(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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[ 121505, 3571]*) (*NotebookOutlinePosition[ 150202, 4490]*) (* CellTagsIndexPosition[ 150158, 4486]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Target-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["\<\ 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["\<\ In this notebook I derive the response functions for electroproduction of \ pseudoscalar mesons from polarized nucleon targets. This notebook is quite \ similar to that for recoil polarization, differing primarily in the choice of \ density matrices; therefore, some of the shared output is suppressed.\ \>", "Text"], Cell[CellGroupData[{ Cell["References", "Subsubsection"], 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};\), "\[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, \[Pi] - \[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[TextData[{ "It is also useful to employ the ", StyleBox["photon basis", FontSlant->"Italic"], " in which ", Cell[BoxData[ \(TraditionalForm\`\(z\&^\)\)]], " is along the momentum transfer and ", Cell[BoxData[ \(TraditionalForm\`\(y\&^\)\)]], " is normal to the electron-scattering plane." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(xyzBasis = Thread[{S\&^, N\&^, L\&^} \[Rule] IdentityMatrix[3]]\)], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \({\(S\&^\) \[Rule] {1, 0, 0}, \(N\&^\) \[Rule] {0, 1, 0}, \(L\&^\) \[Rule] {0, 0, 1}}\)], "Output", CellLabel->"Out[28]="] }, 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[29]:="], 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[32]:="], Cell[BoxData[ \(\(phases[ JW]\ = \ {\[Alpha] \[Rule] \[Pi] - \[Phi]\/2, \[Beta] \[Rule] \ \[Pi] + \[Phi], \[Gamma] \[Rule] \[Pi], \[Delta] \[Rule] \[Pi]};\)\)], "Input",\ CellLabel->"In[35]:="] }, Open ]], Cell[CellGroupData[{ Cell["Virtual photon polarization vectors", "Subsection"], Cell[BoxData[ \(\(a\&\[RightVector] = {a\_x, a\_y, a\_z};\)\)], "Input", CellLabel->"In[36]:="], 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[37]:="] }, 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[40]:="], 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[41]:="], Cell["\<\ The transition operator can now be transformed into the helicity basis.\ \>", "Text"], Cell[BoxData[ \(\(T[helicity] = HermitianConjugate[\[Chi]\_f] . T[spin] . \[Chi]\_i // MySimplify;\)\)], "Input", CellLabel->"In[42]:="] }, 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[43]:="], 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[44]:="], Cell[BoxData[ \(\(T[helicity] /. Fto\[ScriptCapitalF];\)\)], "Input", CellLabel->"In[45]:="] }, Open ]], Cell[CellGroupData[{ Cell["Transition matrices with general phases", "Subsection"], Cell[BoxData[ \(\(T\_0 = \((\(\(T[helicity] /. rule[a\_0]\) /. Fto\[ScriptCapitalF]\) /. trigToExp[\[Phi]])\) // Simplify;\)\)], "Input", CellLabel->"In[46]:="], Cell[BoxData[ \(\(T\_1 = \((\(\(T[helicity] /. rule[a\_1]\) /. Fto\[ScriptCapitalF]\) /. trigToExp[\[Phi]])\) // Simplify;\)\)], "Input", CellLabel->"In[47]:="], Cell[BoxData[ \(\(T\_\(-1\) = \((\(\(T[helicity] /. rule[a\_\(-1\)]\) /. Fto\[ScriptCapitalF]\) /. trigToExp[\[Phi]])\) // Simplify;\)\)], "Input", CellLabel->"In[48]:="] }, Open ]], Cell[CellGroupData[{ Cell["Transition matrices using Jacob-Wick phases", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(T\_0 /. phases[JW] // Simplify\)], "Input", CellLabel->"In[49]:="], 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[49]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(T\_1 /. phases[JW] // Simplify\)], "Input", CellLabel->"In[50]:="], 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[50]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(T\_\(-1\) /. phases[JW] // Simplify\)], "Input", CellLabel->"In[51]:="], 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[51]="] }, 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[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[52]:="], Cell[CellGroupData[{ Cell[BoxData[ \(HelicityToCGLN\ /. phases[JW] // Simplify\)], "Input", CellLabel->"In[53]:="], 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[53]="] }, 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[54]:="] }, 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[57]:="], 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[58]:="], 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["Target-Polarization Response Functions in Target Basis", "Section"], Cell[CellGroupData[{ Cell["\<\ Target polarization vector, projection operator, and density matrix\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(P\&^ = \(P\_L\) L\&^ + \(P\_N\) N\&^ + \(P\_S\) S\&^ /. TargetBasis\)], "Input", CellLabel->"In[64]:="], Cell[BoxData[ \({Sin[\[Phi]]\ P\_N - Cos[\[Phi]]\ P\_S, \(-Cos[\[Phi]]\)\ P\_N - Sin[\[Phi]]\ P\_S, P\_L}\)], "Output", CellLabel->"Out[64]="] }, Open ]], Cell[BoxData[ \(\(\[DoubleStruckCapitalP] = \(1\/2\) \((IdentityMatrix[2] + P\&^ . \[Sigma]\&\[RightVector])\) /. trigToExp[\[Phi]] // Simplify;\)\)], "Input", CellLabel->"In[65]:="], Cell[CellGroupData[{ Cell[BoxData[ \(\[DoubleStruckCapitalP] // MatrixForm\)], "Input", CellLabel->"In[66]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2\ \((1 + P\_L)\)\), \(1\/2\ \[ImaginaryI]\ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \[Phi]\)\ \((P\_N + \[ImaginaryI]\ P\_S)\)\)}, {\(\(-\(1\/2\)\)\ \[ImaginaryI]\ \[ExponentialE]\^\(\[ImaginaryI]\ \ \[Phi]\)\ \((P\_N - \[ImaginaryI]\ P\_S)\)\), \(1\/2\ \((1 - P\_L)\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output", CellLabel->"Out[66]//MatrixForm="] }, Open ]], Cell[BoxData[{ \(\(\[Rho]\_f = \(1\/2\) IdentityMatrix[2];\)\), "\n", \(\(\[Rho]\_i = HermitianConjugate[\[Chi]\_i] . \[DoubleStruckCapitalP] . \[Chi]\_i // Simplify;\)\)}], "Input", CellLabel->"In[67]:="], Cell[CellGroupData[{ Cell[BoxData[ \(\[Rho]\_i // MatrixForm\)], "Input", CellLabel->"In[69]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2\ \((1 - P\_L)\)\), \(1\/2\ \[ExponentialE]\^\(\[ImaginaryI]\ \((\ \[Delta] + \[Phi])\)\)\ \((\[ImaginaryI]\ P\_N + P\_S)\)\)}, {\(1\/2\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \((\[Delta] + \ \[Phi])\)\)\ \((\(-\[ImaginaryI]\)\ P\_N + P\_S)\)\), \(1\/2\ \((1 + P\_L)\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output", CellLabel->"Out[69]//MatrixForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[Rho]\_i /. phases[JW] // Simplify\) // MatrixForm\)], "Input", CellLabel->"In[70]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2\ \((1 - P\_L)\)\), \(\(-\(1\/2\)\)\ \[ExponentialE]\^\(\ \[ImaginaryI]\ \[Phi]\)\ \((\[ImaginaryI]\ P\_N + P\_S)\)\)}, {\(1\/2\ \[ImaginaryI]\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \[Phi]\)\ \((P\_N + \[ImaginaryI]\ P\_S)\)\), \(1\/2\ \((1 + P\_L)\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output", CellLabel->"Out[70]//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[71]:="], Cell[BoxData[ \(Abs[H\_5]\^2 + Abs[H\_6]\^2 + 2\ Im[H\_5\ \(\((H\_6)\)\^*\)]\ P\_N\)], "Output", CellLabel->"Out[71]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[ScriptCapitalW]\_T[TH] // Simplify\) // Collect[#, {P\_x_, f_[a_. \ \[Phi]]}] &\)], "Input", CellLabel->"In[72]:="], 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\ \(\((H\_2)\)\^*\)] - 2\ Im[H\_3\ \(\((H\_4)\)\^*\)])\)\ P\_N\)], "Output", CellLabel->"Out[72]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((\(\[ScriptCapitalW]\_LT[TH] // ExpToTrig\) // MySimplify)\) //. ContractAmplitudeProducts // Collect[#, {P\_x_, f_[a_. \[Phi]]}] &\)], "Input", CellLabel->"In[73]:="], Cell[BoxData[ \(Cos[\[Phi]]\ \((Re[\((H\_1 - H\_4)\)\ \(\((H\_5)\)\^*\)] - 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[73]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((\(\[ScriptCapitalW]\_LTh[TH] // ExpToTrig\) // MySimplify)\) //. ContractAmplitudeProducts // Collect[#, {P\_x_, f_[a_. \[Phi]]}] &\)], "Input", CellLabel->"In[74]:="], 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[74]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((\(\[ScriptCapitalW]\_TT[TH] // ExpToTrig\) // MySimplify)\) // \(Collect[#, {P\_x_, f_[a_. \[Phi]]}] &\) //. ContractAmplitudeProducts\)], "Input", CellLabel->"In[75]:="], 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\ \(\((H\_3)\)\^*\)]\) - Im[H\_2\ \(\((H\_4)\)\^*\)])\)\ P\_N + \((Im[ H\_1\ \(\((H\_3)\)\^*\)] - Im[H\_2\ \(\((H\_4)\)\^*\)])\)\ Sin[ 2\ \[Phi]]\ P\_S\)], "Output", CellLabel->"Out[75]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((\(\[ScriptCapitalW]\_TTh[TH] // ExpToTrig\) // MySimplify)\) // \(Collect[#, {P\_x_, f_[a_. \[Phi]]}] &\) //. ContractAmplitudeProducts\)], "Input", CellLabel->"In[76]:="], Cell[BoxData[ \(1\/2\ \((Abs[H\_1]\^2 - Abs[H\_2]\^2 + Abs[H\_3]\^2 - Abs[H\_4]\^2)\)\ P\_L + 1\/2\ \((\(-2\)\ Re[H\_1\ \(\((H\_2)\)\^*\)] - 2\ Re[H\_3\ \(\((H\_4)\)\^*\)])\)\ P\_S\)], "Output", CellLabel->"Out[76]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[ "\[Phi]-independent response functions in terms of helicity amplitudes"], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_L = \(\(q\^2\/Q\^2\) \[ScriptCapitalW]\_L[TH] // Simplify\) // Collect[#, {P\_x_, f_[a_. \ \[Phi]]}] &;\)\), "\n", \(R\_L[ H] = {\n\tR\_L[0] \[Rule] Select[tempR\_L, FreeQ[#, P\_x_] &], \n\t R\_L[N] \[Rule] Coefficient[tempR\_L, P\_N]\ } // Simplify\)}], "Input", CellLabel->"In[77]:="], Cell[BoxData[ \({R\_L[0] \[Rule] \(q\^2\ \((Abs[H\_5]\^2 + Abs[H\_6]\^2)\)\)\/Q\^2, R\_L[N] \[Rule] \(2\ q\^2\ Im[H\_5\ \(\((H\_6)\)\^*\)]\)\/Q\^2}\)], \ "Output", CellLabel->"Out[78]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_L = \(\(q\^2\/Q\^2\) \[ScriptCapitalW]\_L[TH] // Simplify\) // Collect[#, {P\_x_, f_[a_. \ \[Phi]]}] &;\)\), "\n", \(R\_L[ H] = {\n\tR\_L[0] \[Rule] Select[tempR\_L, FreeQ[#, P\_x_] &], \n\t R\_L[N] \[Rule] Coefficient[tempR\_L, P\_N]\ } // Simplify\)}], "Input", CellLabel->"In[79]:="], Cell[BoxData[ \({R\_L[0] \[Rule] \(q\^2\ \((Abs[H\_5]\^2 + Abs[H\_6]\^2)\)\)\/Q\^2, R\_L[N] \[Rule] \(2\ q\^2\ Im[H\_5\ \(\((H\_6)\)\^*\)]\)\/Q\^2}\)], \ "Output", CellLabel->"Out[80]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_T = \((\(\[ScriptCapitalW]\_T[TH] // ExpToTrig\) // MySimplify)\) // Collect[#, {P\_x_, f_[a_. \ \[Phi]]}] &;\)\), "\n", \(R\_T[ H] = {\n\tR\_T[0] \[Rule] Select[tempR\_T, FreeQ[#, P\_x_] &], \n\t