 - \[ScriptCapitalF]\_4)\))\)\)\/\@2, H\_3 \[Rule] \(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \ \[Alpha] - 2\ \[Gamma] - 2\ \[Delta] - 3\ \[Phi])\)\)\ Sin[\[Theta]\/2]\ Sin[\ \[Theta]]\ \((\(-\[ScriptCapitalF]\_3\) + \[ScriptCapitalF]\_4)\)\)\/\@2, H\_4 \[Rule] \(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \ \[Alpha] - 2\ \[Gamma] - \[Phi])\)\)\ Sin[\[Theta]\/2]\ \((2\ \ \[ScriptCapitalF]\_1 + 2\ \[ScriptCapitalF]\_2 + \((1 + Cos[\[Theta]])\)\ \((\ \[ScriptCapitalF]\_3 + \[ScriptCapitalF]\_4)\))\)\)\/\@2, H\_5 \[Rule] \(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \ \[Alpha] - 2\ \[Gamma] + \[Phi])\)\)\ Q\ Cos[\[Theta]\/2]\ \((\ \[ScriptCapitalF]\_5 + \[ScriptCapitalF]\_6)\)\)\/\[Omega], H\_6 \[Rule] \(\[ExponentialE]\^\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \((2\ \ \[Alpha] - 2\ \[Gamma] - 2\ \[Delta] - \[Phi])\)\)\ Q\ Sin[\[Theta]\/2]\ \ \((\(-\[ScriptCapitalF]\_5\) + \[ScriptCapitalF]\_6)\)\)\/\[Omega]}\)], \ "Output", CellLabel->"Out[58]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(HelicityToCGLN\ /. phases[JW] // Simplify\)], "Input", CellLabel->"In[59]:="], Cell[BoxData[ \({H\_1 \[Rule] \(-\(\(\[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ \((1 + Cos[\[Theta]])\)\ Sin[\[Theta]\/2]\ \ \((\[ScriptCapitalF]\_3 + \[ScriptCapitalF]\_4)\)\)\/\@2\)\), H\_2 \[Rule] \(-\(\(Cos[\[Theta]\/2]\ \((2\ \[ScriptCapitalF]\_1 - 2\ \[ScriptCapitalF]\_2 + \((\(-1\) + Cos[\[Theta]])\)\ \((\[ScriptCapitalF]\_3 - \ \[ScriptCapitalF]\_4)\))\)\)\/\@2\)\), H\_3 \[Rule] \(\[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Phi]\)\ Sin[\ \[Theta]\/2]\ Sin[\[Theta]]\ \((\[ScriptCapitalF]\_3 - \ \[ScriptCapitalF]\_4)\)\)\/\@2, H\_4 \[Rule] \(\[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ Sin[\[Theta]\ \/2]\ \((2\ \[ScriptCapitalF]\_1 + 2\ \[ScriptCapitalF]\_2 + \((1 + Cos[\ \[Theta]])\)\ \((\[ScriptCapitalF]\_3 + \[ScriptCapitalF]\_4)\))\)\)\/\@2, H\_5 \[Rule] \(Q\ Cos[\[Theta]\/2]\ \((\[ScriptCapitalF]\_5 + \ \[ScriptCapitalF]\_6)\)\)\/\[Omega], H\_6 \[Rule] \(\[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ Q\ Sin[\ \[Theta]\/2]\ \((\[ScriptCapitalF]\_5 - \ \[ScriptCapitalF]\_6)\)\)\/\[Omega]}\)], "Output", CellLabel->"Out[59]="] }, Open ]], Cell["\<\ It is useful to express the transition matrices in terms of simplified \ helicity amplitudes whose azimuthal dependencies have been extracted.\ \>", "Text"], Cell[BoxData[{ \(\(TH\_1 = \ {{\(E\^\(I\ \[Phi]\)\) H\_4, \(E\^\(2 I\ \[Phi]\)\) H\_3}, {H\_2, \(E\^\(I\ \[Phi]\)\) H\_1}};\)\), "\n", \(\(TH\_\(-1\) = \ {{\(E\^\(\(-I\)\ \[Phi]\)\) H\_1, \(-H\_2\)}, {\(-E\^\(\(-2\) I\ \[Phi]\)\) H\_3, \(E\^\(\(-I\)\ \[Phi]\)\) H\_4}};\)\), "\n", \(\(TH\_0 = {{H\_5, \(E\^\(I\ \[Phi]\)\) H\_6}, {\(E\^\(\(-I\)\ \[Phi]\)\) H\_6, \(-H\_5\)}};\)\)}], "Input", CellLabel->"In[60]:="] }, Open ]], Cell[CellGroupData[{ Cell["Express response tensor in terms of response functions", "Section"], Cell["\<\ In separate notes, I show that the differential cross section for virtual \ photoexcitation can be expressed in the form\ \>", "Text"], Cell[BoxData[ \(d\[Sigma]\/d\[CapitalOmega]\_N = \ \(p\/K\_\[Gamma]\)[\[Epsilon]\ \ \[ScriptCapitalW]\_L + \[ScriptCapitalW]\_T + \(\@\(\[Epsilon] \((1 + \ \[Epsilon])\)\)\) \[ScriptCapitalW]\_LT + \[Epsilon]\ \[ScriptCapitalW]\_TT + h \(\@\( 1 - \[Epsilon]\^2\)\) \[ScriptCapitalW]\_TTh + \ h \(\@\( \[Epsilon] \((1 - \[Epsilon])\)\)\) \ \[ScriptCapitalW]\_LTh]\)], "DisplayFormula", FontFamily->"Times New Roman"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`p\)]], " is the momentum, ", Cell[BoxData[ \(TraditionalForm\`K\_\[Gamma] = \(W\^2 - m\_N\^2\)\/\(2 W\)\)]], " is the equivalent real photon energy (cm), ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\)]], " is the polarization of the virtual photon, and ", Cell[BoxData[ \(TraditionalForm\`h\)]], " is the electron helicity. The \[Phi]-dependent response functions ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalW]\_i\)]], " are given by the functions defined below. " }], "Text"], Cell[BoxData[ \(\[ScriptCapitalW][ T_, \[Lambda]1_, \[Lambda]2_] := \((\(\((\(-1\))\)\^\(\[Lambda]1 + \ \[Lambda]2\)\) Sum[\((HermitianConjugate[ T\_\[Lambda]2 /. A_\_a_ \[Rule] \(A\_a\^*\)] . \[Rho]\_f . T\_\[Lambda]1 . \[Rho]\_i /. phases[JW])\)\[LeftDoubleBracket]k, k\[RightDoubleBracket], {k, 1, 2}] // Expand)\) //. ContractAmplitudeProducts\)], "Input", CellLabel->"In[63]:="], Cell[BoxData[{ \(\(\[ScriptCapitalW]\_L[T_] := 2\ \[ScriptCapitalW][T, 0, 0];\)\), "\n", \(\(\[ScriptCapitalW]\_T[ T_] := \[ScriptCapitalW][T, 1, 1] + \[ScriptCapitalW][ T, \(-1\), \(-1\)];\)\), "\n", \(\(\[ScriptCapitalW]\_LT[ T_] := \[ScriptCapitalW][T, 0, 1] - \[ScriptCapitalW][T, 0, \(-1\)] + \ \[ScriptCapitalW][T, 1, 0] - \[ScriptCapitalW][ T, \(-1\), 0];\)\), "\n", \(\(\[ScriptCapitalW]\_LTh[ T_] := \[ScriptCapitalW][T, 0, 1] + \[ScriptCapitalW][T, 0, \(-1\)] + \ \[ScriptCapitalW][T, 1, 0] + \[ScriptCapitalW][ T, \(-1\), 0];\)\), "\n", \(\(\[ScriptCapitalW]\_TT[ T_] := \ \(-\((\[ScriptCapitalW][T, 1, \(-1\)] + \ \[ScriptCapitalW][T, \(-1\), 1])\)\);\)\), "\n", \(\(\[ScriptCapitalW]\_TTh[ T_] := \ \[ScriptCapitalW][T, 1, 1] - \ \[ScriptCapitalW][ T, \(-1\), \(-1\)];\)\)}], "Input", CellLabel->"In[64]:="], Cell[TextData[{ "It is well-known that if the polarization vectors are expressed in a basis \ with ", Cell[BoxData[ \(TraditionalForm\`N\&^\)]], " normal to the reaction plane, the azimuthal dependence of the observables \ can be extracted from the response functions. Both the ejectile and target \ bases defined above have this property, as would any other basis related to \ these by a rotation about the normal to the reaction plane. However, the \ precise relationship between the \[Phi]-dependent response functions and the \ more common \[Phi]-independent response functions, ", Cell[BoxData[ \(TraditionalForm\`R\_i\)]], ", depends upon the conventions chosen by a particular author for \ normalizations and the choice between longitudinal and scalar amplitudes \ \[LongDash] unfortunately, many such conventions are found in the literature. \ We have chosen to express the cross section in the form" }], "Text"], Cell[BoxData[ \(d\[Sigma]\/d\[CapitalOmega]\_N = \ \(p\/K\_\[Gamma]\)[\[Epsilon]\_S\ \ \((R\_L[0] + \(\[ScriptCapitalP]\_N\) R\_L[N])\) + \((R\_T[ 0] + \(\[ScriptCapitalP]\_N\) R\_T[N])\)\n\t\t + \(\@\(2 \( \[Epsilon]\_S\) \((1 + \ \[Epsilon])\)\)\) \((R\_LT[0] + \(\[ScriptCapitalP]\_N\) R\_LT[N])\) Cos[\[Phi]] + \(\@\(2 \( \[Epsilon]\_S\) \((1 + \[Epsilon])\)\)\) \ \((\(\[ScriptCapitalP]\_L\) R\_LT[L] + \(\[ScriptCapitalP]\_S\) R\_LT[S])\) Sin[\[Phi]]\n\t\t + \[Epsilon]\ \((R\_TT[ 0] + \(\[ScriptCapitalP]\_N\) R\_TT[N])\) Cos[2 \[Phi]] + \(\@\[Epsilon]\) \((\(\[ScriptCapitalP]\_L\) R\_TT[L] + \(\[ScriptCapitalP]\_S\) R\_TT[S])\) Sin[2 \[Phi]]\n\t\t + h \(\@\( 2 \( \[Epsilon]\_S\) \((1 - \[Epsilon])\)\)\) \((R\_LTh[ 0] + \(\[ScriptCapitalP]\_N\) R\_LTh[N])\) Sin[\[Phi]] + \ h \(\@\( 2 \( \[Epsilon]\_S\) \((1 - \[Epsilon])\)\)\) \((\(\ \[ScriptCapitalP]\_L\) R\_LTh[L] + \(\[ScriptCapitalP]\_S\) R\_LTh[S])\) Cos[\[Phi]]\n\t\t + h \(\@\( 1 - \[Epsilon]\^2\)\) \((\(\[ScriptCapitalP]\_L\) R\_TTh[L] + \(\[ScriptCapitalP]\_S\) R\_TTh[S])\)]\)], "DisplayFormula", FontFamily->"Times New Roman"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\_S = \(Q\^2\/q\^2\) \[Epsilon]\)]], " in the barycentric frame and where ", Cell[BoxData[ \(TraditionalForm \`{\[ScriptCapitalP]\_S, \[ScriptCapitalP]\_N, \[ScriptCapitalP]\_L} \)]], " are components of the nucleon polarization. In this representation it is \ natural to express the ", Cell[BoxData[ \(TraditionalForm\`R\_L\)]], " and ", Cell[BoxData[ \(TraditionalForm\`R\_LT\)]], " response functions in terms of scalar multipole amplitudes. Other \ authors replace ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\_S\)]], " by ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\[Epsilon]\_L = \(Q\^2\/\[Omega]\^2\) \[Epsilon]\), "TraditionalForm"]}], TraditionalForm]]], " and employ longitudinal instead of scalar multipoles. However, \ additional differences in signs and factors of ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], " are common also." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Recoil-Polarization Response Functions", "Section"], Cell[CellGroupData[{ Cell["\<\ Recoil polarization vector, projection operator, and density matrix\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(P\&^ = \(P\_L\) L\&^ + \(P\_N\) N\&^ + \(P\_S\) S\&^ /. EjectileBasis\)], "Input", CellLabel->"In[70]:="], Cell[BoxData[ \({\(-Cos[\[Phi]]\)\ Sin[\[Theta]]\ P\_L + Sin[\[Phi]]\ P\_N + Cos[\[Theta]]\ Cos[\[Phi]]\ P\_S, \(-Sin[\[Theta]]\)\ Sin[\[Phi]]\ \ P\_L - Cos[\[Phi]]\ P\_N + Cos[\[Theta]]\ Sin[\[Phi]]\ P\_S, \(-Cos[\[Theta]]\)\ P\_L - Sin[\[Theta]]\ P\_S}\)], "Output", CellLabel->"Out[70]="] }, Open ]], Cell[BoxData[ \(\(\[DoubleStruckCapitalP] = \(1\/2\) \((IdentityMatrix[2] + P\&^ . \[Sigma]\&\[RightVector])\) /. trigToExp[\[Phi]] // Simplify;\)\)], "Input", CellLabel->"In[71]:="], Cell[CellGroupData[{ Cell[BoxData[ \(\[DoubleStruckCapitalP] // MatrixForm\)], "Input", CellLabel->"In[72]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2\ \((1 - Cos[\[Theta]]\ P\_L - Sin[\[Theta]]\ P\_S)\)\), \(1\/2\ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \[Phi]\)\ \((\(-Sin[\[Theta]]\)\ P\_L + \[ImaginaryI]\ P\_N \ + Cos[\[Theta]]\ P\_S)\)\)}, {\(\(-\(1\/2\)\)\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\)\ \ \((Sin[\[Theta]]\ P\_L + \[ImaginaryI]\ P\_N - Cos[\[Theta]]\ P\_S)\)\), \(1\/2\ \((1 + Cos[\[Theta]]\ P\_L + Sin[\[Theta]]\ P\_S)\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[72]//MatrixForm="] }, Open ]], Cell[BoxData[{ \(\(\[Rho]\_i = \(1\/2\) IdentityMatrix[2];\)\), "\n", \(\(\[Rho]\_f = HermitianConjugate[\[Chi]\_f] . \[DoubleStruckCapitalP] . \[Chi]\_f // Simplify;\)\)}], "Input", CellLabel->"In[73]:="], Cell[CellGroupData[{ Cell[BoxData[ \(\[Rho]\_f // MatrixForm\)], "Input", CellLabel->"In[75]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2\ \((1 + P\_L)\)\), \(1\/2\ \[ImaginaryI]\ \[ExponentialE]\^\(\ \[ImaginaryI]\ \[Beta]\)\ \((P\_N + \[ImaginaryI]\ P\_S)\)\)}, {\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \[Beta]\)\ \((P\_N - \[ImaginaryI]\ P\_S)\)\), \(1\/2\ \((1 \ - P\_L)\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[75]//MatrixForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[Rho]\_f /. phases[JW] // Simplify\) // MatrixForm\)], "Input", CellLabel->"In[76]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2\ \((1 + P\_L)\)\), \(1\/2\ \[ExponentialE]\^\(\[ImaginaryI]\ \ \[Phi]\)\ \((\(-\[ImaginaryI]\)\ P\_N + P\_S)\)\)}, {\(1\/2\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Phi]\)\ \((\ \[ImaginaryI]\ P\_N + P\_S)\)\), \(1\/2\ \((1 - P\_L)\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[76]//MatrixForm="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[ "\[Phi]-dependent response functions in terms of helicity amplitudes"], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\((\[ScriptCapitalW]\_L[TH] // Simplify)\) // Collect[#, {P\_x_, f_[a_. \ \[Phi]]}] &\)], "Input", CellLabel->"In[77]:="], Cell[BoxData[ \(Abs[H\_5]\^2 + Abs[H\_6]\^2 - 2\ Im[H\_5\ \(\((H\_6)\)\^*\)]\ P\_N\)], "Output", CellLabel->"Out[77]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((\[ScriptCapitalW]\_T[TH] // Simplify)\) // Collect[#, {P\_x_, f_[a_. \ \[Phi]]}] &\)], "Input", CellLabel->"In[78]:="], Cell[BoxData[ \(1\/2\ \((Abs[H\_1]\^2 + Abs[H\_2]\^2 + Abs[H\_3]\^2 + Abs[H\_4]\^2)\) + 1\/2\ \((2\ Im[H\_1\ \(\((H\_3)\)\^*\)] + 2\ Im[H\_2\ \(\((H\_4)\)\^*\)])\)\ P\_N\)], "Output", CellLabel->"Out[78]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((\(\[ScriptCapitalW]\_LT[TH] // ExpToTrig\) // MyFullSimplify)\) //. ContractAmplitudeProducts // Collect[#, {P\_x_, f_[a_. \[Phi]]}] &\)], "Input", CellLabel->"In[79]:="], Cell[BoxData[ \(Cos[\[Phi]]\ \((Re[H\_1\ \(\((H\_5)\)\^*\)] - Re[H\_4\ \(\((H\_5)\)\^*\)] - Re[\((H\_2 + H\_3)\)\ \(\((H\_6)\)\^*\)])\) + \((Im[\((H\_1 + H\_4)\)\ \(\((H\_5)\)\^*\)] - Im[\((H\_2 - H\_3)\)\ \(\((H\_6)\)\^*\)])\)\ Sin[\[Phi]]\ P\_L + Cos[\[Phi]]\ \((\(-Im[\((H\_2 + H\_3)\)\ \(\((H\_5)\)\^*\)]\) - Im[\((H\_1 - H\_4)\)\ \(\((H\_6)\)\^*\)])\)\ P\_N + \((Im[\((H\_2 - H\_3)\)\ \(\((H\_5)\)\^*\)] + Im[\((H\_1 + H\_4)\)\ \(\((H\_6)\)\^*\)])\)\ Sin[\[Phi]]\ P\_S\)], \ "Output", CellLabel->"Out[79]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((\(\[ScriptCapitalW]\_LTh[TH] // ExpToTrig\) // MyFullSimplify)\) //. ContractAmplitudeProducts // Collect[#, {P\_x_, f_[a_. \[Phi]]}] &\)], "Input", CellLabel->"In[80]:="], Cell[BoxData[ \(\((\(-Im[\((H\_1 - H\_4)\)\ \(\((H\_5)\)\^*\)]\) + Im[\((H\_2 + H\_3)\)\ \(\((H\_6)\)\^*\)])\)\ Sin[\[Phi]] + Cos[\[Phi]]\ \((\(-Re[\((H\_1 + H\_4)\)\ \(\((H\_5)\)\^*\)]\) + Re[\((H\_2 - H\_3)\)\ \(\((H\_6)\)\^*\)])\)\ P\_L + \((\(-Re[\((H\_2 + H\_3)\)\ \(\((H\_5)\)\^*\)]\) - Re[\((H\_1 - H\_4)\)\ \(\((H\_6)\)\^*\)])\)\ Sin[\[Phi]]\ P\_N + Cos[\[Phi]]\ \((\(-Re[\((H\_2 - H\_3)\)\ \(\((H\_5)\)\^*\)]\) - Re[\((H\_1 + H\_4)\)\ \(\((H\_6)\)\^*\)])\)\ P\_S\)], "Output", CellLabel->"Out[80]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((\(\[ScriptCapitalW]\_TT[TH] // ExpToTrig\) // MyFullSimplify)\) //. ContractAmplitudeProducts // Collect[#, {P\_x_, f_[a_. \[Phi]]}] &\)], "Input", CellLabel->"In[81]:="], Cell[BoxData[ \(Cos[ 2\ \[Phi]]\ \((Re[H\_2\ \(\((H\_3)\)\^*\)] - Re[H\_1\ \(\((H\_4)\)\^*\)])\) + \((Im[H\_2\ \(\((H\_3)\)\^*\)] - Im[H\_1\ \(\((H\_4)\)\^*\)])\)\ Sin[2\ \[Phi]]\ P\_L + Cos[2\ \[Phi]]\ \((Im[H\_1\ \(\((H\_2)\)\^*\)] + Im[H\_3\ \(\((H\_4)\)\^*\)])\)\ P\_N + \((\(-Im[ H\_1\ \(\((H\_2)\)\^*\)]\) + Im[H\_3\ \(\((H\_4)\)\^*\)])\)\ Sin[2\ \[Phi]]\ P\_S\)], "Output",\ CellLabel->"Out[81]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((\[ScriptCapitalW]\_TTh[TH] // MyFullSimplify)\) //. ContractAmplitudeProducts // Collect[#, {P\_x_, f_[a_. \[Phi]]}] &\)], "Input", CellLabel->"In[82]:="], Cell[BoxData[ \(1\/2\ \((\(-Abs[H\_1]\^2\) - Abs[H\_2]\^2 + Abs[H\_3]\^2 + Abs[H\_4]\^2)\)\ P\_L + \((Re[H\_1\ \(\((H\_3)\)\^*\)] + Re[H\_2\ \(\((H\_4)\)\^*\)])\)\ P\_S\)], "Output", CellLabel->"Out[82]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[ "\[Phi]-independent response functions in terms of helicity amplitudes"], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_L = \(\[ScriptCapitalW]\_L[TH] // MySimplify\) // Collect[#, {P\_x_, f_[a_. \ \[Phi]]}] &;\)\), "\n", \(R\_L[ H] = {R\_L[0] \[Rule] Select[tempR\_L, FreeQ[#, P\_x_] &], R\_L[N] \[Rule] Coefficient[tempR\_L, P\_N]\ } // Simplify\)}], "Input", CellLabel->"In[83]:="], Cell[BoxData[ \({R\_L[0] \[Rule] Abs[H\_5]\^2 + Abs[H\_6]\^2, R\_L[N] \[Rule] \(-2\)\ Im[H\_5\ \(\((H\_6)\)\^*\)]}\)], "Output", CellLabel->"Out[84]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_T = \((\(\[ScriptCapitalW]\_T[TH] // MySimplify //. ContractAmplitudeProducts\) // Collect[#, {P\_x_, f_[a_. \ \[Phi]]}] &)\);\)\), "\n", \(R\_T[ H] = {R\_T[0] \[Rule] Select[tempR\_T, FreeQ[#, P\_x_] &], R\_T[N] \[Rule] Coefficient[tempR\_T, P\_N]\ } // Simplify\)}], "Input", CellLabel->"In[85]:="], Cell[BoxData[ \({R\_T[0] \[Rule] 1\/2\ \((Abs[H\_1]\^2 + Abs[H\_2]\^2 + Abs[H\_3]\^2 + Abs[H\_4]\^2)\), R\_T[N] \[Rule] Im[H\_1\ \(\((H\_3)\)\^*\)] + Im[H\_2\ \(\((H\_4)\)\^*\)]}\)], "Output", CellLabel->"Out[86]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_LT = \((\(\[ScriptCapitalW]\_LT[TH] // ExpToTrig\) // MySimplify)\) //. ContractAmplitudeProducts // Collect[#, {P\_x_, f_[a_. \[Phi]]}] &;\)\), "\n", \(R\_LT[ H] = {\n\t R\_LT[0] \[Rule] Select[tempR\_LT, FreeQ[#, P\_x_] &]/Cos[\[Phi]], \n\t R\_LT[N] \[Rule] Coefficient[tempR\_LT, P\_N]/Cos[\[Phi]], \n\t R\_LT[L] \[Rule] Coefficient[tempR\_LT, P\_L]/Sin[\[Phi]], \ \n\t R\_LT[S] \[Rule] Coefficient[tempR\_LT, P\_S]/Sin[\[Phi]]} // Simplify\)}], "Input", CellLabel->"In[87]:="], Cell[BoxData[ \({R\_LT[0] \[Rule] Re[\((H\_1 - H\_4)\)\ \(\((H\_5)\)\^*\)] - Re[\((H\_2 + H\_3)\)\ \(\((H\_6)\)\^*\)], R\_LT[N] \[Rule] \(-Im[\((H\_2 + H\_3)\)\ \(\((H\_5)\)\^*\)]\) - Im[\((H\_1 - H\_4)\)\ \(\((H\_6)\)\^*\)], R\_LT[L] \[Rule] Im[\((H\_1 + H\_4)\)\ \(\((H\_5)\)\^*\)] - Im[\((H\_2 - H\_3)\)\ \(\((H\_6)\)\^*\)], R\_LT[S] \[Rule] Im[\((H\_2 - H\_3)\)\ \(\((H\_5)\)\^*\)] + Im[\((H\_1 + H\_4)\)\ \(\((H\_6)\)\^*\)]}\)], "Output", CellLabel->"Out[88]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_LTh = \((\(\[ScriptCapitalW]\_LTh[TH] // ExpToTrig\) // MySimplify)\) //. ContractAmplitudeProducts // Collect[#, {P\_x_, f_[a_. \[Phi]], f_[a_. \ \[Theta]]}] &;\)\), "\n", \(R\_LTh[ H] = {\n\t R\_LTh[0] \[Rule] Select[tempR\_LTh, FreeQ[#, P\_x_] &]/Sin[\[Phi]], \n\t R\_LTh[N] \[Rule] Coefficient[tempR\_LTh, P\_N]/Sin[\[Phi]], \n\t R\_LTh[L] \[Rule] Coefficient[tempR\_LTh, P\_L]/Cos[\[Phi]], \ \n\t R\_LTh[S] \[Rule] Coefficient[tempR\_LTh, P\_S]/Cos[\[Phi]]} // Simplify\)}], "Input", CellLabel->"In[89]:="], Cell[BoxData[ \({R\_LTh[0] \[Rule] \(-Im[\((H\_1 - H\_4)\)\ \(\((H\_5)\)\^*\)]\) + Im[\((H\_2 + H\_3)\)\ \(\((H\_6)\)\^*\)], R\_LTh[N] \[Rule] \(-Re[\((H\_2 + H\_3)\)\ \(\((H\_5)\)\^*\)]\) - Re[\((H\_1 - H\_4)\)\ \(\((H\_6)\)\^*\)], R\_LTh[L] \[Rule] \(-Re[\((H\_1 + H\_4)\)\ \(\((H\_5)\)\^*\)]\) + Re[\((H\_2 - H\_3)\)\ \(\((H\_6)\)\^*\)], R\_LTh[S] \[Rule] \(-Re[\((H\_2 - H\_3)\)\ \(\((H\_5)\)\^*\)]\) - Re[\((H\_1 + H\_4)\)\ \(\((H\_6)\)\^*\)]}\)], "Output", CellLabel->"Out[90]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_TT = \((\(\[ScriptCapitalW]\_TT[TH] // ExpToTrig\) // MySimplify)\) // Collect[#, {P\_x_, f_[a_. \[Phi]]}] &;\)\), "\n", \(R\_TT[ H] = {\n\t R\_TT[0] \[Rule] Select[tempR\_TT, FreeQ[#, P\_x_] &]/Cos[2 \[Phi]], \n\t R\_TT[N] \[Rule] Coefficient[tempR\_TT, P\_N]/Cos[2 \[Phi]], \n\t R\_TT[L] \[Rule] Coefficient[tempR\_TT, P\_L]/Sin[2 \[Phi]], \ \n\t R\_TT[S] \[Rule] Coefficient[tempR\_TT, P\_S]/Sin[2 \[Phi]]} // Simplify //. ContractAmplitudeProducts\)}], "Input", CellLabel->"In[91]:="], Cell[BoxData[ \({R\_TT[0] \[Rule] Re[H\_2\ \(\((H\_3)\)\^*\)] - Re[H\_1\ \(\((H\_4)\)\^*\)], R\_TT[N] \[Rule] Im[H\_1\ \(\((H\_2)\)\^*\)] + Im[H\_3\ \(\((H\_4)\)\^*\)], R\_TT[L] \[Rule] Im[H\_2\ \(\((H\_3)\)\^*\)] - Im[H\_1\ \(\((H\_4)\)\^*\)], R\_TT[S] \[Rule] \(-Im[H\_1\ \(\((H\_2)\)\^*\)]\) + Im[H\_3\ \(\((H\_4)\)\^*\)]}\)], "Output", CellLabel->"Out[92]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_TTh = \((\(\[ScriptCapitalW]\_TTh[TH] // ExpToTrig\) // MySimplify)\) // Collect[#, {P\_x_, f_[a_. \[Phi]]}] &;\)\), "\n", \(R\_TTh[ H] = {\n\t R\_TTh[0] \[Rule] Select[tempR\_TTh, FreeQ[#, P\_x_] &], \n\t R\_TTh[N] \[Rule] Coefficient[tempR\_TTh, P\_N], \n\t R\_TTh[L] \[Rule] Coefficient[tempR\_TTh, P\_L], \ \n\t R\_TTh[S] \[Rule] Coefficient[tempR\_TTh, P\_S]} // Simplify\)}], "Input", CellLabel->"In[93]:="], Cell[BoxData[ \({R\_TTh[0] \[Rule] 0, R\_TTh[N] \[Rule] 0, R\_TTh[L] \[Rule] 1\/2\ \((\(-Abs[H\_1]\^2\) - Abs[H\_2]\^2 + Abs[H\_3]\^2 + Abs[H\_4]\^2)\), R\_TTh[S] \[Rule] Re[H\_1\ \(\((H\_3)\)\^*\)] + Re[H\_2\ \(\((H\_4)\)\^*\)]}\)], "Output", CellLabel->"Out[94]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[ "\[Phi]-independent response functions in terms of CGLN amplitudes"], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_L = \(\[ScriptCapitalW]\_L[T] // Simplify\) // Collect[#, {P\_x_, f_[a_. \ \[Phi]]}] &;\)\), "\n", \(R\_L[\[ScriptCapitalF]] = {\n\t R\_L[0] \[Rule] Select[tempR\_L, FreeQ[#, P\_x_] &], \n\t R\_L[N] \[Rule] Coefficient[tempR\_L, P\_N]\ } // FullSimplify\)}], "Input", CellLabel->"In[95]:="], Cell[BoxData[ \({R\_L[ 0] \[Rule] \(Q\^2\ \((Abs[\[ScriptCapitalF]\_5]\^2 + Abs[\ \[ScriptCapitalF]\_6]\^2 + 2\ Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_5\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)])\)\)\/\[Omega]\^2, R\_L[N] \[Rule] \(2\ Q\^2\ Im[\[ScriptCapitalF]\_5\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)]\ Sin[\[Theta]]\)\/\[Omega]\^2}\)], "Output", CellLabel->"Out[96]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_T = \((\(\[ScriptCapitalW]\_T[T] // Simplify\) // Collect[#, {P\_x_, f_[a_. \ \[Phi]]}] &)\);\)\), "\n", \(R\_T[\[ScriptCapitalF]] = {\n\t R\_T[0] \[Rule] Select[tempR\_T, FreeQ[#, P\_x_] &], \n\t R\_T[N] \[Rule] Coefficient[tempR\_T, P\_N]\ } // FullSimplify\)}], "Input", CellLabel->"In[97]:="], Cell[BoxData[ \({R\_T[0] \[Rule] Abs[\[ScriptCapitalF]\_1]\^2 + Abs[\[ScriptCapitalF]\_2]\^2 - 2\ Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_2)\)\^*\)] + 1\/2\ \((Abs[\[ScriptCapitalF]\_3]\^2 + Abs[\[ScriptCapitalF]\_4]\^2 + 2\ \((Re[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)] + Re[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_3\ \(\((\ \[ScriptCapitalF]\_4)\)\^*\)])\))\)\ Sin[\[Theta]]\^2, R\_T[N] \[Rule] Sin[\[Theta]]\ \((\(-2\)\ Im[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_2)\)\^*\)] - Im[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_3)\)\^*\)] + Cos[\[Theta]]\ \((Im[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_3)\)\^*\)] - Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)])\) + Im[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Im[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_4)\)\^*\)]\ \ Sin[\[Theta]]\^2)\)}\)], "Output", CellLabel->"Out[98]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_LT = \(\(\[ScriptCapitalW]\_LT[T] // ExpToTrig\) // MySimplify\) // \(Collect[#, {P\_x_, f_[a_. \[Phi]]}] &\) //. ContractAmplitudeProducts;\)\), "\n", \(R\_LT[\[ScriptCapitalF]] = {\n\t R\_LT[0] \[Rule] Select[tempR\_LT, FreeQ[#, P\_x_] &]/Cos[\[Phi]], \n\t R\_LT[N] \[Rule] Coefficient[tempR\_LT, P\_N]/Cos[\[Phi]], \n\t R\_LT[L] \[Rule] Coefficient[tempR\_LT, P\_L]/Sin[\[Phi]], \ \n\t R\_LT[S] \[Rule] Coefficient[tempR\_LT, P\_S]/Sin[\[Phi]]} // FullSimplify\)}], "Input", CellLabel->"In[99]:="], Cell[BoxData[ \({R\_LT[ 0] \[Rule] \(-\(\(1\/\[Omega]\)\((\@2\ Q\ \ \((Re[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] + Re[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] \ + Re[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_6)\)\^*\)] + Cos[\[Theta]]\ \((Re[\[ScriptCapitalF]\_4\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Re[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_6)\)\ \^*\)])\) + Re[\[ScriptCapitalF]\_4\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\)\ Sin[\[Theta]])\)\)\), R\_LT[N] \[Rule] \(\(1\/\[Omega]\)\((\@2\ Q\ \((Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_5)\)\^*\)] + Cos[\[Theta]]\ \((\(-Im[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)]\) + Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\) - Im[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_6)\)\^*\)] + \ \((Im[\[ScriptCapitalF]\_4\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] - Im[\[ScriptCapitalF]\_3\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\)\ Sin[\[Theta]]\^2)\))\)\), R\_LT[L] \[Rule] \(\@2\ Q\ \((Im[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Im[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)])\)\ Sin[\[Theta]]\)\/\[Omega], R\_LT[S] \[Rule] \(\@2\ Q\ \((Im[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] - Im[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)] + Cos[\[Theta]]\ \((\(-Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_5)\)\^*\)]\) + Im[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)])\))\)\)\/\[Omega]}\)], "Output", CellLabel->"Out[100]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_LTh = \(\(\[ScriptCapitalW]\_LTh[T] // ExpToTrig\) // MySimplify\) // Collect[#, {P\_x_, f_[a_. \[Phi]], f_[a_. \ \[Theta]]}] &;\)\), "\n", \(R\_LTh[\[ScriptCapitalF]] = {\n\t R\_LTh[0] \[Rule] Select[tempR\_LTh, FreeQ[#, P\_x_] &]/Sin[\[Phi]], \n\t R\_LTh[N] \[Rule] Coefficient[tempR\_LTh, P\_N]/Sin[\[Phi]], \n\t R\_LTh[L] \[Rule] Coefficient[tempR\_LTh, P\_L]/Cos[\[Phi]], \ \n\t R\_LTh[S] \[Rule] Coefficient[tempR\_LTh, P\_S]/Cos[\[Phi]]} // FullSimplify\)}], "Input", CellLabel->"In[101]:="], Cell[BoxData[ \({R\_LTh[ 0] \[Rule] \(\(1\/\[Omega]\)\((\@2\ Q\ \((Im[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_5)\)\^*\)] + Im[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] + Im[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_6)\)\^*\)] + Cos[\[Theta]]\ \((Im[\[ScriptCapitalF]\_4\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Im[\[ScriptCapitalF]\_3\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\) + Im[\[ScriptCapitalF]\_4\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\)\ Sin[\[Theta]])\)\), R\_LTh[N] \[Rule] \(\(1\/\[Omega]\)\((\@2\ Q\ \ \((Re[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] + Cos[\[Theta]]\ \((\(-Re[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)]\) + Re[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\) - Re[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_6)\)\^*\)] + \ \((Re[\[ScriptCapitalF]\_4\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] - Re[\[ScriptCapitalF]\_3\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\)\ Sin[\[Theta]]\^2)\))\)\), R\_LTh[L] \[Rule] \(-\(\(\@2\ Q\ \((Re[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Re[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\)\ Sin[\[Theta]]\)\/\[Omega]\)\), R\_LTh[S] \[Rule] \(\@2\ Q\ \((\(-Re[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)]\) + Re[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)] + Cos[\[Theta]]\ \((Re[\[ScriptCapitalF]\_1\ \(\ \((\[ScriptCapitalF]\_5)\)\^*\)] - Re[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)])\))\)\)\/\[Omega]}\)], "Output", CellLabel->"Out[102]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_TT = \(\(\[ScriptCapitalW]\_TT[T] // ExpToTrig\) // MySimplify\) // Collect[#, {P\_x_, f_[a_. \[Phi]]}] &;\)\), "\n", \(R\_TT[\[ScriptCapitalF]] = {\n\t R\_TT[0] \[Rule] Select[tempR\_TT, FreeQ[#, P\_x_] &]/Cos[2 \[Phi]], \n\t R\_TT[N] \[Rule] Coefficient[tempR\_TT, P\_N]/Cos[2 \[Phi]], \n\t R\_TT[L] \[Rule] Coefficient[tempR\_TT, P\_L]/Sin[2 \[Phi]], \ \n\t R\_TT[S] \[Rule] Coefficient[tempR\_TT, P\_S]/Sin[2 \[Phi]]} // FullSimplify\)}], "Input", CellLabel->"In[103]:="], Cell[BoxData[ \({R\_TT[0] \[Rule] 1\/2\ \((Abs[\[ScriptCapitalF]\_3]\^2 + Abs[\[ScriptCapitalF]\_4]\^2 + 2\ \((Re[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)] + Re[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_3\ \(\((\ \[ScriptCapitalF]\_4)\)\^*\)])\))\)\ Sin[\[Theta]]\^2, R\_TT[N] \[Rule] Sin[\[Theta]]\ \((\(-Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)]\) + Cos[\[Theta]]\ \((Im[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_3)\)\^*\)] - Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)])\) + Im[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Im[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_4)\)\^*\)]\ \ Sin[\[Theta]]\^2)\), R\_TT[L] \[Rule] \(-\((Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)] + Im[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)])\)\)\ Sin[\[Theta]]\^2, R\_TT[S] \[Rule] \((\(-Im[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)]\) + Im[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Cos[\[Theta]]\ \((Im[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_3)\)\^*\)] - Im[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)])\))\)\ Sin[\[Theta]]}\)], "Output", CellLabel->"Out[104]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_TTh = \(\(\[ScriptCapitalW]\_TTh[T] // ExpToTrig\) // MySimplify\) // Collect[#, {P\_x_, f_[a_. \[Phi]]}] &;\)\), "\n", \(R\_TTh[\[ScriptCapitalF]] = {\n\t R\_TTh[0] \[Rule] Select[tempR\_TTh, FreeQ[#, P\_x_] &], \n\t R\_TTh[N] \[Rule] Coefficient[tempR\_TTh, P\_N], \n\t R\_TTh[L] \[Rule] Coefficient[tempR\_TTh, P\_L], \ \n\t R\_TTh[S] \[Rule] Coefficient[tempR\_TTh, P\_S]} // FullSimplify\)}], "Input", CellLabel->"In[105]:="], Cell[BoxData[ \({R\_TTh[0] \[Rule] 0, R\_TTh[N] \[Rule] 0, R\_TTh[L] \[Rule] \(-\((Abs[\[ScriptCapitalF]\_1]\^2 + Abs[\[ScriptCapitalF]\_2]\^2)\)\)\ Cos[\[Theta]] + 2\ Re[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_2)\)\^*\)] + \ \((Re[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_3)\)\^*\)] + Re[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)])\)\ Sin[\[Theta]]\^2, R\_TTh[S] \[Rule] \((\(-Abs[\[ScriptCapitalF]\_1]\^2\) + Abs[\[ScriptCapitalF]\_2]\^2 + Re[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_3)\)\^*\)] - Re[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Cos[\[Theta]]\ \((\(-Re[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_3)\)\^*\)]\) + Re[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)])\))\)\ Sin[\[Theta]]}\)], "Output", CellLabel->"Out[106]="] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Multipole expansions", "Section"], Cell[CellGroupData[{ Cell["Multipole expansion of CGLN amplitudes", "Subsection"], Cell["\<\ Here we quote the traditional expansions of CGLN amplitudes in terms of \ multipoles. The expressions were obtained from Dennery and are widely quoted \ in the literature, but I have not checked them independently. [Actually, I \ did derive amplitudes 1-4 a long time ago, but my notes are somewhat sketchy \ and have not been reviewed recently. Nevertheless, I am confident that these \ expressions are correct.]