diff --git a/.cspell.json b/.cspell.json index ec942002..15b6b7ab 100644 --- a/.cspell.json +++ b/.cspell.json @@ -119,6 +119,7 @@ "bdist", "bgcolor", "boldsymbol", + "cbff", "celltoolbar", "clim", "cmap", @@ -136,6 +137,7 @@ "csqrt", "cstride", "darkred", + "dasharray", "displaystyle", "docstrings", "dotprint", @@ -156,6 +158,7 @@ "gridspec", "hasattr", "heatmap", + "histtype", "imag", "infty", "iplt", @@ -173,11 +176,14 @@ "kutschke", "lambdifier", "lambdifygenerated", + "linecap", + "linejoin", "linestyle", "linewidth", "linkcheck", "linspace", "livereveal", + "lstrip", "marangotto", "markdownlint", "mathbb", @@ -193,7 +199,9 @@ "nbformat", "nbmake", "ncols", + "ndarray", "nonlocal", + "nonumber", "noqa", "noreply", "nrows", @@ -222,6 +230,7 @@ "richman", "rpartition", "rstride", + "rstrip", "rtfd", "rules's", "savefig", @@ -237,6 +246,7 @@ "spflueger", "startswith", "subslide", + "substack", "suptitle", "symplot", "theano", @@ -252,6 +262,8 @@ "xlim", "xlink", "xreplace", + "xtick", + "xticklabels", "xticks", "ylabel", "ylim", diff --git a/.github/workflows/notebooks.yml b/.github/workflows/notebooks.yml index 1307603c..4bad6ff2 100644 --- a/.github/workflows/notebooks.yml +++ b/.github/workflows/notebooks.yml @@ -11,9 +11,9 @@ jobs: - uses: actions/checkout@master - uses: actions/setup-python@master with: - python-version: "3.7" + python-version: "3.8" - name: Install dependencies run: | sudo apt-get -y install graphviz - pip install -c .constraints/py3.7.txt .[test] + pip install -c .constraints/py3.8.txt .[test] - run: pytest --nbmake diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml index fffd607f..2c10fdd9 100644 --- a/.pre-commit-config.yaml +++ b/.pre-commit-config.yaml @@ -53,7 +53,8 @@ repos: (?x)^( docs/report/004.*| docs/report/005.*| - docs/report/007.* + docs/report/007.*| + docs/report/013.* )$ - id: set-nb-cells diff --git a/docs/conf.py b/docs/conf.py index 452119cf..67edbeed 100644 --- a/docs/conf.py +++ b/docs/conf.py @@ -271,6 +271,9 @@ def get_minor_version(package_name: str) -> str: "report/010*", "report/011*", "report/012*", + "report/013*", + "report/014*", + "report/015*", ] nb_output_stderr = "remove" nb_render_priority = { diff --git a/docs/report/013.ipynb b/docs/report/013.ipynb new file mode 100644 index 00000000..7f948a51 --- /dev/null +++ b/docs/report/013.ipynb @@ -0,0 +1,10021 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "hideCode": true, + "hideOutput": true, + "hidePrompt": true, + "jupyter": { + "source_hidden": true + }, + "slideshow": { + "slide_type": "skip" + }, + "tags": [ + "remove-cell" + ] + }, + "outputs": [], + "source": [ + "%config InlineBackend.figure_formats = ['svg']\n", + "import os\n", + "\n", + "STATIC_WEB_PAGE = {\"EXECUTE_NB\", \"READTHEDOCS\"}.intersection(os.environ)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "```{autolink-concat}\n", + "```" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# [TR-013] Spin alignment with data" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "```{autolink-skip}\n", + "```" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + ":::{seealso}\n", + "\n", + "- {doc}`/report/014`\n", + "- {doc}`/report/015`\n", + "\n", + ":::" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "jupyter": { + "source_hidden": true + }, + "tags": [ + "hide-cell" + ] + }, + "outputs": [], + "source": [ + "%pip install -q ampform==0.13.0 qrules[viz]==0.9.7 tensorwaves[jax,pwa]==0.4.2" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "tags": [] + }, + "source": [ + "In this report, we attempt to check the effect of activating spin alignment ([ampform#245](https://ampform--245.org.readthedocs.build/en/245/usage/helicity/spin-alignment.html)) and compare it with [Figure 2](https://downloads.hindawi.com/journals/ahep/2020/6674595.pdf#page=9) in {cite}`marangottoHelicityAmplitudesGeneric2020`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "jupyter": { + "source_hidden": true + }, + "tags": [ + "hide-cell" + ] + }, + "outputs": [], + "source": [ + "import logging\n", + "import warnings\n", + "\n", + "LOGGER = logging.getLogger()\n", + "LOGGER.setLevel(logging.ERROR)\n", + "warnings.filterwarnings(\"ignore\")" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "tags": [] + }, + "source": [ + "## Phase space sample" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "jupyter": { + "source_hidden": true + }, + "tags": [ + "hide-cell" + ] + }, + "outputs": [], + "source": [ + "import qrules\n", + "\n", + "PDG = qrules.load_pdg()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "tags": [] + }, + "outputs": [], + "source": [ + "from tensorwaves.data import (\n", + " TFPhaseSpaceGenerator,\n", + " TFUniformRealNumberGenerator,\n", + ")\n", + "\n", + "phsp_generator = TFPhaseSpaceGenerator(\n", + " initial_state_mass=PDG[\"Lambda(c)+\"].mass,\n", + " final_state_masses={\n", + " 0: PDG[\"p\"].mass,\n", + " 1: PDG[\"K-\"].mass,\n", + " 2: PDG[\"pi+\"].mass,\n", + " },\n", + ")\n", + "rng = TFUniformRealNumberGenerator(seed=0)\n", + "phsp_momenta = phsp_generator.generate(1_000_000, rng)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Generate transitions" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "jupyter": { + "source_hidden": true + }, + "tags": [ + "hide-cell" + ] + }, + "outputs": [], + "source": [ + "from qrules.particle import ParticleCollection, create_particle\n", + "\n", + "particle_db = ParticleCollection()\n", + "particle_db.add(PDG[\"Lambda(c)+\"])\n", + "particle_db.add(PDG[\"p\"])\n", + "particle_db.add(PDG[\"K-\"])\n", + "particle_db.add(PDG[\"pi+\"])\n", + "\n", + "particle_db.add(\n", + " create_particle(\n", + " PDG[\"K*(892)0\"],\n", + " name=\"K*\",\n", + " latex=\"K^*\",\n", + " )\n", + ")\n", + "particle_db.add(\n", + " create_particle(\n", + " PDG[\"Lambda(1405)\"],\n", + " name=\"Lambda*\",\n", + " latex=R\"\\Lambda^*\",\n", + " )\n", + ")\n", + "particle_db.add(\n", + " create_particle(\n", + " PDG[\"Delta(1232)++\"],\n", + " name=\"Delta*++\",\n", + " latex=R\"\\Delta^*\",\n", + " )\n", + ")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "tags": [] + }, + "outputs": [], + "source": [ + "reaction = qrules.generate_transitions(\n", + " initial_state=(\"Lambda(c)+\", [-0.5, +0.5]),\n", + " final_state=[\"p\", \"K-\", \"pi+\"],\n", + " formalism=\"helicity\",\n", + " particle_db=particle_db,\n", + ")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "jupyter": { + "source_hidden": true + }, + "tags": [ + "hide-input", + "full-width" + ] + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "g0_edge0\n", + "0: p\n", + "\n", + "\n", + "\n", + "g0_edge1\n", + "1: K-\n", + "\n", + "\n", + "\n", + "g0_edge2\n", + "2: pi+\n", + "\n", + "\n", + "\n", + "g0_edge-1\n", + "Lambda(c)+\n", + "\n", + "\n", + "\n", + "g0_node0\n", + "\n", + "\n", + "\n", + "g0_edge-1->g0_node0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node0->g0_edge1\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1\n", + "\n", + "\n", + "\n", + "g0_node0->g0_node1\n", + "\n", + "Delta*++\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge2\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "g0_edge0\n", + "0: p\n", + "\n", + "\n", + "\n", + "g0_edge1\n", + "1: K-\n", + "\n", + "\n", + "\n", + "g0_edge2\n", + "2: pi+\n", + "\n", + "\n", + "\n", + "g0_edge-1\n", + "Lambda(c)+\n", + "\n", + "\n", + "\n", + "g0_node0\n", + "\n", + "\n", + "\n", + "g0_edge-1->g0_node0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node0->g0_edge2\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1\n", + "\n", + "\n", + "\n", + "g0_node0->g0_node1\n", + "\n", + "Lambda*\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge1\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "g0_edge0\n", + "0: p\n", + "\n", + "\n", + "\n", + "g0_edge1\n", + "1: K-\n", + "\n", + "\n", + "\n", + "g0_edge2\n", + "2: pi+\n", + "\n", + "\n", + "\n", + "g0_edge-1\n", + "Lambda(c)+\n", + "\n", + "\n", + "\n", + "g0_node0\n", + "\n", + "\n", + "\n", + "g0_edge-1->g0_node0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node0->g0_edge0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1\n", + "\n", + "\n", + "\n", + "g0_node0->g0_node1\n", + "\n", + "K*\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge1\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge2\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import