R\_T[N] \[Rule] Coefficient[tempR\_T, P\_N]\ } // Simplify\)}], "Input", CellLabel->"In[81]:="], 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\ \(\((H\_2)\)\^*\)]\) - Im[H\_3\ \(\((H\_4)\)\^*\)]}\)], "Output", CellLabel->"Out[82]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_LT = \((\(\(\@\(1\/2\)\) \(q\/Q\) \[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[83]:="], Cell[BoxData[ \({R\_LT[ 0] \[Rule] \(q\ \((Re[\((H\_1 - H\_4)\)\ \(\((H\_5)\)\^*\)] - \ Re[\((H\_2 + H\_3)\)\ \(\((H\_6)\)\^*\)])\)\)\/\(\@2\ Q\), R\_LT[N] \[Rule] \(q\ \((Im[\((H\_2 + H\_3)\)\ \(\((H\_5)\)\^*\)] + Im[\ \((H\_1 - H\_4)\)\ \(\((H\_6)\)\^*\)])\)\)\/\(\@2\ Q\), R\_LT[L] \[Rule] \(-\(\(q\ \((Im[\((H\_1 + H\_4)\)\ \(\((H\_5)\)\^*\)] + Im[\((H\_2 - H\_3)\)\ \(\((H\_6)\)\^*\)])\)\)\/\(\@2\ Q\)\)\), R\_LT[S] \[Rule] \(q\ \((Im[\((H\_2 - H\_3)\)\ \(\((H\_5)\)\^*\)] - Im[\ \((H\_1 + H\_4)\)\ \(\((H\_6)\)\^*\)])\)\)\/\(\@2\ Q\)}\)], "Output", CellLabel->"Out[84]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_LTh = \((\(\(\@\(1\/2\)\) \(q\/Q\) \[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[85]:="], Cell[BoxData[ \({R\_LTh[ 0] \[Rule] \(q\ \((\(-Im[\((H\_1 - H\_4)\)\ \(\((H\_5)\)\^*\)]\) + \ Im[\((H\_2 + H\_3)\)\ \(\((H\_6)\)\^*\)])\)\)\/\(\@2\ Q\), R\_LTh[N] \[Rule] \(q\ \((Re[\((H\_2 + H\_3)\)\ \(\((H\_5)\)\^*\)] + \ Re[\((H\_1 - H\_4)\)\ \(\((H\_6)\)\^*\)])\)\)\/\(\@2\ Q\), R\_LTh[L] \[Rule] \(q\ \((Re[\((H\_1 + H\_4)\)\ \(\((H\_5)\)\^*\)] + \ Re[\((H\_2 - H\_3)\)\ \(\((H\_6)\)\^*\)])\)\)\/\(\@2\ Q\), R\_LTh[S] \[Rule] \(q\ \((\(-Re[\((H\_2 - H\_3)\)\ \(\((H\_5)\)\^*\)]\) \ + Re[\((H\_1 + H\_4)\)\ \(\((H\_6)\)\^*\)])\)\)\/\(\@2\ Q\)}\)], "Output", CellLabel->"Out[86]="] }, 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[87]:="], Cell[BoxData[ \({R\_TT[0] \[Rule] Re[H\_2\ \(\((H\_3)\)\^*\)] - Re[H\_1\ \(\((H\_4)\)\^*\)], R\_TT[N] \[Rule] \(-Im[H\_1\ \(\((H\_3)\)\^*\)]\) - Im[H\_2\ \(\((H\_4)\)\^*\)], R\_TT[L] \[Rule] Im[H\_2\ \(\((H\_3)\)\^*\)] + Im[H\_1\ \(\((H\_4)\)\^*\)], R\_TT[S] \[Rule] Im[H\_1\ \(\((H\_3)\)\^*\)] - Im[H\_2\ \(\((H\_4)\)\^*\)]}\)], "Output", CellLabel->"Out[88]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_TTh = \((\(\[ScriptCapitalW]\_TTh[TH] // ExpToTrig\) // MySimplify)\) //. ContractAmplitudeProducts // 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[89]:="], 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\ \(\((H\_2)\)\^*\)]\) - Re[H\_3\ \(\((H\_4)\)\^*\)]}\)], "Output", CellLabel->"Out[90]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[ "\[Phi]-independent response functions in terms of CGLN amplitudes"], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_L = \(\(q\^2\/Q\^2\) \[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]\ } // Simplify\)}], "Input", CellLabel->"In[91]:="], 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[92]="] }, 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]\ } // Simplify //. ContractAmplitudeProducts\)}], "Input", CellLabel->"In[93]:="], Cell[BoxData[ \({R\_T[0] \[Rule] 1\/4\ \((4\ Abs[\[ScriptCapitalF]\_1]\^2 + 4\ Abs[\[ScriptCapitalF]\_2]\^2 + Abs[\[ScriptCapitalF]\_3]\^2 + Abs[\[ScriptCapitalF]\_4]\^2 - Abs[\[ScriptCapitalF]\_3]\^2\ Cos[2\ \[Theta]] - Abs[\[ScriptCapitalF]\_4]\^2\ Cos[2\ \[Theta]] - 8\ Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_2)\)\^*\)] + 2\ Re[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_3)\)\^*\)] \ - 2\ Cos[2\ \[Theta]]\ Re[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)] + 2\ Re[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_4)\)\^*\)] \ - 2\ Cos[2\ \[Theta]]\ Re[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_3\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)] - Cos[3\ \[Theta]]\ Re[\[ScriptCapitalF]\_3\ \(\((\ \[ScriptCapitalF]\_4)\)\^*\)])\), R\_T[N] \[Rule] 1\/2\ \((2\ Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)] - 2\ Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_3)\)\^*\)] + 2\ Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_4)\)\^*\)] - 2\ Im[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_4)\)\^*\)] \ - Im[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Cos[2\ \[Theta]]\ Im[\[ScriptCapitalF]\_3\ \(\((\ \[ScriptCapitalF]\_4)\)\^*\)])\)\ Sin[\[Theta]]}\)], "Output", CellLabel->"Out[94]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_LT = \((\(\(\@\(1\/2\)\) \(q\/Q\) \[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]]} // Simplify\)}], "Input", CellLabel->"In[95]:="], Cell[BoxData[ \({R\_LT[ 0] \[Rule] \(-\(\(1\/\[Omega]\)\((q\ \((Re[\[ScriptCapitalF]\_2\ \(\ \((\[ScriptCapitalF]\_5)\)\^*\)] + Re[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] \ + Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_4\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] \ + Re[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_6)\)\^*\)] + Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_3\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)] + Re[\[ScriptCapitalF]\_4\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\)\ Sin[\[Theta]])\)\)\), R\_LT[N] \[Rule] \(-\(\(1\/\(2\ \[Omega]\)\)\((q\ \((2\ Im[\ \[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] - 