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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[ 25734, 860]*) (*NotebookOutlinePosition[ 26537, 888]*) (* CellTagsIndexPosition[ 26493, 884]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Two Pion Potentials", "Title", TextAlignment->Center, TextJustification->0, FontSize->18], Cell["\<\ In this notebook we'll study the two-pion potentials defined by: 1) Friar, nucl-th/9901082 2) Rentmeester et al (Nijmegan), nucl-th/9901054 3) Kaiser, Brockmann, Weise (eventually)\ \>", "Text"], Cell["\<\ Summary conclusions: 1) Friar and Nijmegan agree, given Friar's prescription for relating the \ potentials 2) Nijmegan and Kaiser et al. agree in each term of the two-pion potential, \ up to an overall factor of M/E (which is set to 1 below).\ \>", "Text"], Cell[CellGroupData[{ Cell["Clear symbols", "Section", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ In order to avoid interference from symbols defined in other \ notebooks, we first Remove all symbols. We assume that the relevant symbols \ are in the Global` context.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Clear[\"Global`*\"]", "Input", AspectRatioFixed->True], Cell["Remove[\"Global`*\"]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(Off[General::spell1]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["General Definitions and Substitutions", "Section", FontSize->14], Cell[BoxData[ \(c04t\ = \ c0t\ + \ 4\ c4t; \.03\)], "Input"], Cell[BoxData[ \(\(c0t\ = \ 1/gA^2; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Partial Waves", "Section", FontSize->14], Cell["\<\ Here we define matrix elements for different partial waves. How shall we define the mixed elements?\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The tensor operator is defined here as:\n ", Cell[BoxData[ \(TraditionalForm\`S\_12\)]], "= 3 (\\sigma_1 \\cdot \\hat r)(\\sigma_2 \\cdot \\hat r) - \\sigma_1 \ \\cdot \\sigma_2\n We need to check this formula!" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(tensorS12[L_, Lp_, S_, J_]\ := \n\t Which[S == 0, \ 0, \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ S == 1, \ \ \n\t\ \ Which[L == J + 1, \ \ Which[Lp == J + 1, \ \(-2\) J \((J + 2)\)/\((2 J + 1)\), \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Lp == J, \ 0, \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Lp == J - 1, \ 6 Sqrt[J \((J + 1)\)]/\((2 J + 1)\), \n \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ True, \ 0], \n \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ L == J, \ \ \ \ \ \ \ Which[Lp == J + 1, \ 0, \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Lp == J, \ 2, \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Lp == J - 1, \ 0, \n \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ True, \ 0], \n \t\t\t\ \ \ \ \ \ \ \ \ \ \ L == J - 1, \ \ \ \ Which[Lp == J + 1, \ 6 Sqrt[J \((J + 1)\)]/\((2 J + 1)\), \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Lp == J, \ 0, \n \t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Lp == J - 1, \ \(-2\) J \((J - 1)\)/\((2 J + 1)\), \n \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ True, \ 0], \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ True, \ 0\n\t\ ], \n \t\t\ \ \ \ \ \ \ \ \ \ True, \ 0\n]\)], "Input"], Cell[BoxData[""], "Input"], Cell[BoxData[ \(\(pwBuild[L_, S_, J_, T_, name_]\ := \ \((\n\ \ \ tau12\ = \ 2 T \((T + 1)\) - 3; \n\ \ \ sigma12\ = \ 2 S \((S + 1)\) - 3; \n\t\ LdotS\ = \ 1/2 \((\ J \((J + 1)\)\ - \ L \((L + 1)\)\ - S \((S + 1)\)\ )\); \n\t\ S12\ = \ tensorS12[L, L, S, J]; \ \n\ \ \ \ Print["\", name]; \n\t\ \ Print[\*"\"\<\!\(\[Tau]\_12\) = \>\"", tau12]; \n\t\ \ Print[\*"\"\<\!\(\[Sigma]\_12\) = \>\"", sigma12]; \n\t\ \ Print["\", LdotS]; \n\ \t\ Print[\*"\"\<\!\(S\_12\) = \>\"", S12])\)\ \)\)], "Input"], Cell[BoxData[ \(pw1S0\ := \ pwBuild[0, 0, 0, 1, "\<1S0\>"]\)], "Input"], Cell[BoxData[ \(pw1P1\ := \ pwBuild[1, 0, 1, 1, "\<1P1\>"]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(pw1P1\)], "Input"], Cell[BoxData[ InterpretationBox[\("partial wave: "\[InvisibleSpace]"1P1"\), SequenceForm[ "partial wave: ", "1P1"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("\!\(\[Tau]\_12\) = "\[InvisibleSpace]1\), SequenceForm[ "\!\(\[Tau]\_12\) = ", 1], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("\!\(\[Sigma]\_12\) = "\[InvisibleSpace]\(-3\)\), SequenceForm[ "\!\(\[Sigma]\_12\) = ", -3], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("L\[CenterDot]S = "\[InvisibleSpace]0\), SequenceForm[ "L\[CenterDot]S = ", 0], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("\!\(S\_12\) = "\[InvisibleSpace]0\), SequenceForm[ "\!\(S\_12\) = ", 0], Editable->False]], "Print"] }, Open ]], Cell[BoxData[ \(pwClear\ := \ \((S12 =. ; \ sigma12 =. ; \ tau12 =. ; \ LdotS =. ; \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Print["\"])\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(pwClear\)], "Input"], Cell[BoxData[ \("Partial wave variables have been cleared."