\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(CGLN1[\[ScriptL]_]\), " ", "=", " ", RowBox[{"{", "\n", "\t", RowBox[{ RowBox[{\(\[ScriptCapitalF]\_1\), "\[Rule]", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{\((\[ScriptL]\ \(M\_+\)[\[ScriptL]] + \(E\_+\)[\ \[ScriptL]])\), " ", RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] + 1\), "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], " ", "+", " ", RowBox[{\((\((\[ScriptL] + 1)\) \(M\_-\)[\[ScriptL]] + \(E\_-\)[\ \[ScriptL]])\), RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] - 1\), "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}], ",", "\n", "\t", RowBox[{\(\[ScriptCapitalF]\_2\), "\[Rule]", RowBox[{"Sum", "[", RowBox[{ RowBox[{\((\((\[ScriptL] + 1)\) \(M\_+\)[\[ScriptL]]\ + \ \[ScriptL]\ \(M\ \_-\)[\[ScriptL]])\), RowBox[{ SubsuperscriptBox["P", "\[ScriptL]", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}], ",", "\n", "\t", RowBox[{\(\[ScriptCapitalF]\_3\), "\[Rule]", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{\((\(E\_+\)[\[ScriptL]]\ - \ \ \(M\_+\)[\[ScriptL]])\), " ", RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] + 1\), "\[DoublePrime]", MultilineFunction->None], "[", "x", "]"}]}], " ", "+", " ", RowBox[{\((\(E\_-\)[\[ScriptL]] + \(M\_-\)[\[ScriptL]])\), RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] - 1\), "\[DoublePrime]", MultilineFunction->None], "[", "x", "]"}]}]}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}], ",", "\n", "\t", RowBox[{\(\[ScriptCapitalF]\_4\), "\[Rule]", RowBox[{"Sum", "[", RowBox[{ RowBox[{\((\(M\_+\)[\[ScriptL]] - \(E\_+\)[\[ScriptL]] - \ \(M\_-\)[\[ScriptL]] - \(E\_-\)[\[ScriptL]])\), RowBox[{ SubsuperscriptBox["P", "\[ScriptL]", "\[DoublePrime]", MultilineFunction->None], "[", "x", "]"}]}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}], ",", "\n", "\t", RowBox[{\(\[ScriptCapitalF]\_5\), "\[Rule]", RowBox[{\(\[Omega]\/q\), RowBox[{"Sum", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\((\[ScriptL] + 1)\), \(\(S\_+\)[\[ScriptL]]\), RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] + 1\), "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], " ", "-", " ", RowBox[{"\[ScriptL]", " ", \(\(S\_-\)[\[ScriptL]]\), RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] - 1\), "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], ")"}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}]}], ",", "\n", "\t", RowBox[{\(\[ScriptCapitalF]\_6\), "\[Rule]", RowBox[{\(\[Omega]\/q\), RowBox[{"Sum", "[", RowBox[{ RowBox[{"(", RowBox[{\((\[ScriptL]\ \(S\_-\)[\[ScriptL]] - \ \((\ \[ScriptL] + 1)\) \(S\_+\)[\[ScriptL]])\), RowBox[{ SubsuperscriptBox["P", "\[ScriptL]", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], ")"}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}]}]}], "\n", "\t\t", "}"}]}], ";"}]], "Input", CellLabel->"In[107]:="], Cell[TextData[{ "Note that is simplest to allow the sums to begin with ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL] = 0\)]], " and to eliminate nonphysical amplitudes later." }], "Text"], Cell[BoxData[ \(\(rule[ mp] = {\(M\_-\)[0] \[Rule] 0, \(E\_-\)[0] \[Rule] 0, \(S\_-\)[0] \[Rule] 0, \(M\_+\)[0] \[Rule] 0, \(E\_-\)[1] \[Rule] 0, \(0\^*\) \[Rule] 0};\)\)], "Input", CellLabel->"In[108]:="], Cell["\<\ It is useful to combine these rules with expansions for the complex \ conjugates. We include rules for expanding products of sums also.\ \>", "Text"], Cell[BoxData[ \(\(ExpandAmplitudeProducts = \ {\[IndentingNewLine]Abs[\[ScriptCapitalF]\_a_]\^2 \[Rule] \(\[ScriptCapitalF]\ \_a\) \(\((\[ScriptCapitalF]\_a)\)\^*\), \[IndentingNewLine]Re[\((A_. \ \[ScriptCapitalF]\_a_ + B_. \ \[ScriptCapitalF]\_b_)\)\ \(\((C_. \ \ \[ScriptCapitalF]\_c_\ + \ D_. \[ScriptCapitalF]\_d_)\)\^*\)] \[Rule] A\ C\ Re[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_c)\)\^*\ \)] + A\ D\ Re[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_d)\)\^*\)] + B\ C\ Re[\(\[ScriptCapitalF]\_b\) \ \(\((\[ScriptCapitalF]\_c)\)\^*\)] + B\ D\ Re[\(\[ScriptCapitalF]\_b\) \ \(\((\[ScriptCapitalF]\_d)\)\^*\)], Im[\((A_. \[ScriptCapitalF]\_a_ + B_. \ \[ScriptCapitalF]\_b_)\)\ \(\((C_. \ \ \[ScriptCapitalF]\_c_\ + \ D_. \[ScriptCapitalF]\_d_)\)\^*\)] \[Rule] A\ C\ Im[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_c)\)\^*\ \)] + A\ D\ Im[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_d)\)\^*\)] + B\ C\ Im[\(\[ScriptCapitalF]\_b\) \ \(\((\[ScriptCapitalF]\_c)\)\^*\)] + B\ D\ Im[\(\[ScriptCapitalF]\_b\) \ \(\((\[ScriptCapitalF]\_d)\)\^*\)], Re[\((A_. \[ScriptCapitalF]\_a_ + B_. \ \[ScriptCapitalF]\_b_)\)\ \(\((C_. \ \ \[ScriptCapitalF]\_c_)\)\^*\)\ ] \[Rule] A\ C\ Re[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_c)\)\^*\ \)] + B\ C\ Re[\(\[ScriptCapitalF]\_b\) \(\((\[ScriptCapitalF]\_c)\)\^*\)], Im[\((A_. \[ScriptCapitalF]\_a_ + B_. \ \[ScriptCapitalF]\_b_)\)\ \(\((C_. \ \ \[ScriptCapitalF]\_c_)\)\^*\)\ ] \[Rule] A\ C\ Im[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_c)\)\^*\ \)] + B\ C\ Im[\(\[ScriptCapitalF]\_b\) \ \(\((\[ScriptCapitalF]\_c)\)\^*\)]};\)\)], "Input", CellLabel->"In[109]:="], Cell[BoxData[ \(\(ExpandCGLN\ = \ Join[{Sum[A_, {\[ScriptL]1_, \[ScriptL]1min_, \[ScriptL]1max_}] Sum[B_, {\[ScriptL]2_, \[ScriptL]2min_, \[ScriptL]2max_}] \ \[RuleDelayed] MySum[Expand[ A\ B], {\[ScriptL]1, \[ScriptL]1min, \[ScriptL]1max}, {\ \[ScriptL]2, \[ScriptL]2min, \[ScriptL]2max}]\ }, CGLN1[\[ScriptL]1], CGLN1[\[ScriptL]2] /. {\[ScriptCapitalF]\_a_ \[Rule] \(\((\ \[ScriptCapitalF]\_a)\)\^*\), \(M\_+\)[ a_] \[Rule] \(\((\(M\_+\)[a])\)\^*\), \(E\_+\)[ a_] \[Rule] \(\((\(E\_+\)[a])\)\^*\), \(S\_+\)[ a_] \[Rule] \(\((\(S\_+\)[a])\)\^*\), \(M\_-\)[ a_] \[Rule] \(\((\(M\_-\)[a])\)\^*\), \(E\_-\)[ a_] \[Rule] \(\((\(E\_-\)[a])\)\^*\), \(S\_-\)[ a_] \[Rule] \(\((\(S\_-\)[a])\)\^*\)}];\)\)], "Input", CellLabel->"In[110]:="], Cell["\<\ The following rules expand the Legendre polynomials and their derivatives.\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"ExpandLegendre", " ", "=", " ", RowBox[{"{", RowBox[{\(P\_\[Lambda]_[x] \[Rule] LegendreP[\[Lambda], x]\), ",", RowBox[{ RowBox[{ SubsuperscriptBox["P", "\[Lambda]_", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Rule]", \(D[LegendreP[\[Lambda], x], x]\)}], ",", RowBox[{ RowBox[{ SubsuperscriptBox["P", "\[Lambda]_", "\[DoublePrime]", MultilineFunction->None], "[", "x", "]"}], "\[Rule]", \(D[LegendreP[\[Lambda], x], x, x]\)}]}], "}"}]}], ";"}]], "Input", CellLabel->"In[111]:="], Cell[TextData[{ "The following rules are designed to expand expressions to the form ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\(a\_i\) f[A\ \(B\^*\)]\)]], " where ", Cell[BoxData[ \(TraditionalForm\`f\)]], " is ", Cell[BoxData[ FormBox[ StyleBox["Re", FontSlant->"Italic"], TraditionalForm]]], ", ", StyleBox["Im", FontSlant->"Italic"], ", or ", StyleBox["Abs", FontSlant->"Italic"], " and where amplitudes products ", Cell[BoxData[ \(TraditionalForm\`A\ \(B\^*\)\)]], " are unique with a prescribed ordering hierarchy." }], "Text"], Cell[BoxData[ \(AnyMP = \(M\_-\) | \(M\_+\) | \(E\_-\) | \(E\_+\) | \(S\_-\) | \ \(S\_+\); \(AnyMP\_-\) = \(M\_-\) | \(E\_-\) | \(S\_-\); \(AnyMP\_+\) = \ \(M\_+\) | \(E\_+\) | \(S\_+\);\)], "Input", CellLabel->"In[112]:="], Cell[BoxData[ \(\(ExpandMultipoleProducts = {\[IndentingNewLine]\(\((a_ \((x : AnyMP)\)[ m_])\)\^*\) \[Rule] a\ \(\((x[m])\)\^*\), \[IndentingNewLine]Re[ a_. \ \ x_\ \(\((x_)\)\^*\)] \[Rule] a\ Abs[x]\^2, \ Im[a_. \ x_\ \(\((x_)\)\^*\)] \[Rule] 0, \ Abs[f_[\[Theta]]]\^2 \[Rule] f[\[Theta]]\^2, Abs[a_. \ \((b_\ + \ c_)\)]\^2 \[Rule] \(Abs[ a]\^2\) \((Abs[b]\^2 + Abs[c]\^2 + 2 Re[b\ \(c\^*\)])\), Abs[a_. \ \((x : \((AnyMP)\))\)[m_]]\^2 \[Rule] \(a\^2\) Abs[x[m]]\^2, \[IndentingNewLine]\((f : \((Re | Im)\))\)[ a_. \ \((b_\ + \ c_)\)] \[Rule] f[a\ b] + f[a\ c], \((f : \((Re | Im)\))\)[ a_. \ \(\((\((b_\ + \ c_)\) d_. )\)\^*\)] \[Rule] f[a\ \(\((b\ d)\)\^*\)] + f[a\ \(\((c\ d)\)\^*\)], \((f : \((Re | Im)\))\)[ a_\ \((x : AnyMP)\)[ m_] \(\((\((y : AnyMP)\)[n_])\)\^*\)] \[Rule] a\ f[\(\((y[n])\)\^*\) x[m]], Re[\(\((\((x : \((\(M\_-\) | \(M\_+\))\))\)[ m_])\)\^*\) \((y : \((\(E\_-\) | \(E\_+\) | \(S\_-\) \ | \(S\_+\))\))\)[n_]] \[Rule] Re[\(\((y[n])\)\^*\) x[m]], Im[\(\((\((x : \((\(M\_-\) | \(M\_+\))\))\)[ m_])\)\^*\) \((y : \((\(E\_-\) | \(E\_+\) | \(S\_-\) \ | \(S\_+\))\))\)[n_]] \[Rule] \(-Im[\(\((y[n])\)\^*\) x[m]]\), Re[\(\((\((x : \((\(E\_-\) | \(E\_+\))\))\)[ m_])\)\^*\) \((y : \((\(S\_-\) | \(S\_+\))\))\)[ n_]] \[Rule] Re[\(\((y[n])\)\^*\) x[m]], Im[\(\((\((x : \((\(E\_-\) | \(E\_+\))\))\)[ m_])\)\^*\) \((y : \((\(S\_-\) | \(S\_+\))\))\)[ n_]] \[Rule] \(-Im[\(\((y[n])\)\^*\) x[m]]\), \ Re[\(\((x : \(M\_+\)[m_])\)\^*\) \(y : \(M\_-\)[n_]\)] \[Rule] Re[\(y\^*\) x], \[IndentingNewLine]Im[\(\((x : \(M\_+\)[ m_])\)\^*\) \(y : \(M\_-\)[ n_]\)] \[Rule] \(-Im[\(y\^*\) x]\), \[IndentingNewLine]Re[\(\((x : \(E\_+\)[ m_])\)\^*\) \(y : \(E\_-\)[n_]\)] \[Rule] Re[\(y\^*\) x], \[IndentingNewLine]Im[\(\((x : \(E\_+\)[ m_])\)\^*\) \(y : \(E\_-\)[ n_]\)] \[Rule] \(-Im[\(y\^*\) x]\), \[IndentingNewLine]Re[\(\((x : \(S\_+\)[ m_])\)\^*\) \(y : \(S\_-\)[n_]\)] \[Rule] Re[\(y\^*\) x], \[IndentingNewLine]Im[\(\((x : \(S\_+\)[ m_])\)\^*\) \(y : \(S\_-\)[ n_]\)] \[Rule] \(-Im[\(y\^*\) x]\), Re[\((x : AnyMP)\)[m_]\ \(\((\((x : AnyMP)\)[n_])\)\^*\)] /; \((n < m)\) \[Rule] Re[\(x[m]\^*\) x[n]], Im[\((x : AnyMP)\)[m_]\ \(\((\((x : AnyMP)\)[n_])\)\^*\)] /; \((n < m)\) \[Rule] \(-Im[\(x[m]\^*\) x[n]]\)\n};\)\)], "Input", CellLabel->"In[113]:="] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Functions which perform multipole expansion of response functions\ \>", "Subsection"], Cell[TextData[{ "To obtain tractable expressions, it is necessary to specify the maximum \ angular momentum. The following function constructs a multipole expansion \ for a specific response function and attempts to perform simplification. It \ is useful to express the expansions in terms of ", Cell[BoxData[ \(TraditionalForm\`x = Cos[\[Theta]\_\[Pi]]\)]], " and for some response functions it is helpful to extract factors of \ either ", Cell[BoxData[ \(TraditionalForm\`Sin[\[Theta]\_\[Pi]]\)]], " or ", Cell[BoxData[ \(TraditionalForm\`Sin[\[Theta]\_\[Pi]]\^2\)]], "." }], "Text"], Cell[BoxData[ \(ExpandR[R\_\[Alpha]_[a_], lmax_Integer /; \((lmax \[GreaterEqual] 0)\), n_Integer: 0] := \ Module[{\[ScriptCapitalR]}, \n\t\t\[ScriptCapitalR] = \ \(TrigExpand[\(\((\(\(R\_\[Alpha][a] /. R\_\[Alpha][\[ScriptCapitalF]]\) //. ExpandCGLN\) /. {\[ScriptL]max \[Rule] lmax, MySum \[Rule] Sum})\) //. ExpandMultipoleProducts\) //. rule[mp]] /. {Cos[\[Theta]] \[Rule] x, Sin[\[Theta]] \[Rule] \@\(1 - x\^2\)}\) /. ExpandLegendre; \[IndentingNewLine]Collect[ MySimplify[\(\(Sin[\[Theta]]\^n\) \[ScriptCapitalR]\)\/\((1 - x\^2)\ \)\^\(n/2\)], {Sin[\[Theta]], x}, MySimplify]]\)], "Input", CellLabel->"In[114]:="], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_T[0], 1]\)], "Input", CellLabel->"In[115]:="], Cell[BoxData[ \(Abs[\(E\_+\)[0]]\^2 + 9\/2\ Abs[\(E\_+\)[1]]\^2 + Abs[\(M\_-\)[1]]\^2 + 5\/2\ Abs[\(M\_+\)[1]]\^2 + x\ \((\(-2\)\ Re[\(M\_-\)[1]\ \(\(E\_+\)[0]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) + 3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 3\/2\ x\^2\ \((3\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 - 6\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\) + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)]\)], "Output", CellLabel->"Out[115]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify /@ CoefficientList[ExpandR[R\_T[0], 1], x]\)], "Input", CellLabel->"In[116]:="], Cell[BoxData[ \({Abs[\(E\_+\)[0]]\^2 + 9\/2\ Abs[\(E\_+\)[1]]\^2 + Abs[\(M\_-\)[1]]\^2 + 5\/2\ Abs[\(M\_+\)[1]]\^2 + 3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[ 1]\ \(\(M\_-\)[1]\^*\)], \(-2\)\ Re[\(M\_-\)[ 1]\ \(\(E\_+\)[0]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)], 3\/2\ \((3\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 - 6\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)}\)], "Output", CellLabel->"Out[116]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_TT[0], 1, 2]\)], "Input", CellLabel->"In[117]:="], Cell[BoxData[ \(1\/2\ \((9\ Abs[\(E\_+\)[1]]\^2 - 3\ \((Abs[\(M\_+\)[1]]\^2 + 2\ \((\(-Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)]\) + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\))\)\ Sin[\[Theta]]\^2\)], \ "Output", CellLabel->"Out[117]="] }, Open ]], Cell["\<\ The following function eliminates terms which do not involve particular \ multipoles. \ \>", "Text"], Cell[BoxData[{ \(\(Clear[MyFreeQ];\)\), "\n", \(\(MyFreeQ[x_, y_List] := And @@ \((\(FreeQ[x, #] &\) /@ y\ )\);\)\), "\n", \(MyFreeQ[x_, y_] := FreeQ[x, y]\)}], "Input", CellLabel->"In[118]:="], Cell[BoxData[ \(AbbreviateMultipoleExpansion[expr_, choices_] := Collect[MySimplify[\((If[MyFreeQ[#, choices], 0, #] &)\) /@ \ Expand[expr]], x, MySimplify]\)], "Input", CellLabel->"In[121]:="], Cell[TextData[{ "For example, we can enforce ", Cell[BoxData[ \(TraditionalForm\`\(M\_+\)[1]\)]], " dominance for \[Pi] production near ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\)]], " resonance." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(AbbreviateMultipoleExpansion[\ ExpandR[R\_T[0], 1], \(M\_+\)[1]]\)], "Input", CellLabel->"In[122]:="], Cell[BoxData[ \(5\/2\ Abs[\(M\_+\)[1]]\^2 + 2\ x\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)] - 3\/2\ x\^2\ \((Abs[\(M\_+\)[1]]\^2 - 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\)], "Output", CellLabel->"Out[122]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Results for selected response functions", "Subsection"], Cell[CellGroupData[{ Cell[TextData[{ "Expansions through ", StyleBox["s", FontSlant->"Italic"], " and ", StyleBox["p", FontSlant->"Italic"], " waves" }], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_L[0], 1]\)], "Input", CellLabel->"In[123]:="], Cell[BoxData[ \(\(2\ Q\^2\ x\ \((Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[0]] + 4\ \ Re[\(\(S\_+\)[1]\^*\)\ \(S\_+\)[0]])\)\)\/q\^2 + \(Q\^2\ \ \((Abs[\(S\_-\)[1]]\^2 + Abs[\(S\_+\)[0]]\^2 + 4\ Abs[\(S\_+\)[1]]\^2 - 4\ \ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[1]])\)\)\/q\^2 + \(12\ Q\^2\ x\^2\ \((Abs[\(S\ \_+\)[1]]\^2 + Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[1]])\)\)\/q\^2\)], "Output", CellLabel->"Out[123]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_T[0], 1]\)], "Input", CellLabel->"In[124]:="], Cell[BoxData[ \(Abs[\(E\_+\)[0]]\^2 + 9\/2\ Abs[\(E\_+\)[1]]\^2 + Abs[\(M\_-\)[1]]\^2 + 5\/2\ Abs[\(M\_+\)[1]]\^2 + x\ \((\(-2\)\ Re[\(M\_-\)[1]\ \(\(E\_+\)[0]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) + 3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 3\/2\ x\^2\ \((3\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 - 6\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\) + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)]\)], "Output", CellLabel->"Out[124]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_TT[0], 1, 2]\)], "Input", CellLabel->"In[125]:="], Cell[BoxData[ \(1\/2\ \((9\ Abs[\(E\_+\)[1]]\^2 - 3\ \((Abs[\(M\_+\)[1]]\^2 + 2\ \((\(-Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)]\) + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\))\)\ Sin[\[Theta]]\^2\)], \ "Output", CellLabel->"Out[125]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_LT[0], 1, 1]\)], "Input", CellLabel->"In[126]:="], Cell[BoxData[ \(\((\(-\(\(1\/q\)\((\@2\ Q\ \((Re[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] + 3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ Re[\(E\_+\)[ 0]\ \(\(S\_+\)[ 1]\^*\)])\))\)\)\) - \(\(1\/q\)\((6\ \@2\ Q\ \ x\ \((Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[126]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_LTh[0], 1, 1]\)], "Input", CellLabel->"In[127]:="], Cell[BoxData[ \(\((\(\(1\/q\)\((\@2\ Q\ \((Im[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] + 3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] - Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ Im[\(E\_+\)[ 0]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) + \(\(1\/q\)\((6\ \@2\ Q\ x\ \ \((Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - Im[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[127]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpR\_LTh[N] = ExpandR[R\_LTh[N], 1]\)], "Input", CellLabel->"In[128]:="], Cell[BoxData[ \(\(\(1\/q\)\((\@2\ Q\ x\ \((Re[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] + 3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 4\ Re[\(E\_+\)[ 0]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) + \(\(1\/q\)\((\@2\ Q\ \((\(-3\)\ \ Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 2\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 2\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) + \(\(1\/q\)\((6\ \@2\ Q\ x\^2\ \ \((Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\))\)\)\)], "Output", CellLabel->"Out[128]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpR\_LTh[L] = ExpandR[R\_LTh[L], 1, 1]\)], "Input", CellLabel->"In[129]:="], Cell[BoxData[ \(\((\(-\(\(3\ \@2\ Q\ x\ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ Re[\(E\_+\)[ 0]\ \(\(S\_+\)[ 1]\^*\)])\)\)\/q\)\) - \(\(1\/q\)\((\@2\ Q\ \ \((Re[\(M\_-\)[1]\ \(\(S\_-\)[1]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] - 2\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 4\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) - \(18\ \@2\ Q\ x\^2\ \ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\ \)\)\/q)\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[129]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpR\_LTh[S] = ExpandR[R\_LTh[S], 1]\)], "Input", CellLabel->"In[130]:="], Cell[BoxData[ \(\(3\ \@2\ Q\ x\^2\ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + \ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\ \)\)\/q + \(\(1\/q\)\((\@2\ Q\ \((Re[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ \((Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(E\_+\)[ 0]\ \(\(S\_+\)[ 1]\^*\)])\))\))\)\) + \(\(1\/q\)\((\@2\ Q\ x\ \ \((3\ Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] - 6\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 4\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 14\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) + \(18\ \@2\ Q\ x\^3\ \((Re[\(E\_+\ \)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\)\/q\)], \ "Output", CellLabel->"Out[130]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpR\_TTh[L] = ExpandR[R\_TTh[L], 1]\)], "Input", CellLabel->"In[131]:="], Cell[BoxData[ \(2\ Re[\(M\_-\)[1]\ \(\(E\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] - 3\ x\^2\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) - 18\ x\^3\ \((Abs[\(E\_+\)[1]]\^2 + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)])\) + x\ \((\(-Abs[\(E\_+\)[0]]\^2\) + 9\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_-\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 + 6\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 12\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\)], "Output", CellLabel->"Out[131]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpR\_TTh[S] = ExpandR[R\_TTh[S], 1, 1] // FullSimplify\)], "Input", CellLabel->"In[132]:="], Cell[BoxData[ \(\((\(-Abs[\(E\_+\)[0]]\^2\) + Abs[\(M\_-\)[1]]\^2 - 2\ Abs[\(M\_+\)[1]]\^2 + 3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 3\ x\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] + 6\ x\ \((Abs[\(E\_+\)[1]]\^2 + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)])\))\) + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[132]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_L[N], 1, 1] // FullSimplify\)], "Input", CellLabel->"In[133]:="], Cell[BoxData[ \(\(2\ Q\^2\ \((Im[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[0]] - 2\ Im[\(\(S\_+\)[1]\ \^*\)\ \(S\_+\)[0]] + 6\ x\ Im[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[1]])\)\ Sin[\ \[Theta]]\)\/q\^2\)], "Output", CellLabel->"Out[133]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_T[N], 1, 1] // FullSimplify\)], "Input", CellLabel->"In[134]:="], Cell[BoxData[ \(\((2\ Im[\(M\_-\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 3\ \((Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] - 3\ x\ Im[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + x\ Im[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[134]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_LT[N], 1] // FullSimplify\)], "Input", CellLabel->"In[135]:="], Cell[BoxData[ \(\(\(1\/q\)\((\@2\ Q\ \((\(-3\)\ Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] + 6\ Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 2\ Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 2\ Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + x\ \((Im[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] + 3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 4\ Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)] + 6\ x\ \((Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\))\))\)\)\)], "Output", CellLabel->"Out[135]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_LT[L], 1, 1] // FullSimplify\)], "Input", CellLabel->"In[136]:="], Cell[BoxData[ \(\(\(1\/q\)\((\@2\ Q\ \((Im[\(M\_-\)[1]\ \(\(S\_-\)[1]\^*\)] + 2\ Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] - 2\ \((Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + 2\ Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\) + 3\ x\ \((Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ \((Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)] + 3\ x\ \((Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\))\))\)\ \ Sin[\[Theta]])\)\)\)], "Output", CellLabel->"Out[136]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_LT[S], 1, 0] // FullSimplify\)], "Input", CellLabel->"In[137]:="], Cell[BoxData[ \(\(-\(\(1\/q\)\((\@2\ Q\ \((Im[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ \((Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\) + x\ \((3\ Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] - 2\ \((3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 2\ Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + 7\ Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\) + 3\ x\ \((Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ \((Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)] + 3\ x\ \((Im[\(E\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\))\))\))\))\)\)\)\)], \ "Output", CellLabel->"Out[137]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_TT[N], 1, 1] // FullSimplify\)], "Input", CellLabel->"In[138]:="], Cell[BoxData[ \(\((\(-3\)\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) + 3\ x\ \((Im[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 4\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Im[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[138]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_TT[L], 1, 2] // FullSimplify\)], "Input", CellLabel->"In[139]:="], Cell[BoxData[ \(\(-3\)\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] + 6\ x\ Im[\(M\_+\)[ 1]\ \(\(E\_+\)[1]\^*\)])\)\ Sin[\[Theta]]\^2\)], "Output", CellLabel->"Out[139]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_TT[S], 1, 1] // FullSimplify\)], "Input", CellLabel->"In[140]:="], Cell[BoxData[ \(\((3\ x\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] + 6\ x\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)])\) - 3\ \((Im[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 2\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Im[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[140]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`P\_33\)]], " dominance" }], "Subsubsection"], Cell[TextData[{ "Here we display expansions based upon ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\)]], " dominance, retaining only ", StyleBox["s", FontSlant->"Italic"], " and ", StyleBox["p", FontSlant->"Italic"], " waves." }], "Text"], Cell[BoxData[ \(mpBrief[P\_33, R_, n_Integer: 0] := AbbreviateMultipoleExpansion[ ExpandR[R, 1, n], {\(M\_+\)[1], \(E\_+\)[1], \(S\_+\)[1]}]\)], "Input",\ CellLabel->"In[141]:="], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_L[0]]\)], "Input", CellLabel->"In[142]:="], Cell[BoxData[ \(\(8\ Q\^2\ x\ Re[\(\(S\_+\)[1]\^*\)\ \(S\_+\)[0]]\)\/q\^2 + \(4\ Q\^2\ \ \((Abs[\(S\_+\)[1]]\^2 - Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[1]])\)\)\/q\^2 + \ \(12\ Q\^2\ x\^2\ \((Abs[\(S\_+\)[1]]\^2 + Re[\(\(S\_-\)[1]\^*\)\ \ \(S\_+\)[1]])\)\)\/q\^2\)], "Output", CellLabel->"Out[142]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_T[0]]\)], "Input", CellLabel->"In[143]:="], Cell[BoxData[ \(9\/2\ Abs[\(E\_+\)[1]]\^2 + 5\/2\ Abs[\(M\_+\)[1]]\^2 + 2\ x\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) + 3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 3\/2\ x\^2\ \((3\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 - 6\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\) + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)]\)], "Output", CellLabel->"Out[143]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_LT[0], 1]\)], "Input", CellLabel->"In[144]:="], Cell[BoxData[ \(\(-\(\(\@2\ Q\ \((3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ Re[\(E\_+\)[ 0]\ \(\(S\_+\)[ 1]\^*\)])\)\ Sin[\[Theta]]\)\/q\)\) - \ \(\(1\/q\)\((6\ \@2\ Q\ x\ \((Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\)\ Sin[\[Theta]])\)\)\)], "Output", CellLabel->"Out[144]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_TT[0], 2]\)], "Input", CellLabel->"In[145]:="], Cell[BoxData[ \(3\/2\ \((3\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 - 2\ \((\(-Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)]\) + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\^2\)], "Output", CellLabel->"Out[145]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_TT[S], 1] // Simplify\)], "Input", CellLabel->"In[146]:="], Cell[BoxData[ \(3\ \((x\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) - Im[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 2\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\ x\^2\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - Im[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[146]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_LT[N]]\)], "Input", CellLabel->"In[147]:="], Cell[BoxData[ \(\(\@2\ Q\ x\ \((3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \ \(\(S\_+\)[0]\^*\)] + 4\ Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\)\)\/q + \ \(\(1\/q\)\((\@2\ Q\ \((\(-3\)\ Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 6\ Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 2\ Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 2\ Im[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) + \(\(1\/q\)\((6\ \@2\ Q\ x\^2\ \ \((Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\))\)\)\)], "Output", CellLabel->"Out[147]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_LT[S]]\)], "Input", CellLabel->"In[148]:="], Cell[BoxData[ \(\(2\ \@2\ Q\ \((Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \ \(\(S\_+\)[1]\^*\)])\)\)\/q - \(3\ \@2\ Q\ x\^2\ \((Im[\(E\_+\)[1]\ \ \(\(S\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ Im[\(E\_+\)[0]\ \ \(\(S\_+\)[1]\^*\)])\)\)\/q - \(18\ \@2\ Q\ x\^3\ \((Im[\(E\_+\)[1]\ \ \(\(S\_+\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\)\/q - \ \(\(1\/q\)\((\@2\ Q\ x\ \((3\ Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - 2\ \((3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 2\ Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + 7\ Im[\(M\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)])\))\))\)\)\)], "Output", CellLabel->"Out[148]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_LTh[S]]\)], "Input", CellLabel->"In[149]:="], Cell[BoxData[ \(\(-\(\(2\ \@2\ Q\ \((Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(E\_+\)[ 0]\ \(\(S\_+\)[ 1]\^*\)])\)\)\/q\)\) + \(3\ \@2\ Q\ x\^2\ \ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] \ + 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\)\)\/q + \(18\ \@2\ Q\ x\^3\ \((Re[\ \(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \ \(\(S\_+\)[1]\^*\)])\)\)\/q + \(\(1\/q\)\((\@2\ Q\ x\ \((3\ Re[\(E\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - 2\ \((3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 2\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + 7\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)])\))\))\)\)\)], "Output", CellLabel->"Out[149]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_LTh[L], 1] // Simplify\)], "Input", CellLabel->"In[150]:="], Cell[BoxData[ \(\(-\(\(1\/q\)\((\@2\ Q\ \((2\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 3\ x\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 3\ x\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 6\ x\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)] + 18\ x\^2\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 2\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 4\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 18\ x\^2\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\)\ Sin[\[Theta]])\)\)\)\)], "Output", CellLabel->"Out[150]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_TTh[S], 1] // Simplify\)], "Input", CellLabel->"In[151]:="], Cell[BoxData[ \(\((\(-2\)\ Abs[\(M\_+\)[1]]\^2 - 3\ x\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) + 3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 18\ x\^2\ \((Abs[\(E\_+\)[1]]\^2 + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)])\) + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[151]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_TTh[L]]\)], "Input", CellLabel->"In[152]:="], Cell[BoxData[ \(Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] - 3\ x\^2\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) - 18\ x\^3\ \((Abs[\(E\_+\)[1]]\^2 + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)])\) + x\ \((9\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 + 2\ \((3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\))\)\)], "Output", CellLabel->"Out[152]="] }, Open ]], Cell[TextData[{ "It is also useful to examine severely truncated multipole expansions for \ parallel versus antiparallel kinematics. Here we assume ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance and define a few functions which facilitate comparisons between \ parallel (", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_\[Pi] = \[Pi]\)]], ") and antiparallel (", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_\[Pi] = 0\)]], ") kinematics." }], "Text"], Cell[BoxData[ \(mpVeryBrief[P\_33, R_, n_Integer: 0] := AbbreviateMultipoleExpansion[ ExpandR[R, 1, n], {\(M\_+\)[1]}]\)], "Input", CellLabel->"In[153]:="], Cell[BoxData[ \(fbsum[P\_33, R_, n_Integer: 0] := \n\t\t\((mpVeryBrief[P\_33, R, n] + mpVeryBrief[P\_33, R, n] /. x \[Rule] \(-x\))\) // Simplify\)], "Input", CellLabel->"In[154]:="], Cell[BoxData[ \(fbdiff[P\_33, R_, n_Integer: 0] := \n\t\t\((mpVeryBrief[P\_33, R, n] - mpVeryBrief[P\_33, R, n] /. x \[Rule] \(-x\))\) // Simplify\)], "Input", CellLabel->"In[155]:="], Cell[BoxData[ \(fb[P\_33, R_, n_Integer: 0] := Module[{f, b}, \n\t\tf = mpVeryBrief[P\_33, R, n] /. x \[Rule] \(-x\); \n\t\tb = mpVeryBrief[P\_33, R, n]; \n\t\t\((f - b)\)/\((f + b)\) // Simplify]\)], "Input", CellLabel->"In[156]:="], Cell[TextData[{ "Selected response functions for ", Cell[BoxData[ \(TraditionalForm\`M\_\(1 + \)\)]], " dominance are given below." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_T[0]]\)], "Input", CellLabel->"In[157]:="], Cell[BoxData[ \(5\/2\ Abs[\(M\_+\)[1]]\^2 + 2\ x\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)] - 3\/2\ x\^2\ \((Abs[\(M\_+\)[1]]\^2 - 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\)], "Output", CellLabel->"Out[157]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_LT[0], 1] // FullSimplify\)], "Input", CellLabel->"In[158]:="], Cell[BoxData[ \(\(\@2\ Q\ \((Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 6\ x\ Re[\(M\_+\)[1]\ \ \(\(S\_+\)[1]\^*\)])\)\ Sin[\[Theta]]\)\/q\)], "Output", CellLabel->"Out[158]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_TT[0], 2] // FullSimplify\)], "Input", CellLabel->"In[159]:="], Cell[BoxData[ \(\(-\(3\/2\)\)\ \((Abs[\(M\_+\)[1]]\^2 + 2\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\^2\)], "Output", CellLabel->"Out[159]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_LT[N]] // FullSimplify\)], "Input", CellLabel->"In[160]:="], Cell[BoxData[ \(\(\@2\ Q\ \((Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + x\ Im[\(M\_+\)[1]\ \ \(\(S\_+\)[0]\^*\)] + 2\ \((\(-1\) + 3\ x\^2)\)\ Im[\(M\_+\)[1]\ \ \(\(S\_+\)[1]\^*\)])\)\)\/q\)], "Output", CellLabel->"Out[160]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_LTh[S]] // FullSimplify\)], "Input", CellLabel->"In[161]:="], Cell[BoxData[ \(\(\(1\/q\)\((\@2\ Q\ \((x\ Re[\(M\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] + \((\(-2\) + 3\ x\^2)\)\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ x\ \((\(-7\) + 9\ x\^2)\)\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)])\))\)\)\)], "Output", CellLabel->"Out[161]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_TTh[S], 1] // FullSimplify\)], "Input", CellLabel->"In[162]:="], Cell[BoxData[ \(\((\(-2\)\ Abs[\(M\_+\)[1]]\^2 - 3\ x\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + \((6 - 18\ x\^2)\)\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[162]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_LTh[L], 1] // FullSimplify\)], "Input", CellLabel->"In[163]:="], Cell[BoxData[ \(\(-\(\(1\/q\)\((\@2\ Q\ \((2\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 3\ x\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ \((\(-2\) + 9\ x\^2)\)\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\)\ Sin[\[Theta]])\)\)\)\)], "Output", CellLabel->"Out[163]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_TTh[L]] // FullSimplify\)], "Input", CellLabel->"In[164]:="], Cell[BoxData[ \(\(-x\)\ Abs[\(M\_+\)[1]]\^2 + \((1 - 3\ x\^2)\)\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 2\ x\ \((\((6 - 9\ x\^2)\)\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\)], "Output", CellLabel->"Out[164]="] }, Open ]], Cell[TextData[{ "Observing that the ", Cell[BoxData[ \(TraditionalForm\`S\_\(\(0\)\(+\)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`S\_\(\(1\)\(+\)\)\)]], " contributions to ", Cell[BoxData[ \(TraditionalForm\`R\_LT\)]], " have opposite symmetries with respect to \[Theta]\[Rule]\[Pi]-\[Theta], \ we can separate those terms using" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fbsum[P\_33, R\_LT[0], 1] // FullSimplify\)], "Input", CellLabel->"In[165]:="], Cell[BoxData[ \(\(2\ \@2\ Q\ \((Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 6\ x\ \ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ Sin[\[Theta]]\)\/q\)], "Output", CellLabel->"Out[165]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fbdiff[P\_33, R\_LT[0]]\)], "Input", CellLabel->"In[166]:="], Cell[BoxData[ \(0\)], "Output", CellLabel->"Out[166]="] }, Open ]], Cell[TextData[{ "Similarly, observing that the ", Cell[BoxData[ \(TraditionalForm\`S\_\(0 + \)\)]], " contribution to ", Cell[BoxData[ \(TraditionalForm\`R\_LTh[S]\)]], " changes sign relative to the ", Cell[BoxData[ \(TraditionalForm\`S\_\(1 - \)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`S\_\(1 + \)\)]], " terms" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_LTh[S]] /. x \[Rule] \(-x\)\)], "Input", CellLabel->"In[167]:="], Cell[BoxData[ \(\(-\(\(2\ \@2\ Q\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 0]\^*\)]\)\/q\)\) + \(3\ \@2\ Q\ x\^2\ Re[\(M\_+\)[1]\ \ \(\(S\_+\)[0]\^*\)]\)\/q - \(\@2\ Q\ x\ \((Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\ \)] - 14\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\)\/q - \(18\ \@2\ Q\ x\^3\ \ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\)\/q\)], "Output", CellLabel->"Out[167]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_LTh[S]]\)], "Input", CellLabel->"In[168]:="], Cell[BoxData[ \(\(-\(\(2\ \@2\ Q\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 0]\^*\)]\)\/q\)\) + \(3\ \@2\ Q\ x\^2\ Re[\(M\_+\)[1]\ \ \(\(S\_+\)[0]\^*\)]\)\/q + \(\@2\ Q\ x\ \((Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\ \)] - 14\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\)\/q + \(18\ \@2\ Q\ x\^3\ \ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\)\/q\)], "Output", CellLabel->"Out[168]="] }, Open ]], Cell[TextData[{ "we find that the ", Cell[BoxData[ \(TraditionalForm\`S\_\(0 + \)\)]], " contribution can be extracted by averaging ", Cell[BoxData[ \(TraditionalForm\`R\_LTh[S]\)]], " for parallel and antiparallel kinematics." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fbsum[P\_33, R\_LTh[S]] // FullSimplify\)], "Input", CellLabel->"In[169]:="], Cell[BoxData[ \(\(-\(\(1\/q\)\((2\ \@2\ Q\ \((x\ Re[\(M\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] + \((2 - 3\ x\^2)\)\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ x\ \((\(-7\) + 9\ x\^2)\)\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)])\))\)\)\)\)], "Output", CellLabel->"Out[169]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fbdiff[P\_33, R\_LTh[S]]\)], "Input", CellLabel->"In[170]:="], Cell[BoxData[ \(0\)], "Output", CellLabel->"Out[170]="] }, Open ]], Cell["Finally,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_TTh[L]]\)], "Input", CellLabel->"In[171]:="], Cell[BoxData[ \(Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 3\ x\^2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 18\ x\^3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + x\ \((\(-Abs[\(M\_+\)[1]]\^2\) + 2\ \((6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\))\)\)], "Output", CellLabel->"Out[171]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fbsum[P\_33, R\_TTh[L]]\)], "Input", CellLabel->"In[172]:="], Cell[BoxData[ \(2\ \((x\ Abs[\(M\_+\)[1]]\^2 + \((1 - 3\ x\^2)\)\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 2\ x\ \((\((\(-6\) + 9\ x\^2)\)\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\))\)\)], "Output", CellLabel->"Out[172]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fbdiff[P\_33, R\_TTh[L]]\)], "Input", CellLabel->"In[173]:="], Cell[BoxData[ \(0\)], "Output", CellLabel->"Out[173]="] }, Open ]], Cell[TextData[{ "the ", Cell[BoxData[ \(TraditionalForm\`E\_\(0 + \)\)]], " contribution can be obtained by averaging ", Cell[BoxData[ \(TraditionalForm\`R\_TTh[L]\)]], ". Alternatively, the following expressions" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fbsum[P\_33, R\_LTh[L], 1]\)], "Input", CellLabel->"In[174]:="], Cell[BoxData[ \(\(\(1\/q\)\((2\ \@2\ Q\ \((\(-2\)\ Re[\(M\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] + 3\ x\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ \((2 - 9\ x\^2)\)\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\)\ Sin[\[Theta]])\)\)\)], "Output", CellLabel->"Out[174]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fbsum[P\_33, R\_TTh[S], 1]\)], "Input", CellLabel->"In[175]:="], Cell[BoxData[ \(2\ \((\(-2\)\ Abs[\(M\_+\)[1]]\^2 + 3\ x\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 18\ x\^2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[175]="] }, Open ]], Cell["\<\ yield redundant determinations of these interference products; this \ redundancy can be used to test for and minimize model dependence.\ \>", "Text"], Cell["\<\ However, we must still assess the contributions of other multipoles more \ quantitatively.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(1\/2\) \((mpBrief[P\_33, R\_LTh[S], 0] + mpBrief[P\_33, R\_LTh[S]] /. x \[Rule] \(-x\))\) // Simplify\)], "Input", CellLabel->"In[176]:="], Cell[BoxData[ \(\(\(1\/q\)\((\@2\ Q\ \((\(-3\)\ x\ Re[\(E\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] - x\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 3\ x\^2\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 3\ x\^2\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)] + 6\ x\^2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)] + 6\ x\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 18\ x\^3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 4\ x\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + 14\ x\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 18\ x\^3\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\))\)\)\)], "Output",\ CellLabel->"Out[176]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(1\/2\) \((mpBrief[P\_33, R\_TTh[L], 0] + mpBrief[P\_33, R\_TTh[L]] /. x \[Rule] \(-x\))\) // Simplify\)], "Input", CellLabel->"In[177]:="], Cell[BoxData[ \(1\/2\ \((2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] - 6\ x\^2\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) + 36\ x\^3\ \((Abs[\(E\_+\)[1]]\^2 + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)])\) - 2\ x\ \((9\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 + 2\ \((3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[ 1]\ \(\(M\_-\)[1]\^*\)])\))\))\)\)], "Output", CellLabel->"Out[177]="] }, Open ]], Cell["\<\ More generally, it is advantageous to measure response functions at \ complementary angles in order to exploit the fact that even and odd partial \ waves have opposite symmetries with respect to \[Theta]\[Rule]\[Pi]-\[Theta].