2\ Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Im[\[ScriptCapitalF]\_4\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] \ - Cos[2\ \[Theta]]\ Im[\[ScriptCapitalF]\_4\ \ \(\((\[ScriptCapitalF]\_5)\)\^*\)] + 2\ Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)] - 2\ Im[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_6)\)\^*\ \)] - Im[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_6)\)\^*\)] + Cos[2\ \[Theta]]\ Im[\[ScriptCapitalF]\_3\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)])\))\)\)\), R\_LT[L] \[Rule] \(-\(\(q\ \((Im[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\)\ Sin[\[Theta]]\)\/\[Omega]\)\), R\_LT[S] \[Rule] \(-\(\(q\ \((Im[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] - Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)] - Im[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\)\)\/\[Omega]\)\)}\)], "Output", CellLabel->"Out[96]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_LTh = \((\(\(\@\(1\/2\)\) \(q\/Q\) \[ScriptCapitalW]\_LTh[T] // ExpToTrig\) // MySimplify)\) // \(Collect[#, {P\_x_, f_[a_. \[Phi]], f_[a_. \ \[Theta]]}] &\) //. ContractAmplitudeProducts;\)\), "\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]]} // Simplify\)}], "Input", CellLabel->"In[97]:="], Cell[BoxData[ \({R\_LTh[ 0] \[Rule] \(\(1\/\[Omega]\)\((q\ \((Im[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Im[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] + Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_4\ \ \(\((\[ScriptCapitalF]\_5)\)\^*\)] + Im[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_6)\)\^*\)] + Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_3\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)] + Im[\[ScriptCapitalF]\_4\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\)\ Sin[\[Theta]])\)\), R\_LTh[N] \[Rule] \(-\(\(1\/\(2\ \[Omega]\)\)\((q\ \((2\ Re[\ \[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] - 2\ Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Re[\[ScriptCapitalF]\_4\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] \ - Cos[2\ \[Theta]]\ Re[\[ScriptCapitalF]\_4\ \ \(\((\[ScriptCapitalF]\_5)\)\^*\)] + 2\ Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)] - 2\ Re[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_6)\)\^*\ \)] - Re[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_6)\)\^*\)] + Cos[2\ \[Theta]]\ Re[\[ScriptCapitalF]\_3\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)])\))\)\)\), R\_LTh[L] \[Rule] \(q\ \((Re[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Re[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)])\)\ Sin[\[Theta]]\)\/\[Omega], R\_LTh[S] \[Rule] \(q\ \((Re[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] - Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)] - Re[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)])\)\)\/\[Omega]}\)], "Output", CellLabel->"Out[98]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_TT = \((\(\[ScriptCapitalW]\_TT[T] // ExpToTrig\) // MySimplify)\) // \(Collect[#, {P\_x_, f_[a_. \[Phi]]}] &\) //. ContractAmplitudeProducts;\)\), "\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]]} // Simplify\)}], "Input", CellLabel->"In[99]:="], 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] 1\/2\ \((4\ Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_2)\)\^*\)] + 2\ Im[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_3)\)\^*\)] \ - 2\ Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)] + 2\ Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_4)\)\^*\)] - 2\ Im[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_4)\)\^*\)] \ - Im[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Cos[2\ \[Theta]]\ Im[\[ScriptCapitalF]\_3\ \(\((\ \[ScriptCapitalF]\_4)\)\^*\)])\)\ Sin[\[Theta]], R\_TT[L] \[Rule] \((Im[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_3)\ \)\^*\)] + Im[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_4)\)\^*\)])\)\ \ Sin[\[Theta]]\^2, R\_TT[S] \[Rule] \((2\ Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_2)\)\^*\)] + Im[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_3)\)\^*\)] - Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)] + Cos[\[Theta]]\ Im[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)] - Im[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_4)\)\^*\)])\)\ \ Sin[\[Theta]]}\)], "Output", CellLabel->"Out[100]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(tempR\_TTh = \((\(\[ScriptCapitalW]\_TTh[T] // ExpToTrig\) // MySimplify)\) // \(Collect[#, {P\_x_, f_[a_. \[Phi]]}] &\) //. ContractAmplitudeProducts;\)\), "\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]} // Simplify\)}], "Input", CellLabel->"In[101]:="], Cell[BoxData[ \({R\_TTh[0] \[Rule] 0, R\_TTh[N] \[Rule] 0, R\_TTh[L] \[Rule] 1\/2\ \((\(-2\)\ Abs[\[ScriptCapitalF]\_1]\^2 - 2\ Abs[\[ScriptCapitalF]\_2]\^2 + 4\ Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_2)\)\^*\)] - Re[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_3)\)\^*\)] + Cos[2\ \[Theta]]\ Re[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_3)\)\^*\)] - Re[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Cos[2\ \[Theta]]\ Re[\[ScriptCapitalF]\_1\ \(\((\ \[ScriptCapitalF]\_4)\)\^*\)])\), R\_TTh[S] \[Rule] \((\(-Re[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\ \_3)\)\^*\)]\) + Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)] - Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Re[\[ScriptCapitalF]\_2\ \(\((\[ScriptCapitalF]\_4)\)\^*\)])\)\ \ Sin[\[Theta]]}\)], "Output", CellLabel->"Out[102]="] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Parallel/antiparallel kinematics", "Section"], Cell[CellGroupData[{ Cell["Nucleon parallel to q", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Hparallel = \(HelicityToCGLN /. phases[JW]\) /. {\[Theta] \[Rule] \[Pi], \[Phi] \[Rule] 0} // Simplify\)], "Input", CellLabel->"In[103]:="], Cell[BoxData[ \({H\_1 \[Rule] 0, H\_2 \[Rule] 0, H\_3 \[Rule] 0, H\_4 \[Rule] \@2\ \((\[ScriptCapitalF]\_1 + \[ScriptCapitalF]\_2)\), H\_5 \[Rule] 0, H\_6 \[Rule] \(Q\ \((\[ScriptCapitalF]\_5 - \[ScriptCapitalF]\_6)\)\)\/\ \[Omega]}\)], "Output", CellLabel->"Out[103]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(R\_L[H] /. Hparallel\)], "Input", CellLabel->"In[104]:="], Cell[BoxData[ \({R\_L[ 0] \[Rule] \(q\^2\ Abs[\(Q\ \((\[ScriptCapitalF]\_5 - \ \[ScriptCapitalF]\_6)\)\)\/\[Omega]]\^2\)\/Q\^2, R\_L[N] \[Rule] 0}\)], "Output", CellLabel->"Out[104]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(R\_T[H] /. Hparallel\)], "Input", CellLabel->"In[105]:="], Cell[BoxData[ \({R\_T[0] \[Rule] Abs[\[ScriptCapitalF]\_1 + \[ScriptCapitalF]\_2]\^2, R\_T[N] \[Rule] 0}\)], "Output", CellLabel->"Out[105]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(R\_LT[H] /. 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Hparallel\) /. {\[Phi] \[Rule] 0} // MySimplify\)], "Input", CellLabel->"In[142]:="], Cell[BoxData[ \(\(-Abs[\[ScriptCapitalF]\_1 + \[ScriptCapitalF]\_2]\^2\)\ P\_z\)], \ "Output", CellLabel->"Out[142]="] }, 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[143]:="], 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[144]:="], 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[145]:="], 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[146]:="], 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[147]:="], 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[148]:="], 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[149]:="] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Functions which perform multipole expansion of response functions\ \>", "Subsection"], Cell["\<\ 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.\ \>", "Text"], Cell[BoxData[ \(ExpandR[R\_\[Alpha]_[a_], lmax_Integer /; \((lmax \[GreaterEqual] 0)\)] := \ Module[{\[ScriptCapitalR]}, \n\t\t\[ScriptCapitalR] = \(\(\((\(\(R\_\ \[Alpha][a] /. R\_\[Alpha][\[ScriptCapitalF]]\) //. ExpandCGLN\) /. {\[ScriptL]max \[Rule] lmax, MySum \[Rule] Sum})\) //. rule[mp]\) /. ExpandLegendre\) /. x \[Rule] Cos[\[Theta]] // Simplify; \n MySimplify /@ \((\[ScriptCapitalR] //. ExpandMultipoleProducts // Collect[#, {Abs[_], Re[_], Im[_]}] &\ )\)]\)], "Input", CellLabel->"In[150]:="], 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[151]:="], Cell[BoxData[ \(AbbreviateMultipoleExpansion[expr_, choices_] := Simplify /@ \((\(\((If[MyFreeQ[#, choices], 0, #] &)\) /@ \ Expand[expr] // Simplify\) // Collect[#, {Abs[_], Re[_], Im[_]}] &)\)\)], "Input", CellLabel->"In[154]:="], Cell[TextData[{ "For example, we can enforce ", Cell[BoxData[ \(TraditionalForm\`\(M\_+\)[1]\)]], " 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[155]:="], Cell[BoxData[ \(1\/4\ Abs[\(M\_+\)[1]]\^2\ \((7 - 3\ Cos[2\ \[Theta]])\) + 2\ Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\/2\ \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 1\/2\ \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(M\_-\)[1]\^*\)]\)], "Output", CellLabel->"Out[155]="] }, 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[ \(ExpandR[R\_L[0], 1] // FullSimplify\)], "Input", CellLabel->"In[156]:="], Cell[BoxData[ \(Abs[\(S\_-\)[1]]\^2 + Abs[\(S\_+\)[0]]\^2 + 2\ \((Abs[\(S\_+\)[1]]\^2\ \((5 + 3\ Cos[2\ \[Theta]])\) + Cos[\[Theta]]\ \((Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[0]] + 4\ Re[\(\(S\_+\)[1]\^*\)\ \(S\_+\)[0]])\) + \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[ 1]])\)\)], "Output", CellLabel->"Out[156]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_T[0], 1] // FullSimplify\)], "Input", CellLabel->"In[157]:="], Cell[BoxData[ \(Abs[\(E\_+\)[0]]\^2 + 1\/4\ \((4\ Abs[\(M\_-\)[1]]\^2 + Abs[\(M\_+\)[1]]\^2\ \((7 - 3\ Cos[2\ \[Theta]])\) + 9\ Abs[\(E\_+\)[1]]\^2\ \((3 + Cos[2\ \[Theta]])\) + 8\ Cos[\[Theta]]\ \((\(-Re[\(M\_-\)[1]\ \(\(E\_+\)[0]\^*\)]\) + Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) - 2\ \((1 + 3\ Cos[2\ \[Theta]])\)\ \((3\ Re[\(M\_-\)[ 1]\ \(\(E\_+\)[1]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\))\)\)], "Output", CellLabel->"Out[157]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_L[N], 1] // FullSimplify\)], "Input", CellLabel->"In[158]:="], Cell[BoxData[ \(\(-2\)\ \((Im[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[0]] - 2\ Im[\(\(S\_+\)[1]\^*\)\ \(S\_+\)[0]] + 6\ Cos[\[Theta]]\ Im[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[ 