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pw1S0\)], "Input"], Cell[BoxData[ InterpretationBox[\("partial wave: "\[InvisibleSpace]"1S0"\), SequenceForm[ "partial wave: ", "1S0"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("\!\(\[Tau]\_12\) = "\[InvisibleSpace]1\), SequenceForm[ "\!\(\[Tau]\_12\) = ", 1], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("\!\(\[Sigma]\_12\) = "\[InvisibleSpace]\(-3\)\), SequenceForm[ "\!\(\[Sigma]\_12\) = ", -3], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("L\[CenterDot]S = "\[InvisibleSpace]0\), SequenceForm[ "L\[CenterDot]S = ", 0], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("\!\(S\_12\) = "\[InvisibleSpace]0\), SequenceForm[ "\!\(S\_12\) = ", 0], Editable->False]], "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Nijmegan Stuff", "Section", FontSize->14], Cell[BoxData[ \(\(xi\ = \ 1; \)\)], "Input"], Cell[BoxData[ \(\(f\ = \ \[Alpha]; \)\)], "Input"], Cell[BoxData[ \(\(rel\ = \ 1; \)\)], "Input"], Cell[BoxData[ \(\(avC\ = \ {3/4, 9 + 48\ c1t\ + \ 24 c3t, \ 27 + 96 c1t + 96 c3t, \n \t\t\ \ \ \ \ \ \ \ \ \ \ 99/2 + 48\ c1t\ + \ 240\ c3t, \ 54 + 288\ c3t, \ 27\ + \ 144\ c3t}; \)\)], "Input"], Cell[BoxData[ \(\(avS\ = \ {0, \(-3\), \(-9\), \(-33\)/2, \(-18\), \(-9\)}; \)\)], "Input"], Cell[BoxData[ \(\(avT\ = \ {0, 3/2, 27/4, 15, 18, 9}; \)\)], "Input"], Cell[BoxData[ \(\(avSO\ = \ \ {0, \ 0, \ \(-12\), \ \(-36\), \ \(-48\), \ \(-24\)}; \)\)], "Input"], Cell[BoxData[ \(\(awC\ = \ {3/2, 4 - 2\ c0t, \ 14 - 8 c0t, \ 31 - 20\ c0t, \ 36 - 24\ c0t, \n \t\t\ \ \ \ \ \ \ \ \ \ \ 18 - 12\ c0t}; \)\)], "Input"], Cell[BoxData[ \(\(awS\ = \ {0, \(-2\)/3, \(-14\)/3 + 8\ c04t/3, \ \(-31\)/3 + 20\ c04t/3, \n \t\t\ \ \ \ \ \ \ \ \ \ \(-12\) + 8\ c04t, \ \(-6\) + 4\ c04t}; \)\)], "Input"], Cell[BoxData[ \(\(awT\ = \ {0, \ 1/3, \ 17/6 - 4 c04t/3, 26/3 - 16 c04t/3, 12 - 8 c04t, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6 - 4 c04t}; \)\)], "Input"], Cell[BoxData[ \(\(awSO\ = \ {0, 0, 0, 8 - 8 c0t, 16 - 16 c0t, 8 - 8 c0t}; \)\)], "Input"], Cell[BoxData[ \(\(parray\ = \ {1/x, 1/x^2, 1/x^3, 1/x^4, 1/x^5, 1/x^6}; \)\)], "Input"], Cell[BoxData[ \(\(vC1\ = \ 0; \)\)], "Input"], Cell[BoxData[ \(\(vS1\ = \ 12\ K0/x^3\ + \ \((12 + 8 x^2)\) K1/x^4; \)\)], "Input"], Cell[BoxData[ \(\(vT1\ = \ \(-12\)\ K0/x^3\ - \ \((15 + 4\ x^2)\) K1/x^4; \)\)], "Input"], Cell[BoxData[ \(\(vSO1\ = \ 0; \)\)], "Input"], Cell[BoxData[ \(\(wC1\ = \ \((c0t^2\ + \ 10 c0t\ - \ 23\ - \ 4 x^2)\)\ K0/x^3\ + \n \t\t\ \ \ \ \ \ \((c0t^2\ + \ 10 c0t\ - \ 23\ + \ \((4 c0t\ - 12)\)\ x^2)\)\ K1/x^4; \)\)], "Input"], Cell[BoxData[ \(\(wS1\ = \ 0; \)\)], "Input"], Cell[BoxData[ \(\(wT1\ = \ 0; \)\)], "Input"], Cell[BoxData[ \(\(wSO1\ = \ 0; \)\)], "Input"], Cell[BoxData[ \(\(vC2\ = \ Exp[\(-2\) x]\ *\ Apply[Plus, avC*parray]; \)\)], "Input"], Cell[BoxData[ \(\(vS2\ = \ Exp[\(-2\) x]\ *\ Apply[Plus, avS*parray]; \)\)], "Input"], Cell[BoxData[ \(\(vT2\ = \ Exp[\(-2\) x]\ *\ Apply[Plus, avT*parray]; \)\)], "Input"], Cell[BoxData[ \(\(vSO2\ = \ Exp[\(-2\) x]\ *\ Apply[Plus, avSO*parray]; \)\)], "Input"], Cell[BoxData[ \(\(wC2\ = \ Exp[\(-2\) x]\ *\ Apply[Plus, awC*parray]; \)\)], "Input"], Cell[BoxData[ \(\(wS2\ = \ Exp[\(-2\) x]\ *\ Apply[Plus, awS*parray]; \)\)], "Input"], Cell[BoxData[ \(\(wT2\ = \ Exp[\(-2\) x]\ *\ Apply[Plus, awT*parray]; \)\)], "Input"], Cell[BoxData[ \(\(wSO2\ = \ Exp[\(-2\) x]\ *\ Apply[Plus, awSO*parray]; \)\)], "Input"], Cell[BoxData[ \(\(vC\ = \ \((2/Pi)\) vC1\ + \ \((m\[Pi]/M)\) vC2; \)\)], "Input"], Cell[BoxData[ \(\(wC\ = \ \((2/Pi)\) wC1\ + \ \((m\[Pi]/M)\) wC2; \)\)], "Input"], Cell[BoxData[ \(\(vS\ = \ \((2/Pi)\) vS1\ \ + \ \((m\[Pi]/M)\) vS2; \)\)], "Input"], Cell[BoxData[ \(\(wS\ = \ \((2/Pi)\) wS1\ + \ \((m\[Pi]/M)\) wS2; \)\)], "Input"], Cell[BoxData[ \(\(vT\ = \ \((2/Pi)\) vT1\ \ + \ \((m\[Pi]/M)\) vT2; \)\)], "Input"], Cell[BoxData[ \(\(wT\ = \ \((2/Pi)\) wT1\ + \ \((m\[Pi]/M)\) wT2; \)\)], "Input"], Cell[BoxData[ \(\(vSO\ = \ \((2/Pi)\) vSO1\ \ + \ \((m\[Pi]/M)\) vSO2; \)\)], "Input"], Cell[BoxData[ \(\(wSO\ = \ \((2/Pi)\) wSO1\ + \ \((m\[Pi]/M)\) wSO2; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(V2\[Pi]\ = \ Simplify[\ f^2\ xi^2\ rel\ m\[Pi]\ \((\ \((vC + \ tau12\ wC)\)\ \n \t\t\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \ sigma12\ \((vS + tau12\ wS)\)\ )\)\ ]\)], "Input"], Cell[BoxData[ \(\(-\(1\/\(4\ gA\^4\ M\ \[Pi]\ x\^6\)\(( E\^\(\(-2\)\ x\)\ m\[Pi]\ \((\(-8\)\ E\^\(2\ x\)\ M\ x\^2\ \((K1 + K0\ x)\) + 8\ gA\^2\ \((\(-2\)\ E\^\(2\ x\)\ M\ x\^2\ \((5\ K1 + 5\ K0\ x + 2\ K1\ x\^2)\) + m\[Pi]\ \[Pi]\ \((12 + 24\ x + 20\ x\^2 + 8\ x\^3 + x\^4)\))\) - gA\^4\ \(( \(-8\)\ E\^\(2\ x\)\ M\ x\^2\ \((59\ K1 + 59\ K0\ x + 36\ K1\ x\^2 + 4\ K0\ x\^3)\) + m\[Pi]\ \[Pi]\ \((360 + 576\ c3t - 192\ c4t + 720\ x + 1152\ c3t\ x - 384\ c4t\ x + 644\ x\^2 + 192\ c1t\ x\^2 + 960\ c3t\ x\^2 - 320\ c4t\ x\^2 + 328\ x\^3 + 384\ c1t\ x\^3 + 384\ c3t\ x\^3 - 128\ c4t\ x\^3 + 96\ x\^4 + 192\ c1t\ x\^4 + 96\ c3t\ x\^4 + 9\ x\^5) \))\))\)\ \[Alpha]\^2)\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(delV2\[Pi]\ = \ Simplify[\ f^2\ xi^2\ rel\ m\[Pi]\ \((m\[Pi]/M)\)\n \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((\ \((vC2 + tau12\ wC2)\)\ + \ sigma12\ \((vS2 + tau12\ wS2)\)\ ) \)\ ]\)], "Input"], Cell[BoxData[ \(\(1\/\(4\ gA\^2\ M\ x\^6\)\(( E\^\(\(-2\)\ x\)\ m\[Pi]\^2\ \((\(-8\)\ \((12 + 24\ x + 20\ x\^2 + 8\ x\^3 + x\^4)\) + gA\^2\ \(( 360 - 192\ c4t + 720\ x - 384\ c4t\ x + 644\ x\^2 + 192\ c1t\ x\^2 - 320\ c4t\ x\^2 + 328\ x\^3 + 384\ c1t\ x\^3 - 128\ c4t\ x\^3 + 96\ x\^4 + 192\ c1t\ x\^4 + 9\ x\^5 + 96\ c3t\ \((6 + 12\ x + 10\ x\^2 + 4\ x\^3 + x\^4)\))\))\)\ \[Alpha]\^2)\)\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Friar Stuff", "Section", FontSize->14], Cell[BoxData[ \(\(f0\ = \ Sqrt[\[Alpha]]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(g\ = \ \(-Exp[\(-x\)]\)\ \((1/x^2\ + \ 1/x^3)\)\)], "Input"], Cell[BoxData[ \(\(-E\^\(-x\)\)\ \((1\/x\^3 + 1\/x\^2)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(gp\ = \ Simplify[\ D[g, x]\ ]\)], "Input"], Cell[BoxData[ \(\(E\^\(-x\)\ \((3 + 3\ x + x\^2)\)\)\/x\^4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(gpp\ = \ Simplify[D[gp, x]]\)], "Input"], Cell[BoxData[ \(\(-\(\(E\^\(-x\)\ \((12 + 12\ x + 5\ x\^2 + x\^3)\)\)\/x\^5\)\)\)], "Output"] }, Open ]], Cell[BoxData[ \(\(\(delV2\[Pi]MNL\ = f0^4\ m\[Pi]^2\ \ /\((4 M)\)\ *\ \((\n\t\t\t 3\ \((\ 3\ x\ g\ \((4 gp\ + \ x\ gpp)\)\ - \ x^2\ gp\ \((2\ gp\ + \ x\ gpp)\)\ + \n\t\t\t\t\t 32\ c3t\ \((3\ g^2\ + \ x\ gp\ \((2 g\ + \ x\ gp)\)\ )\)\ + \ 64\ c1t\ x^2\ g^2\ )\)\n\t\t\t - \ 2\ tau12\ \((7\ x\ g \((4\ gp\ + \ x\ gpp)\)\ + \ 3\ x^2\ gp\ \((2\ gp\ + \ x\ gpp)\)\n \t\t\t\t\t\t\ \ \ \ \ \ + \ 4/gA^2\ \((3\ g^2\ + \ x\ gp\ \((2\ g\ + \ x\ gp)\)\ ) \)\ )\)\n\t\t\t + \ 1/3\ \((9\ + \ 2\ tau12)\)\ \((\ S12*\((x\ g\ gp\ + \ x^2\ \((gp^2\ + \ g\ gpp)\))\)\n \t\t\t\t\t\t - \ 2\ sigma12\ \((4\ x\ g\ gp\ + \ x^2\ \((gp^2\ + \ g\ gpp)\)\ )\)\ ) \)\n\t\t\t + \ 16/3\ tau12\ \((4\ c4t\ + \ 1/gA^2)\)\ \((S12*\((x\ g\ gp)\)\ - \ \n \t\t\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sigma12\ \((3\ g^2\ + \ 2\ x\ g\ gp)\)\ )\)\ )\)\ ; \)\n \t\t\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(V\[Pi]sq4M\ = \ Simplify[f0^4\ m\[Pi]^2\ /\ \((12\ M)\)\ *\ \((3\ - \ 2\ tau12)\)\ *\ \((\n\t\t\ \ 2\ S12\ \((x\ g\ gp)\)\ - \ 2\ sigma12\ \((3\ g^2\ + \ 2\ x\ g\ gp)\)\ + \n \t\t\t\t\ \ \ \ \ 3\ \((3\ g^2\ + \ 2\ x\ g\ gp\ \ + \ x^2\ gp^2)\)\ )\)\ ]\)], "Input"], Cell[BoxData[ \(\(E\^\(\(-2\)\ x\)\ m\[Pi]\^2\ \[Alpha]\^2\)\/\(4\ M\ x\^2\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Compare Nijmegan and Friar!", "Section", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[delV2\[Pi]\ - \ \((delV2\[Pi]MNL\ + \ 3\ V\[Pi]sq4M)\)\ ] \)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Kaiser Potentials", "Section", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(r\ = \ x/m\[Pi]\)], "Input"], Cell[BoxData[ \(x\/m\[Pi]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(VCkai\ = \ Simplify[\ 3\ gA^2\ /\ \((32\ Pi^2\ fpi^4)\)\ Exp[\(-2\) x]/r^6\ \ *\ \((\n\t\t\((2 c1\ + \ 3 gA^2/\((16 M)\))\) x^2 \((1 + x)\)^2\ + \ gA^2\ x^5/\((32\ M)\)\ + \n\t\t\t\t \((c3\ + \ 3 gA^2/\((16 M)\))\) \((6\ + \ 12 x\ + \ 10 x^2\ + \ 4 x^3\ + \ x^4)\)\ ) \)\ ]\)], "Input"], Cell[BoxData[ \(\(1\/\(32\ fpi\^4\ \[Pi]\^2\ x\^6\)\(( 3\ E\^\(\(-2\)\ x\)\ gA\^2\ m\[Pi]\^6\ \((\(gA\^2\ x\^5\)\/\(32\ M\) + \((2\ c1 + \(3\ gA\^2\)\/\(16\ M\))\)\ x\^2\ \((1 + x)\)\^2 + \((c3 + \(3\ gA\^2\)\/\(16\ M\))\)\ \((6 + 12\ x + 10\ x\^2 + 4\ x\^3 + x\^4)\))\))\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(WCkai\ = \ m\[Pi]/\((128\ Pi^3\ fpi^4\ r^4)\)\ *\ \((\n\t\t\ \((1\ + \ 2 gA^2 \((5 + 2 x^2)\)\ - \ gA^4 \((23 + 12 x^2)\)) \) K1\n\t\t\t + \ x \((1 + 10\ gA^2\ - \ gA^4 \((23 + 4 x^2)\))\) K0\n\t\t\t + \ gA^2\ Pi\ Exp[\(-2\) x]/\((4\ M\ r)\)\ *\ \((2\ \((3 gA^2 - 2)\)*\n\t\t\t\t\t\t\t \((6/x\ + \ 12\ + \ 10 x\ + \ 4 x^2\ + \ x^3)\)\n \t\t\t\t\t\t + \ gA^2\ x \((2\ + \ 4 x\ + \ 2 x^2\ + \ 3 x^3)\)\ )\)\ ) \)\)], "Input"], Cell[BoxData[ \(\(1\/\(128\ fpi\^4\ \[Pi]\^3\ x\^4\)\(( m\[Pi]\^5\ \((K0\ x\ \((1 + 10\ gA\^2 - gA\^4\ \((23 + 4\ x\^2)\))\) + K1\ \((1 + 2\ gA\^2\ \((5 + 2\ x\^2)\) - gA\^4\ \((23 + 12\ x\^2)\))\) + \(1\/\(4\ M\ x\)\(( E\^\(\(-2\)\ x\)\ gA\^2\ m\[Pi]\ \[Pi]\ \((2\ \((\(-2\) + 3\ gA\^2)\)\ \((12 + 6\/x + 10\ x + 4\ x\^2 + x\^3)\) + gA\^2\ x\ \((2 + 4\ x + 2\ x\^2 + 3\ x\^3)\))\))\)\))\)) \)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(VSkai\ = \ gA^4\ m\[Pi]/\((32\ Pi^3\ fpi^4\ r^4)\)\ *\ \((\n\t\t\ \ 3\ x\ K0\ + \ \((3\ + 2 x^2)\)\ K1\ - \ \n\t\t\t 3\ Pi\ Exp[\(-2\) x]/\((16\ M\ r)\)\ *\ \((6/x\ + \ 12\ + \ 11 x\ + \ 6 x^2\ + \ 2\ x^3\ )\)\ ) \)\)], "Input"], Cell[BoxData[ \(\(gA\^4\ m\[Pi]\^5\ \((3\ K0\ x + K1\ \((3 + 2\ x\^2)\) - 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