\ \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`S\_11\)]], " dominance" }], "Subsubsection"], Cell[TextData[{ "Here I compare multipole expansions for ", Cell[BoxData[ \(TraditionalForm\`S\_11\)]], " dominance with results from KDT, retaining just ", Cell[BoxData[ \(TraditionalForm\`S\_11\)]], ", ", Cell[BoxData[ \(TraditionalForm\`P\_11\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`D\_13\)]], " multipoles. Note that my ", Cell[BoxData[ \(TraditionalForm\`{S\&^, N\&^, L\&^}\)]], " correspond to their ", Cell[BoxData[ \(TraditionalForm \`{\(-x\&^\^\[Prime]\), y\&^\^\[Prime], \(-z\&^\^\[Prime]\)}\)]], " ." }], "Text"], Cell[BoxData[ \(mpBrief[S\_11, R_, n_Integer: 0] := AbbreviateMultipoleExpansion[ ExpandR[R, 2, n], {\(E\_+\)[0], \(S\_+\)[0]}]\)], "Input", CellLabel->"In[178]:="], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[S\_11, R\_L[0]] // FullSimplify\)], "Input", CellLabel->"In[179]:="], Cell[BoxData[ \(\(\(1\/q\^2\)\((Q\^2\ \((Abs[\(S\_+\)[0]]\^2 + 2\ x\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[0]] + 4\ \((\(-1\) + 3\ x\^2)\)\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[0]] + 8\ x\ Re[\(\(S\_+\)[1]\^*\)\ \(S\_+\)[0]] + 9\ \((\(-1\) + 3\ x\^2)\)\ Re[\(\(S\_+\)[2]\^*\)\ \(S\_+\)[ 0]])\))\)\)\)], "Output", CellLabel->"Out[179]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[S\_11, R\_T[0]]\)], "Input", CellLabel->"In[180]:="], Cell[BoxData[ \(Abs[\(E\_+\)[0]]\^2 - Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)] + 3\ Re[\(M\_-\)[2]\ \(\(E\_+\)[0]\^*\)] - 3\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)] + x\ \((\(-2\)\ Re[\(M\_-\)[1]\ \(\(E\_+\)[0]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) - 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[2]\^*\)] + 3\ x\^2\ \((Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)] - 3\ Re[\(M\_-\)[2]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[2]\^*\)])\)\)], "Output", CellLabel->"Out[180]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[S\_11, R\_LT[0], 1] // Simplify\)], "Input", CellLabel->"In[181]:="], Cell[BoxData[ \(\(-\(\(1\/q\)\((\@2\ Q\ \((Re[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] + 6\ x\ Re[\(E\_+\)[0]\ \(\(S\_-\)[2]\^*\)] - 3\ x\ Re[\(E\_-\)[2]\ \(\(S\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 12\ x\ Re[\(E\_+\)[2]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] + 3\ x\ Re[\(M\_-\)[2]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 3\ x\ Re[\(M\_+\)[2]\ \(\(S\_+\)[0]\^*\)] - 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)] - 9\ x\ Re[\(E\_+\)[ 0]\ \(\(S\_+\)[ 2]\^*\)])\)\ Sin[\[Theta]])\)\)\)\)], "Output", CellLabel->"Out[181]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[S\_11, R\_TT[0], 2]\)], "Input", CellLabel->"In[182]:="], Cell[BoxData[ \(\(-3\)\ \((Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)] + Re[\(M\_-\)[2]\ \(\(E\_+\)[0]\^*\)] - Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)] + Re[\(E\_+\)[ 0]\ \(\(E\_+\)[2]\^*\)])\)\ Sin[\[Theta]]\^2\)], "Output", CellLabel->"Out[182]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[S\_11, R\_LT[N]]\)], "Input", CellLabel->"In[183]:="], Cell[BoxData[ \(\(\(1\/q\)\((\@2\ Q\ x\ \((Im[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] + 3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 4\ Im[\(E\_+\)[ 0]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) - \(\(1\/\(\@2\ q\)\)\((Q\ \((4\ \ Im[\(E\_+\)[0]\ \(\(S\_-\)[2]\^*\)] + 4\ Im[\(E\_-\)[2]\ \(\(S\_+\)[0]\^*\)] - 2\ Im[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] + 9\ Im[\(E\_+\)[2]\ \(\(S\_+\)[0]\^*\)] + 9\ Im[\(E\_+\)[ 0]\ \(\(S\_+\)[ 2]\^*\)])\))\)\) + \(\(1\/\(\@2\ q\)\)\((3\ Q\ x\^2\ \ \((4\ Im[\(E\_+\)[0]\ \(\(S\_-\)[2]\^*\)] + 2\ Im[\(E\_-\)[2]\ \(\(S\_+\)[0]\^*\)] + 7\ Im[\(E\_+\)[2]\ \(\(S\_+\)[0]\^*\)] - 2\ Im[\(M\_-\)[2]\ \(\(S\_+\)[0]\^*\)] + 2\ Im[\(M\_+\)[2]\ \(\(S\_+\)[0]\^*\)] + 9\ Im[\(E\_+\)[0]\ \(\(S\_+\)[2]\^*\)])\))\)\)\)], "Output", CellLabel->"Out[183]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[S\_11, R\_LTh[S]]\)], "Input", CellLabel->"In[184]:="], Cell[BoxData[ \(\(3\ \@2\ Q\ x\^2\ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + \ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\ \)\)\/q + \(\(1\/q\)\((\@2\ Q\ \((Re[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ \((Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(E\_+\)[ 0]\ \(\(S\_+\)[ 1]\^*\)])\))\))\)\) + \(\(1\/\(\@2\ q\)\)\((Q\ \ x\ \((8\ Re[\(E\_+\)[0]\ \(\(S\_-\)[2]\^*\)] + 2\ Re[\(E\_-\)[2]\ \(\(S\_+\)[0]\^*\)] + 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] - 3\ Re[\(E\_+\)[2]\ \(\(S\_+\)[0]\^*\)] - 6\ Re[\(M\_-\)[2]\ \(\(S\_+\)[0]\^*\)] - 24\ Re[\(M\_+\)[2]\ \(\(S\_+\)[0]\^*\)] - 27\ Re[\(E\_+\)[ 0]\ \(\(S\_+\)[ 2]\^*\)])\))\)\) + \(15\ Q\ x\^3\ \((Re[\(E\_+\)[2]\ \ \(\(S\_+\)[0]\^*\)] + 2\ Re[\(M\_+\)[2]\ \(\(S\_+\)[0]\^*\)] + 3\ Re[\(E\_+\ \)[0]\ \(\(S\_+\)[2]\^*\)])\)\)\/\(\@2\ q\)\)], "Output", CellLabel->"Out[184]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[S\_11, R\_LTh[L], 1] // Simplify\)], "Input", CellLabel->"In[185]:="], Cell[BoxData[ \(\(\(1\/\(\@2\ q\)\)\((Q\ \((4\ Re[\(E\_+\)[0]\ \(\(S\_-\)[2]\^*\)] - 2\ Re[\(E\_-\)[2]\ \(\(S\_+\)[0]\^*\)] - 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] - 6\ x\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[2]\ \(\(S\_+\)[0]\^*\)] - 15\ x\^2\ Re[\(E\_+\)[2]\ \(\(S\_+\)[0]\^*\)] - 6\ Re[\(M\_-\)[2]\ \(\(S\_+\)[0]\^*\)] - 6\ x\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 6\ Re[\(M\_+\)[2]\ \(\(S\_+\)[0]\^*\)] - 30\ x\^2\ Re[\(M\_+\)[2]\ \(\(S\_+\)[0]\^*\)] - 12\ x\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)] + 9\ Re[\(E\_+\)[0]\ \(\(S\_+\)[2]\^*\)] - 45\ x\^2\ Re[\(E\_+\)[ 0]\ \(\(S\_+\)[ 2]\^*\)])\)\ Sin[\[Theta]])\)\)\)], "Output", CellLabel->"Out[185]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[S\_11, R\_TTh[S], 1] // Simplify\)], "Input", CellLabel->"In[186]:="], Cell[BoxData[ \(\((\(-Abs[\(E\_+\)[0]]\^2\) + Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)] - 3\ Re[\(M\_-\)[2]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)] - 3\ x\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) + 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[2]\^*\)] - 15\ x\^2\ \((Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)] + 2\ Re[\(E\_+\)[ 0]\ \(\(E\_+\)[ 2]\^*\)])\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[186]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[S\_11, R\_TTh[L]]\)], "Input", CellLabel->"In[187]:="], Cell[BoxData[ \(2\ Re[\(M\_-\)[1]\ \(\(E\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] - 3\ x\^2\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) - 15\ x\^3\ \((Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)] + 2\ Re[\(E\_+\)[0]\ \(\(E\_+\)[2]\^*\)])\) + x\ \((\(-Abs[\(E\_+\)[0]]\^2\) - 2\ Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)] + 6\ Re[\(M\_-\)[2]\ \(\(E\_+\)[0]\^*\)] + 9\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)] + 18\ Re[\(E\_+\)[0]\ \(\(E\_+\)[2]\^*\)])\)\)], "Output", CellLabel->"Out[187]="] }, Open ]] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Legendre expansions of response functions", "Section"], Cell[CellGroupData[{ Cell["expansion functions", "Subsection"], Cell[BoxData[{ \(\(ToLegendreP[x_, m_?EvenQ] := \ Sum[\(\(\(2\^\(2 n\)\) \((4 n + 1)\) \(m!\) \(\((m\/2 + n)\)!\)\)\/\(\(\((m + 2 n + 1)\)!\) \(\((m\/2 - n)\)!\)\)\) P\_\(2 n\)[x], {n, 0, m\/2}];\)\), "\[IndentingNewLine]", \(\(ToLegendreP[x_, m_?OddQ] := \ Sum[\(\(\(2\^\(2 n + 1\)\) \((4 n + 3)\) \(m!\) \(\((\(m + 1\)\/2 + n)\)!\)\)\/\(\(\((m + 2 n + 2)\)!\) \(\((\(m - 1\)\/2 - n)\)!\)\)\) P\_\(2 n + 1\)[x], {n, 0, m\/2 + 1}];\)\)}], "Input", CellLabel->"In[188]:="], Cell[BoxData[ \(\(xExpand\ := \ x\^n_. \[RuleDelayed] ToLegendreP[x, n];\)\)], "Input",\ CellLabel->"In[190]:="], Cell[BoxData[ \(RtoLegendre[sp, R_, n_] := Collect[MySimplify[\(ExpandR[R, 1, n] /. xExpand\) /. P\_0[x] \[Rule] 1], {Sin[\[Theta]], P\_\[Lambda]_[x]}, MySimplify]\)], "Input", CellLabel->"In[191]:="], Cell[BoxData[ \(RtoLegendre[P\_33, R_, n_] := Collect[MySimplify[\(AbbreviateMultipoleExpansion[ ExpandR[R, 1, n], {\(M\_+\)[1], \(E\_+\)[1], \(S\_+\)[1]}] /. xExpand\) /. P\_0[x] \[Rule] 1], {Sin[\[Theta]], P\_\[Lambda]_[x]}, MySimplify]\)], "Input", CellLabel->"In[192]:="], Cell[BoxData[ \(RtoLegendre[M1, R_, n_] := Collect[\(AbbreviateMultipoleExpansion[ ExpandR[R, 1, n], {\(M\_+\)[1]}] /. xExpand\) /. P\_0[x] \[Rule] 1, {Sin[\[Theta]], P\_\[Lambda]_[x]}, MySimplify]\)], "Input", CellLabel->"In[193]:="], Cell[BoxData[ \(RtoLegendre[R_, lmax_Integer, n_Integer] := Collect[MySimplify[\(ExpandR[R, lmax, n] /. xExpand\) /. P\_0[x] \[Rule] 1], {Sin[\[Theta]], P\_\[Lambda]_[x]}, MySimplify]\)], "Input", CellLabel->"In[194]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`s, p\)]], " expansion" }], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_L[0], 0]\)], "Input", CellLabel->"In[195]:="], Cell[BoxData[ \(\(Q\^2\ \((Abs[\(S\_-\)[1]]\^2 + Abs[\(S\_+\)[0]]\^2 + 8\ \ Abs[\(S\_+\)[1]]\^2)\)\)\/q\^2 + \(2\ Q\^2\ \((Re[\(\(S\_-\)[1]\^*\)\ \ \(S\_+\)[0]] + 4\ Re[\(\(S\_+\)[1]\^*\)\ \(S\_+\)[0]])\)\ P\_1[x]\)\/q\^2 + \ \(8\ Q\^2\ \((Abs[\(S\_+\)[1]]\^2 + Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[1]])\)\ P\ \_2[x]\)\/q\^2\)], "Output", CellLabel->"Out[195]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_T[0], 0]\)], "Input", CellLabel->"In[196]:="], Cell[BoxData[ \(Abs[\(E\_+\)[0]]\^2 + 6\ Abs[\(E\_+\)[1]]\^2 + Abs[\(M\_-\)[1]]\^2 + 2\ Abs[\(M\_+\)[1]]\^2 + \((\(-2\)\ Re[\(M\_-\)[ 1]\ \(\(E\_+\)[0]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\)\ P\_1[ x] + \((3\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 - 6\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ P\_2[x]\)], "Output", CellLabel->"Out[196]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_TT[0], 2] // Collect[#, {Sin[\[Theta]], P\_\[Lambda]_[x]}] &\)], "Input", CellLabel->"In[197]:="], Cell[BoxData[ \(1\/2\ \((9\ Abs[\(E\_+\)[1]]\^2 - 3\ \((Abs[\(M\_+\)[1]]\^2 + 2\ \((\(-Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)]\) + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\))\)\ Sin[\[Theta]]\^2\)], \ "Output", CellLabel->"Out[197]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_LT[0], 1]\)], "Input", CellLabel->"In[198]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(-\(\(1\/q\)\((\@2\ Q\ \((Re[\(E\_+\)[ 0]\ \(\(S\_-\)[1]\^*\)] + 3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ Re[\(E\_+\)[ 0]\ \(\(S\_+\)[ 1]\^*\)])\))\)\)\) - \(\(1\/q\)\((6\ \@2\ Q\ \ \((Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[ x])\)\))\)\)], "Output", CellLabel->"Out[198]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_LTh[0], 1]\)], "Input", CellLabel->"In[199]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(\(1\/q\)\((\@2\ Q\ \((Im[\(E\_+\)[ 0]\ \(\(S\_-\)[1]\^*\)] + 3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] - Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ Im[\(E\_+\)[ 0]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) + \(\(1\/q\)\((6\ \@2\ Q\ \ \((Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[ x])\)\))\)\)], "Output", CellLabel->"Out[199]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_LTh[N], 0]\)], "Input", CellLabel->"In[200]:="], Cell[BoxData[ \(\(\(1\/q\)\((\@2\ Q\ \((\(-Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)]\) - Re[\(M\_-\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] + 8\ Re[\(E\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) + \(\(1\/q\)\((\@2\ Q\ \ \((Re[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] + 3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 4\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[ x])\)\) + \(\(1\/q\)\((4\ \@2\ Q\ \((Re[\(E\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_2[ x])\)\)\)], "Output", CellLabel->"Out[200]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_LTh[L], 1]\)], "Input", CellLabel->"In[201]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(-\(\(1\/q\)\((\@2\ Q\ \((Re[\(M\_-\)[ 1]\ \(\(S\_-\)[1]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 2\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + 2\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\)\) - \(3\ \@2\ Q\ \ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] \ + 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[x]\)\/q - \(12\ \@2\ Q\ \ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\ \)\ P\_2[x]\)\/q)\)\)], "Output", CellLabel->"Out[201]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_LTh[S], 0]\)], "Input", CellLabel->"In[202]:="], Cell[BoxData[ \(\(\(1\/q\)\((\@2\ Q\ \((Re[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 0]\^*\)])\))\)\) + \(\(1\/\(5\ q\)\)\((\@2\ Q\ \((15\ \ Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - 5\ Re[\(M\_-\)[1]\ \(\(S\_-\)[1]\^*\)] + 5\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 5\ Re[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] + 24\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 20\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 16\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[ x])\)\) + \(2\ \@2\ Q\ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + \ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\ \)\ P\_2[x]\)\/q + \(36\ \@2\ Q\ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\ \(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_3[x]\)\/\(5\ q\)\)], "Output", CellLabel->"Out[202]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_TTh[L], 0]\)], "Input", CellLabel->"In[203]:="], Cell[BoxData[ \(2\ Re[\(M\_-\)[1]\ \(\(E\_+\)[0]\^*\)] + \((\(-Abs[\(E\_+\)[0]]\^2\) - 9\/5\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_-\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 + 