1]])\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[158]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandR[R\_T[N], 1] // FullSimplify\)], "Input", CellLabel->"In[159]:="], Cell[BoxData[ \(3\ \((Im[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Im[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] - Cos[\[Theta]]\ \((Im[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 4\ Im[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Im[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[159]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpR\_LTh[L] = ExpandR[R\_LTh[L], 1] // FullSimplify\)], "Input", CellLabel->"In[160]:="], Cell[BoxData[ \(\((Re[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] + Re[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)] + 3\ Cos[\[Theta]]\ \((Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 2\ \((\(-Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\) + Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\))\)\ Sin[\[Theta]]\)], "Output",\ CellLabel->"Out[160]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpR\_LTh[S] = ExpandR[R\_LTh[S], 1] // FullSimplify\)], "Input", CellLabel->"In[161]:="], Cell[BoxData[ \(\(-Re[\(M\_-\)[1]\ \(\(S\_-\)[1]\^*\)]\) - 2\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[0]\ \(\(S\_+\)[0]\^*\)] + Cos[\[Theta]]\ \((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]\^*\)] + 3\ Cos[\[Theta]]\ \((Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 4\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\))\) - \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + 4\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\)], "Output", CellLabel->"Out[161]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpR\_TTh[L] = ExpandR[R\_TTh[L], 1] // FullSimplify\)], "Input", CellLabel->"In[162]:="], Cell[BoxData[ \(1\/2\ \((\(-2\)\ Abs[\(E\_+\)[0]]\^2 - 2\ Abs[\(M\_-\)[1]]\^2 + Abs[\(M\_+\)[1]]\^2\ \((1 - 3\ Cos[2\ \[Theta]])\) - 2\ Cos[\[Theta]]\ \((9\ Abs[\(E\_+\)[1]]\^2\ Cos[\[Theta]] - 2\ Re[\(M\_-\)[1]\ \(\(E\_+\)[0]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) + 3\ \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 12\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(M\_-\)[1]\^*\)])\)\)], "Output", CellLabel->"Out[162]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpR\_TTh[S] = ExpandR[R\_TTh[S], 1] // FullSimplify\)], "Input", CellLabel->"In[163]:="], Cell[BoxData[ \(3\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] + Cos[\[Theta]]\ \((\(-3\)\ Abs[\(E\_+\)[1]]\^2 + Abs[\(M\_+\)[1]]\^2 + Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - Re[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[163]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(AbbreviateMultipoleExpansion[mpR\_LTh[L], \(M\_+\)[1]] // FullSimplify\)], "Input", CellLabel->"In[164]:="], Cell[BoxData[ \(\((2\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 3\ Cos[\[Theta]]\ \((Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 2\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[164]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(AbbreviateMultipoleExpansion[mpR\_LTh[S], \(M\_+\)[1]] // FullSimplify\)], "Input", CellLabel->"In[165]:="], Cell[BoxData[ \(\((\(-2\) + 3\ Cos[\[Theta]]\^2)\)\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 4\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\)], "Output", CellLabel->"Out[165]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(AbbreviateMultipoleExpansion[mpR\_TTh[L], \(M\_+\)[1]] // FullSimplify\)], "Input", CellLabel->"In[166]:="], Cell[BoxData[ \(1\/2\ \((Abs[\(M\_+\)[1]]\^2\ \((1 - 3\ Cos[2\ \[Theta]])\) - 4\ Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 12\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(M\_-\)[1]\^*\)])\)\)], "Output", CellLabel->"Out[166]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(AbbreviateMultipoleExpansion[mpR\_TTh[S], \(M\_+\)[1]] // FullSimplify\)], "Input", CellLabel->"In[167]:="], Cell[BoxData[ \(3\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Cos[\[Theta]]\ \((Abs[\(M\_+\)[1]]\^2 + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - Re[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[167]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`P\_33\)]], " dominance" }], "Subsubsection"], Cell[TextData[{ "Here we display expansions based upon ", Cell[BoxData[ \(TraditionalForm\`\(M\_+\)[1]\)]], " dominance, retaining only ", StyleBox["s", FontSlant->"Italic"], " and ", StyleBox["p", FontSlant->"Italic"], " waves." }], "Text"], Cell[BoxData[ \(mpBrief[P\_33, R_, angle_: \[Theta]] := AbbreviateMultipoleExpansion[ ExpandR[R, 1], {\(M\_+\)[1], \(E\_+\)[1], \(S\_+\)[ 1]}] /. {\[Theta] \[Rule] angle} // MyFullSimplify\)], "Input",\ CellLabel->"In[168]:="], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_L[0]]\)], "Input", CellLabel->"In[169]:="], Cell[BoxData[ \(2\ \((Abs[\(S\_+\)[1]]\^2\ \((5 + 3\ Cos[2\ \[Theta]])\) + 4\ Cos[\[Theta]]\ Re[\(\(S\_+\)[1]\^*\)\ \(S\_+\)[0]] + \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[ 1]])\)\)], "Output", CellLabel->"Out[169]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_T[0]]\)], "Input", CellLabel->"In[170]:="], Cell[BoxData[ \(1\/4\ \((Abs[\(M\_+\)[1]]\^2\ \((7 - 3\ Cos[2\ \[Theta]])\) + 9\ Abs[\(E\_+\)[1]]\^2\ \((3 + Cos[2\ \[Theta]])\) + 8\ Cos[\[Theta]]\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) - 2\ \((1 + 3\ Cos[2\ \[Theta]])\)\ \((3\ Re[\(M\_-\)[ 1]\ \(\(E\_+\)[1]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\))\)\)], "Output", CellLabel->"Out[170]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_LT[0]]\)], "Input", CellLabel->"In[171]:="], Cell[BoxData[ \(\((\(-3\)\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\)\ Sin[\[Theta]] - 3\ \((Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ Sin[ 2\ \[Theta]]\)], "Output", CellLabel->"Out[171]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_TT[0]]\)], "Input", CellLabel->"In[172]:="], 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[172]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_LT[N]]\)], "Input", CellLabel->"In[173]:="], Cell[BoxData[ \(\(-3\)\ Cos[2\ \[Theta]]\ Im[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - Cos[\[Theta]]\ \((3\ Im[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 4\ Im[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\) - 3\ \((3 + Cos[2\ \[Theta]])\)\ Im[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + \((1 + 3\ Cos[2\ \[Theta]])\)\ \((Im[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - Im[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\)], "Output", CellLabel->"Out[173]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_LTh[S]]\)], "Input", CellLabel->"In[174]:="], Cell[BoxData[ \(\(-2\)\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Cos[\[Theta]]\ \((3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 4\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)] + 3\ Cos[\[Theta]]\ \((Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 4\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\))\) - \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + 4\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\)], "Output", CellLabel->"Out[174]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_LTh[L]]\)], "Input", CellLabel->"In[175]:="], Cell[BoxData[ \(\((2\ \((Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)])\) + 3\ Cos[\[Theta]]\ \((Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 2\ \((\(-Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\) + Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\))\)\ Sin[\[Theta]]\)], "Output",\ CellLabel->"Out[175]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_TTh[S]]\)], "Input", CellLabel->"In[176]:="], Cell[BoxData[ \(3\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] + Cos[\[Theta]]\ \((\(-3\)\ Abs[\(E\_+\)[1]]\^2 + Abs[\(M\_+\)[1]]\^2 + Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - Re[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[176]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpBrief[P\_33, R\_TTh[L]]\)], "Input", CellLabel->"In[177]:="], Cell[BoxData[ \(1\/2\ \((Abs[\(M\_+\)[1]]\^2\ \((1 - 3\ Cos[2\ \[Theta]])\) - 2\ Cos[\[Theta]]\ \((9\ Abs[\(E\_+\)[1]]\^2\ Cos[\[Theta]] + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)])\) + 3\ \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 12\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(M\_-\)[1]\^*\)])\)\)], "Output", CellLabel->"Out[177]="] }, 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_, angle_: \[Theta]] := AbbreviateMultipoleExpansion[ ExpandR[R, 1], {\(M\_+\)[1]}] /. {\[Theta] \[Rule] angle} // MyFullSimplify\)], "Input", CellLabel->"In[178]:="], Cell[BoxData[ \(fbsum[P\_33, R_] := \n\t\t\((mpVeryBrief[P\_33, R, \[Theta]] + mpVeryBrief[P\_33, R, \[Theta] - \[Pi]])\) // Simplify\)], "Input",\ CellLabel->"In[179]:="], Cell[BoxData[ \(fbdiff[P\_33, R_] := \n\t\t\((mpVeryBrief[P\_33, R, \[Theta]] - mpVeryBrief[P\_33, R, \[Theta] - \[Pi]])\) // Simplify\)], "Input",\ CellLabel->"In[180]:="], Cell[BoxData[ \(fb[P\_33, R_] := Module[{f, b}, \n\t\tf = mpVeryBrief[P\_33, R, \[Pi]]; \n\t\tb = mpVeryBrief[P\_33, R, 0]; \n\t\t\((f - b)\)/\((f + b)\) // Simplify]\)], "Input", CellLabel->"In[181]:="], 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[182]:="], Cell[BoxData[ \(1\/4\ \((Abs[\(M\_+\)[1]]\^2\ \((7 - 3\ Cos[2\ \[Theta]])\) + 8\ Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 2\ \((1 + 3\ Cos[2\ \[Theta]])\)\ \((\(-3\)\ Re[\(M\_+\)[ 1]\ \(\(E\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\))\)\)], "Output", CellLabel->"Out[182]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_LT[0]]\)], "Input", CellLabel->"In[183]:="], Cell[BoxData[ \(\((Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 6\ Cos[\[Theta]]\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)])\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[183]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_TT[0]]\)], "Input", CellLabel->"In[184]:="], 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[184]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_LT[N]]\)], "Input", CellLabel->"In[185]:="], Cell[BoxData[ \(\(-Im[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)]\) - Cos[\[Theta]]\ Im[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - \((1 + 3\ Cos[2\ \[Theta]])\)\ Im[\(M\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)]\)], "Output", CellLabel->"Out[185]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_LTh[S]]\)], "Input", CellLabel->"In[186]:="], Cell[BoxData[ \(\((\(-2\) + 3\ Cos[\[Theta]]\^2)\)\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 4\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\)], "Output", CellLabel->"Out[186]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_TTh[S]]\)], "Input", CellLabel->"In[187]:="], Cell[BoxData[ \(3\ \((Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + Cos[\[Theta]]\ \((Abs[\(M\_+\)[1]]\^2 + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - Re[\(M\_+\)[ 1]\ \(\(M\_-\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[187]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_LTh[L]]\)], "Input", CellLabel->"In[188]:="], Cell[BoxData[ \(\((2\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 3\ Cos[\[Theta]]\ \((Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 2\ Re[\(M\_+\)[ 1]\ \(\(S\_+\)[ 1]\^*\)])\))\)\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[188]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_TTh[L]]\)], "Input", CellLabel->"In[189]:="], Cell[BoxData[ \(1\/2\ \((Abs[\(M\_+\)[1]]\^2\ \((1 - 3\ Cos[2\ \[Theta]])\) - 4\ Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 12\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(M\_-\)[1]\^*\)])\)\)], "Output", CellLabel->"Out[189]="] }, 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[ \(fbsum[P\_33, R\_LT[0]]\)], "Input", CellLabel->"In[190]:="], Cell[BoxData[ \(6\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\ Sin[2\ \[Theta]]\)], "Output", CellLabel->"Out[190]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fbdiff[P\_33, R\_LT[0]]\)], "Input", CellLabel->"In[191]:="], Cell[BoxData[ \(2\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)]\ Sin[\[Theta]]\)], "Output", CellLabel->"Out[191]="] }, 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], \[Pi]]\)], "Input", CellLabel->"In[192]:="], Cell[BoxData[ \(Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 4\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\)], "Output", CellLabel->"Out[192]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_LTh[S], 0]\)], "Input", CellLabel->"In[193]:="], Cell[BoxData[ \(Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + 4\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\)], "Output", CellLabel->"Out[193]="] }, 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[ \(fbsum[P\_33, R\_LTh[S]]\)], "Input", CellLabel->"In[194]:="], Cell[BoxData[ \(\((\(-1\) + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 8\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\)], "Output", CellLabel->"Out[194]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fbdiff[P\_33, R\_LTh[S]]\)], "Input", CellLabel->"In[195]:="], Cell[BoxData[ \(2\ Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)]\)], "Output", CellLabel->"Out[195]="] }, Open ]], Cell["Finally,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(mpVeryBrief[P\_33, R\_TTh[L]]\)], "Input", CellLabel->"In[196]:="], Cell[BoxData[ \(1\/2\ \((Abs[\(M\_+\)[1]]\^2\ \((1 - 3\ Cos[2\ \[Theta]])\) - 4\ Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] - 12\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(M\_-\)[1]\^*\)])\)\)], "Output", CellLabel->"Out[196]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fbsum[P\_33, R\_TTh[L]]\)], "Input", CellLabel->"In[197]:="], Cell[BoxData[ \(Abs[\(M\_+\)[1]]\^2\ \((1 - 3\ Cos[2\ \[Theta]])\) - 12\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + \((1 + 3\ Cos[2\ \[Theta]])\)\ Re[\(M\_+\)[ 1]\ \(\(M\_-\)[1]\^*\)]\)], "Output", CellLabel->"Out[197]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fbdiff[P\_33, R\_TTh[L]]\)], "Input", CellLabel->"In[198]:="], Cell[BoxData[ \(\(-4\)\ Cos[\[Theta]]\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)]\)], "Output",\ CellLabel->"Out[198]="] }, 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[ \(fbsum[P\_33, R\_LTh[L]]\)], "Input", CellLabel->"In[199]:="], Cell[BoxData[ \(3\ \((Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\ Sin[ 2\ \[Theta]]\)], "Output", CellLabel->"Out[199]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fbsum[P\_33, R\_TTh[S]]\)], "Input", CellLabel->"In[200]:="], Cell[BoxData[ \(3\ \((Abs[\(M\_+\)[1]]\^2 + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)])\)\ Sin[ 2\ \[Theta]]\)], "Output", CellLabel->"Out[200]="] }, 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], \[Pi]])\) // Simplify\)], "Input", CellLabel->"In[201]:="], Cell[BoxData[ \(3\ Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + 4\ \((3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)])\)\)], "Output", CellLabel->"Out[201]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(1\/2\) \((mpBrief[P\_33, R\_TTh[L], 0] + mpBrief[P\_33, R\_TTh[L], \[Pi]])\) // Simplify\)], "Input", CellLabel->"In[202]:="], Cell[BoxData[ \(\(-9\)\ 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[202]="] }, 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. 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