6\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\/5\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ P\_1[x] - 2\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\)\ P\_2[x] - 36\/5\ \((Abs[\(E\_+\)[1]]\^2 + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)])\)\ P\_3[x]\)], "Output", CellLabel->"Out[203]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_TTh[S], 1]\)], "Input", CellLabel->"In[204]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(-Abs[\(E\_+\)[0]]\^2\) - 6\ Abs[\(E\_+\)[1]]\^2 + Abs[\(M\_-\)[1]]\^2 - 2\ Abs[\(M\_+\)[1]]\^2 + 3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)] - 3\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\)\ P\_1[x] - 12\ \((Abs[\(E\_+\)[1]]\^2 + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)])\)\ P\_2[ x])\)\)], "Output", CellLabel->"Out[204]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_L[N], 1]\)], "Input", CellLabel->"In[205]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(2\ Q\^2\ \((Im[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[0]] - 2\ \ Im[\(\(S\_+\)[1]\^*\)\ \(S\_+\)[0]])\)\)\/q\^2 + \(12\ Q\^2\ \ Im[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[1]]\ P\_1[x]\)\/q\^2)\)\)], "Output", CellLabel->"Out[205]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_T[N], 1]\)], "Input", CellLabel->"In[206]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((2\ Im[\(M\_-\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 3\ Im[\(E\_+\)[ 0]\ \(\(E\_+\)[ 1]\^*\)] + \((9\ Im[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 3\ Im[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ P\_1[ x])\)\)], "Output", CellLabel->"Out[206]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_LT[N], 0]\)], "Input", CellLabel->"In[207]:="], Cell[BoxData[ \(\(\(1\/q\)\((\@2\ Q\ \((\(-Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)]\) - Im[\(M\_-\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] + 8\ Im[\(E\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) + \(\(1\/q\)\((\@2\ Q\ \ \((Im[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] + 3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 4\ Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[ x])\)\) + \(\(1\/q\)\((4\ \@2\ Q\ \((Im[\(E\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_2[ x])\)\)\)], "Output", CellLabel->"Out[207]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_LT[L], 1]\)], "Input", CellLabel->"In[208]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(\(1\/q\)\((\@2\ Q\ \((Im[\(M\_-\)[ 1]\ \(\(S\_-\)[1]\^*\)] + 2\ Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] + 6\ Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 2\ Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + 2\ Im[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) + \(3\ \@2\ Q\ \ \((Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] \ + 2\ Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[x]\)\/q + \(12\ \@2\ Q\ \ \((Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\ \)\ P\_2[x]\)\/q)\)\)], "Output", CellLabel->"Out[208]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_LT[S], 0]\)], "Input", CellLabel->"In[209]:="], Cell[BoxData[ \(\(-\(\(1\/q\)\((\@2\ Q\ \((Im[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] - Im[\(M\_+\)[ 1]\ \(\(S\_+\)[ 0]\^*\)])\))\)\)\) - \(\(1\/\(5\ q\)\)\((\@2\ Q\ \ \((15\ Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - 5\ Im[\(M\_-\)[1]\ \(\(S\_-\)[1]\^*\)] + 5\ Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 5\ Im[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] + 24\ Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 20\ Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 16\ Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[ x])\)\) - \(2\ \@2\ Q\ \((Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + \ Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\ \)\ P\_2[x]\)\/q - \(36\ \@2\ Q\ \((Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\ \(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_3[x]\)\/\(5\ q\)\)], "Output", CellLabel->"Out[209]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_TT[N], 1]\)], "Input", CellLabel->"In[210]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(-3\)\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) + 3\ \((Im[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 4\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ P\_1[ x])\)\)], "Output", CellLabel->"Out[210]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_TT[L], 2]\)], "Input", CellLabel->"In[211]:="], Cell[BoxData[ \(Sin[\[Theta]]\^2\ \((\(-3\)\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) - 18\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)]\ P\_1[x])\)\)], "Output", CellLabel->"Out[211]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[sp, R\_TT[S], 1]\)], "Input", CellLabel->"In[212]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(-3\)\ \((Im[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\) + 3\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\)\ P\_1[x] + 12\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)]\ P\_2[x])\)\)], "Output", CellLabel->"Out[212]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ \(TraditionalForm\`P\_33\)]]], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_L[0], 0]\)], "Input", CellLabel->"In[213]:="], Cell[BoxData[ \(\(8\ Q\^2\ Abs[\(S\_+\)[1]]\^2\)\/q\^2 + \(8\ Q\^2\ Re[\(\(S\_+\)[1]\^*\ \)\ \(S\_+\)[0]]\ P\_1[x]\)\/q\^2 + \(8\ Q\^2\ \((Abs[\(S\_+\)[1]]\^2 + Re[\(\ \(S\_-\)[1]\^*\)\ \(S\_+\)[1]])\)\ P\_2[x]\)\/q\^2\)], "Output", CellLabel->"Out[213]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_T[0], 0]\)], "Input", CellLabel->"In[214]:="], Cell[BoxData[ \(2\ \((3\ Abs[\(E\_+\)[1]]\^2 + Abs[\(M\_+\)[1]]\^2)\) + 2\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\)\ P\_1[ x] + \((3\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 - 6\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ P\_2[x]\)], "Output", CellLabel->"Out[214]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_TT[0], 2] // Collect[#, {Sin[\[Theta]], P\_\[Lambda]_[x]}] &\)], "Input", CellLabel->"In[215]:="], Cell[BoxData[ \(3\/2\ \((3\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 - 2\ \((\(-Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)]\) + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\^2\)], "Output", CellLabel->"Out[215]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_LT[0], 1]\)], "Input", CellLabel->"In[216]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(\@2\ Q\ \((\(-3\)\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\ \)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ Re[\(E\_+\)[0]\ \ \(\(S\_+\)[1]\^*\)])\)\)\/q - \(\(1\/q\)\((6\ \@2\ Q\ \((Re[\(E\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[ x])\)\))\)\)], "Output", CellLabel->"Out[216]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_LTh[0], 1]\)], "Input", CellLabel->"In[217]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(\@2\ Q\ \((3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - \ Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\ \)\)\/q + \(\(1\/q\)\((6\ \@2\ Q\ \((Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[ x])\)\))\)\)], "Output", CellLabel->"Out[217]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_LTh[N], 0]\)], "Input", CellLabel->"In[218]:="], Cell[BoxData[ \(\(\@2\ Q\ \((\(-Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)]\) + Re[\(M\_+\)[1]\ \ \(\(S\_-\)[1]\^*\)] + 8\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\)\/q + \(\@2\ \ Q\ \((3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \ \(\(S\_+\)[0]\^*\)] + 4\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[x]\)\/q \ + \(\(1\/q\)\((4\ \@2\ Q\ \((Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_2[ x])\)\)\)], "Output", CellLabel->"Out[218]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_LTh[L], 1]\)], "Input", CellLabel->"In[219]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(-\(\(1\/q\)\((2\ \@2\ Q\ \((Re[\(M\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] + 3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\)\) - \(3\ \@2\ Q\ \ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] \ + 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[x]\)\/q - \(12\ \@2\ Q\ \ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\ \)\ P\_2[x]\)\/q)\)\)], "Output", CellLabel->"Out[219]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_LTh[S], 0]\)], "Input", CellLabel->"In[220]:="], Cell[BoxData[ \(\(\@2\ Q\ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_+\)[1]\ \ \(\(S\_+\)[0]\^*\)])\)\)\/q + \(\(1\/\(5\ q\)\)\((\@2\ Q\ \((15\ Re[\(E\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] + 5\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 4\ \((6\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 5\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 4\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\))\)\ P\_1[ x])\)\) + \(2\ \@2\ Q\ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + \ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\ \)\ P\_2[x]\)\/q + \(36\ \@2\ Q\ \((Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\ \(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_3[x]\)\/\(5\ q\)\)], "Output", CellLabel->"Out[220]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_TTh[L], 0]\)], "Input", CellLabel->"In[221]:="], Cell[BoxData[ \(\((\(-\(9\/5\)\)\ Abs[\(E\_+\)[1]]\^2 - Abs[\(M\_+\)[1]]\^2 + 6\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 6\/5\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ P\_1[x] - 2\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\)\ P\_2[x] - 36\/5\ \((Abs[\(E\_+\)[1]]\^2 + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)])\)\ P\_3[x]\)], "Output", CellLabel->"Out[221]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_TTh[S], 1]\)], "Input", CellLabel->"In[222]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(-6\)\ Abs[\(E\_+\)[1]]\^2 - 2\ Abs[\(M\_+\)[1]]\^2 + 3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)] - 3\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\)\ P\_1[x] - 12\ \((Abs[\(E\_+\)[1]]\^2 + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)])\)\ P\_2[ x])\)\)], "Output", CellLabel->"Out[222]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_L[N], 1]\)], "Input", CellLabel->"In[223]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(-\(\(4\ Q\^2\ Im[\(\(S\_+\)[1]\^*\)\ \(S\_+\)[ 0]]\)\/q\^2\)\) + \(12\ Q\^2\ Im[\(\(S\_-\)[1]\^*\)\ \ \(S\_+\)[1]]\ P\_1[x]\)\/q\^2)\)\)], "Output", CellLabel->"Out[223]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_T[N], 1]\)], "Input", CellLabel->"In[224]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 3\ Im[\(E\_+\)[ 0]\ \(\(E\_+\)[ 1]\^*\)] + \((9\ Im[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 3\ Im[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ P\_1[ x])\)\)], "Output", CellLabel->"Out[224]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_LT[N], 0]\)], "Input", CellLabel->"In[225]:="], Cell[BoxData[ \(\(\@2\ Q\ \((\(-Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)]\) + Im[\(M\_+\)[1]\ \ \(\(S\_-\)[1]\^*\)] + 8\ Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\)\/q + \(\@2\ \ Q\ \((3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \ \(\(S\_+\)[0]\^*\)] + 4\ Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[x]\)\/q \ + \(\(1\/q\)\((4\ \@2\ Q\ \((Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_2[ x])\)\)\)], "Output", CellLabel->"Out[225]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_LT[L], 1]\)], "Input", CellLabel->"In[226]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(\(1\/q\)\((2\ \@2\ Q\ \((Im[\(M\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] + 3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\) + \(3\ \@2\ Q\ \((Im[\(E\_+\)[1]\ \ \(\(S\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ \ Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\)\ P\_1[x]\)\/q + \(12\ \@2\ Q\ \ \((Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\ \)\ P\_2[x]\)\/q)\)\)], "Output", CellLabel->"Out[226]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_LT[S], 0]\)], "Input", CellLabel->"In[227]:="], Cell[BoxData[ \(\(-\(\(\@2\ Q\ \((Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - Im[\(M\_+\)[ 1]\ \(\(S\_+\)[ 0]\^*\)])\)\)\/q\)\) - \(\(1\/\(5\ q\)\)\((\@2\ Q\ \ \((15\ Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 5\ Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 4\ \((6\ Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 5\ Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 4\ Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\))\)\ P\_1[ x])\)\) - \(2\ \@2\ Q\ \((Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + \ Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\ \)\ P\_2[x]\)\/q - \(36\ \@2\ Q\ \((Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Im[\ \(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ P\_3[x]\)\/\(5\ q\)\)], "Output", CellLabel->"Out[227]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_TT[N], 1]\)], "Input", CellLabel->"In[228]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(-3\)\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) + 3\ \((Im[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 4\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ P\_1[ x])\)\)], "Output", CellLabel->"Out[228]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_TT[L], 2]\)], "Input", CellLabel->"In[229]:="], Cell[BoxData[ \(Sin[\[Theta]]\^2\ \((\(-3\)\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) - 18\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)]\ P\_1[x])\)\)], "Output", CellLabel->"Out[229]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[P\_33, R\_TT[S], 1]\)], "Input", CellLabel->"In[230]:="], Cell[BoxData[ \(Sin[\[Theta]]\ \((\(-3\)\ \((Im[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + Im[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\) + 3\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\)\ P\_1[x] + 12\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)]\ P\_2[x])\)\)], "Output", CellLabel->"Out[230]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["M1", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[M1, R\_L[0], 0]\)], "Input", CellLabel->"In[231]:="], Cell[BoxData[ \(0\)], "Output", CellLabel->"Out[231]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RtoLegendre[M1, R\_T[0], 0]\)], "Input", CellLabel->"In[232]:="], Cell[BoxData[ \(2\ Abs[\(M\_+\)[1]]\^2 + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)]\ P\_1[ x] + \((\(-Abs[\(M\_+\)[1]]\^2\) + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 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He also shows figures for many of the angular distributions \ that look qualitatively consistent with the corresponding multipole \ expansions. However, the results that I obtain using ", StyleBox["epiprod", FontSlant->"Italic"], " are often quite different, both algebraically and numerically. For \ example, the figure he gives for ", Cell[BoxData[ \(TraditionalForm\`R\_TTh[L]\)]], " is consistent with the 3 - Cos[2\[Theta]] shape given by his expression, \ while my figure has a ", Cell[BoxData[ \(TraditionalForm\`Cos[\[Theta]]\)]], " distribution that is consistent with the results above. When I checked a \ few special cases, I found that the UMd multipole expansions were often quite \ different from those of Raskin and Donnelly, but consistent with those of \ Drechsel and Tiator from Mainz. On the other hand, Donnelly expressed \ considerable confidence in his expressions and questioned the reliability of \ the Mainz work. Therefore, it became incumbent upon me to resolve this \ discrepancy." }], "Text"], Cell[TextData[{ "There appear to be two important errors in the Raskin and Donnelly paper. \ First, they claim that the ejectile basis is used for recoil polarization but \ from the comparison below I conclude that they actually used the target \ basis. Second, their expressions appear to employ the ejectile angle but I \ find it necessary to replace that angle with the pion angle instead. [Note \ that because \[Theta] is the pion angle in the derivation above, I should \ have had to replace ", Cell[BoxData[ \(TraditionalForm\`Cos[\[Theta]] \[Rule] \(-Cos[\[Theta]]\)\)]], " in order to obtain the correct relative sign between even and odd \ multipoles, but that replacement was not necessary because they actually used \ the pion angle.] To demonstrate these claims, I reproduce the \ helicity-dependent response functions for polarization components in the \ reaction plane given by Lourie. [Also note that Louries formula for ", Cell[BoxData[ \(TraditionalForm\`R\_TTh[L]\)]], " differs from Raskin and Donnelly in the sign of the last term; that \ mistake is corrected here.]" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Lourie[TTh, L]\ = Simplify /@ \((\((\(\(5\/3\) Abs[\(M\_+\)[1]]\^2 - 2 Re[\(\(E\_+\)[1]\^*\) \(M\_+\)[ 1] + \(2\/3\) \(\(M\_-\)[1]\^*\) \(M\_+\)[ 1]] + 2 Re[\(\(E\_+\)[0]\^*\) \(M\_+\)[1]] P\_1[x] + \((\(-\(2\/3\)\) Abs[\(M\_+\)[1]]\^2 + Re[8 \(\( E\_+\)[1]\^*\) \(M\_+\)[ 1] - \(2\/3\) \(\(M\_-\)[ 1]\^*\) \(M\_+\)[1]])\) P\_2[x] /. ExpandLegendre\) /. x \[Rule] Cos[\[Theta]] // Simplify)\) //. ExpandMultipoleProducts // Collect[#, {Abs[_], Re[_], Im[_]}] &\ )\)\)], "Input", CellLabel->"In[249]:="], Cell[BoxData[ \(\(-\(1\/2\)\)\ Abs[\(M\_+\)[1]]\^2\ \((\(-3\) + Cos[2\ \[Theta]])\) + 2\ Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ Cos[2\ \[Theta]]\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 1\/2\ \((3 + Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(M\_-\)[1]\^*\)]\)], "Output", CellLabel->"Out[249]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Lourie[TTh, S]\ = Simplify /@ \((\((\(Sin[\[Theta]] \((Re[\(\(E\_+\)[0]\^*\) \(M\_+\)[ 1]] + \((Abs[\(M\_+\)[1]]\^2 + Re[6 \(\( E\_+\)[1]\^*\) \(M\_+\)[ 1] + \(\(M\_-\)[1]\^*\) \(M\_+\)[ 1]])\) P\_1[x])\) /. ExpandLegendre\) /. x \[Rule] Cos[\[Theta]] // Simplify)\) //. ExpandMultipoleProducts // Collect[#, {Abs[_], Re[_], Im[_]}] &\ )\)\)], "Input", CellLabel->"In[250]:="], Cell[BoxData[ \(Abs[\(M\_+\)[1]]\^2\ Cos[\[Theta]]\ Sin[\[Theta]] + Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)]\ Sin[\[Theta]] + 6\ Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)]\ Sin[\[Theta]] + Cos[\[Theta]]\ Re[\(M\_+\)[ 1]\ \(\(M\_-\)[1]\^*\)]\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[250]="] }, Open ]], Cell[BoxData[ \(\(Lourie[LTh, L]\ = Simplify /@ \((\((\(\(-\@2\) Sin[\[Theta]] \((Re[ 2 \(\( S\_+\)[0]\^*\) \(M\_+\)[ 1] + \((\(\(S\_-\)[1]\^*\) \(M\_+\)[1] + 10 \(\( S\_+\)[1]\^*\) \(M\_+\)[1])\) P\_1[x]])\) /. ExpandLegendre\) /. x \[Rule] Cos[\[Theta]] // Simplify)\) //. ExpandMultipoleProducts // Collect[#, {Abs[_], Re[_], Im[_]}] &\ )\);\)\)], "Input", CellLabel->"In[251]:="], Cell[BoxData[ \(\(Lourie[LTh, S]\ = Simplify /@ \((\((\(\(\@2\) Re[\(5\/3\) \(\(S\_-\)[1]\^*\) \(M\_+\)[ 1] - \(4\/3\) \(\(S\_+\)[1]\^*\) \(M\_+\)[ 1] + \(\(S\_+\)[0]\^*\) \(M\_+\)[1] P\_1[x] - \(2\/3\) \((\(\(S\_-\)[ 1]\^*\) \(M\_+\)[1] - 8 \(\( S\_+\)[1]\^*\) \(M\_+\)[1])\) P\_2[x]] /. ExpandLegendre\) /. x \[Rule] Cos[\[Theta]] // Simplify)\) //. ExpandMultipoleProducts // Collect[#, {Abs[_], Re[_], Im[_]}] &\ )\);\)\)], "Input", CellLabel->"In[252]:="], Cell["\<\ Applying a rotation to the Lourie formulas and inserting the relative \ normalizations between our definitions of the interference response \ functions, we can now obtain results that are consistent with the present \ derivation.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((roty[\(-\[Theta]\)] . {Lourie[LTh, S], 0, Lourie[LTh, L]\ } == \((\(\((\(\@2\) AbbreviateMultipoleExpansion[#, \(M\_+\)[ 1]])\) &\)\ /@ \ {mpR\_LTh[S], 0, mpR\_LTh[L]})\))\) // Simplify\)], "Input", CellLabel->"In[253]:="], Cell[BoxData[ \({\(\(1\/\@2\)\((2\ Cos[\[Theta]]\ Re[\(M\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)] + \((\(-1\) + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[0]\^*\)] + \((\(-Cos[\[Theta]]\) + 9\ Cos[3\ \[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)])\)\), 0, \(-\@2\)\ \((2\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 3\ Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + \((5 + 9\ Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\)\ Sin[\[Theta]]} \[Equal] \ {\(\(1\/q\)\((2\ Q\ \((x\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + \((\(-2\) + 3\ x\^2)\)\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ x\ \((\(-7\) + 9\ x\^2)\)\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)])\))\)\), 0, \(-\(\(1\/q\)\((2\ Q\ \((2\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 3\ x\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ \((\(-2\) + 9\ x\^2)\)\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\)\ Sin[\[Theta]])\)\)\)}\)], "Output",\ CellLabel->"Out[253]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((roty[\(-\[Theta]\)] . {Lourie[TTh, S], 0, Lourie[TTh, L]\ } == \((\(\((\(-AbbreviateMultipoleExpansion[#, \(M\_+\)[ 1]]\))\) &\)\ /@ \ {mpR\_TTh[S], 0, mpR\_TTh[L]})\))\) // Simplify\)], "Input", CellLabel->"In[254]:="], Cell[BoxData[ \({3\ \((\(-x\) + Cos[\[Theta]])\)\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ \((x + Cos[\[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(E\_+\)[1]\^*\)])\)\ Sin[\[Theta]], 0, \((\(-x\) + Cos[\[Theta]])\)\ \((Abs[\(M\_+\)[1]]\^2 + 3\ \((x + Cos[\[Theta]])\)\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 18\ x\^2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 18\ x\ Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 9\ Cos[2\ \[Theta]]\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)} \[Equal] {0, 0, 0}\)], "Output", CellLabel->"Out[254]="] }, Open ]], Cell[TextData[{ "Thus apart from trivial normalization factors we recover Lourie's \ multipole expansions of the helicity-dependent in-plane response functions by \ replacing ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_N\)]], " by ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_\[Pi]\)]], " and rotating to the target basis. I have not checked the \ helicity-independent response functions, but expect the same behavior. \ Furthermore, I incidentally encountered remarks by Dmitrasinovic [Phys. Rev. \ 51, 1528 (1995), footnote 1] indicating that both he and Hanstein [Mainz Ph.D \ thesis] have also found that a rotation is needed to correct the Lourie \ expressions, although he does not mention the replacement of ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_N\)]], " by ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_\[Pi]\)]], ". It turns out that Donnelly remembered a conversation with the Mainz \ group along similar lines, but thought that his results were correct and that \ Mainz had corrected their problem. However, unless I am seriouly misreading \ the Raskin and Donnelly paper, I have to conclude that the MIT results were \ represented incorrectly. Therefore, I will continue to trust ", StyleBox["epiprod", FontSlant->"Italic"], " and will disregard the expressions and figures in Lourie's proposal. I \ will also proceed to develop the target polarization using the present \ formalism and methods. 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Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellFrame->{{0, 0}, {0, 0.25}}, CellMargins->{{18, 30}, {4, 20}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, CellFrameMargins->9, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, LineSpacing->{0.95, 0}, CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->36], Cell[StyleData["Title", 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Cell[StyleData["SectionFirst", "Printout"], FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellMargins->{{18, 30}, {4, 30}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontSize->18, FontWeight->"Bold"], Cell[StyleData["Section", "Printout"], FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSquare]", CellMargins->{{18, 30}, {4, 20}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontSize->14, FontWeight->"Bold"], Cell[StyleData["Subsection", "Printout"], FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{18, 30}, {4, 12}}, CellGroupingRules->{"SectionGrouping", 60}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Subsubsection", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Printout"], FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{18, 10}, 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Be careful when modifying, renaming, or removing these styles, \ because the front end associates special meanings with these style names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{55, 10}, {5, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Input", "Printout"], CellMargins->{{55, 55}, {0, 10}}, ShowCellLabel->False, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->9.5] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{55, 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DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Message", "Printout"], CellMargins->{{55, 55}, {0, 3}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Printout"], CellMargins->{{54, 72}, {2, 10}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", StyleMenuListing->None], Cell[StyleData["Graphics", "Printout"], CellMargins->{{55, 55}, {0, 15}}, ImageSize->{0.0625, 0.0625}, ImageMargins->{{35, Inherited}, {Inherited, 0}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], CellMargins->{{9, Inherited}, {Inherited, Inherited}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontSlant->"Oblique"], Cell[StyleData["CellLabel", "Printout"], CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Unique Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Author"], CellMargins->{{45, Inherited}, {2, 20}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->14, FontWeight->"Bold"], Cell[StyleData["Author", "Printout"], CellMargins->{{36, Inherited}, {2, 30}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Address"], CellMargins->{{45, Inherited}, {2, 2}}, CellGroupingRules->{"TitleGrouping", 30}, PageBreakBelow->False, LineSpacing->{1, 1}, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->12, FontSlant->"Italic"], Cell[StyleData["Address", "Printout"], CellMargins->{{36, Inherited}, {2, 2}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Abstract"], CellMargins->{{45, 75}, {Inherited, 30}}, Hyphenation->True, LineSpacing->{1, 0}], Cell[StyleData["Abstract", "Printout"], CellMargins->{{36, 67}, {Inherited, 50}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Reference"], CellMargins->{{18, 40}, {2, 2}}, TextJustification->1, Hyphenation->True, LineSpacing->{1, 0}], Cell[StyleData["Reference", "Printout"], CellMargins->{{18, 40}, {Inherited, 0}}, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext ButtonBoxes. 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Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard ButtonFunctions, for use \ in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], 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HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", AutoSpacing->False, ScriptLevel->1, ScriptBaselineShifts->{0.6, Automatic}, SingleLetterItalics->False, ZeroWidthTimes->True], Cell[StyleData["ChemicalFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellMargins->{{18, 10}, {Inherited, 6}}, Hyphenation->False, LanguageCategory->"Formula", FontFamily->"Courier"], Cell[StyleData["Program", "Printout"], CellMargins->{{18, 30}, {Inherited, 4}}, FontSize->9.5] }, Closed]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Notation Package Styles", "Section", GeneratedCell->True, CellTags->"NotationPackage"], Cell["\<\ The cells below define certain styles needed by the Notation package. 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(******************************************************************* End of Mathematica Notebook file. *******************************************************************)