graphviz\n", + "\n", + "n = len(reaction.transitions)\n", + "for t in reaction.transitions[:: n // 3]:\n", + " dot = qrules.io.asdot([t], collapse_graphs=True, size=3.5)\n", + " graph = graphviz.Source(dot)\n", + " display(graph)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Distribution without alignment" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Amplitude model formulated following [Appendix C](https://downloads.hindawi.com/journals/ahep/2020/6674595.pdf#page=13):" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{m_{A}=-1/2}^{1/2} \\sum_{m_{0}=-1/2}^{1/2} \\sum_{m_{1}=0} \\sum_{m_{2}=0}{\\left|{{A^{01}}_{m_{A},m_{0},m_{1},m_{2}} + {A^{02}}_{m_{A},m_{0},m_{1},m_{2}} + {A^{12}}_{m_{A},m_{0},m_{1},m_{2}}}\\right|^{2}}$" + ], + "text/plain": [ + "PoolSum(Abs(A^01[m_A, m0, m1, m2] + A^02[m_A, m0, m1, m2] + A^12[m_A, m0, m1, m2])**2, (m_A, (1/2, -1/2)), (m0, (1/2, -1/2)), (m1, (0,)), (m2, (0,)))" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "import ampform\n", + "from ampform.dynamics.builder import RelativisticBreitWignerBuilder\n", + "\n", + "builder = ampform.get_builder(reaction)\n", + "builder.align_spin = False\n", + "builder.stable_final_state_ids = list(reaction.final_state)\n", + "builder.scalar_initial_state_mass = True\n", + "bw_builder = RelativisticBreitWignerBuilder()\n", + "for name in reaction.get_intermediate_particles().names:\n", + " builder.set_dynamics(name, bw_builder)\n", + "standard_model = builder.formulate()\n", + "standard_model.intensity" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "jupyter": { + "source_hidden": true + }, + "tags": [ + "hide-input", + "full-width" + ] + }, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{eqnarray}\n", + "{A^{01}}_{- \\frac{1}{2},- \\frac{1}{2},0,0} & = & \\frac{C_{\\Lambda_{c}^{+} \\to \\Lambda^*_{+1/2} \\pi^{+}_{0}; \\Lambda^* \\to K^{-}_{0} p_{+1/2}} \\Gamma_{\\Lambda^*} m_{\\Lambda^*} D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(- \\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right)}{- i \\Gamma_{\\Lambda^*} m_{\\Lambda^*} - m_{01}^{2} + \\left(m_{\\Lambda^*}\\right)^{2}} \\nonumber\\\\\n", + "& & + \\frac{C_{\\Lambda_{c}^{+} \\to \\Lambda^*_{-1/2} \\pi^{+}_{0}; \\Lambda^* \\to K^{-}_{0} p_{+1/2}} \\Gamma_{\\Lambda^*} m_{\\Lambda^*} D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right)}{- i \\Gamma_{\\Lambda^*} m_{\\Lambda^*} - m_{01}^{2} + \\left(m_{\\Lambda^*}\\right)^{2}} \n", + "\\end{eqnarray}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{eqnarray}\n", + "{A^{01}}_{- \\frac{1}{2},\\frac{1}{2},0,0} & = & \\frac{C_{\\Lambda_{c}^{+} \\to \\Lambda^*_{+1/2} \\pi^{+}_{0}; \\Lambda^* \\to K^{-}_{0} p_{+1/2}} \\Gamma_{\\Lambda^*} m_{\\Lambda^*} D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(- \\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right)}{- i \\Gamma_{\\Lambda^*} m_{\\Lambda^*} - m_{01}^{2} + \\left(m_{\\Lambda^*}\\right)^{2}} \\nonumber\\\\\n", + "& & + \\frac{C_{\\Lambda_{c}^{+} \\to \\Lambda^*_{-1/2} \\pi^{+}_{0}; \\Lambda^* \\to K^{-}_{0} p_{+1/2}} \\Gamma_{\\Lambda^*} m_{\\Lambda^*} D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right)}{- i \\Gamma_{\\Lambda^*} m_{\\Lambda^*} - m_{01}^{2} + \\left(m_{\\Lambda^*}\\right)^{2}} \n", + "\\end{eqnarray}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{eqnarray}\n", + "{A^{01}}_{\\frac{1}{2},- \\frac{1}{2},0,0} & = & \\frac{C_{\\Lambda_{c}^{+} \\to \\Lambda^*_{+1/2} \\pi^{+}_{0}; \\Lambda^* \\to K^{-}_{0} p_{+1/2}} \\Gamma_{\\Lambda^*} m_{\\Lambda^*} D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(- \\phi_{01},\\theta_{01},0\\right)}{- i \\Gamma_{\\Lambda^*} m_{\\Lambda^*} - m_{01}^{2} + \\left(m_{\\Lambda^*}\\right)^{2}} \\nonumber\\\\\n", + "& & + \\frac{C_{\\Lambda_{c}^{+} \\to \\Lambda^*_{-1/2} \\pi^{+}_{0}; \\Lambda^* \\to K^{-}_{0} p_{+1/2}} \\Gamma_{\\Lambda^*} m_{\\Lambda^*} D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi_{01},\\theta_{01},0\\right)}{- i \\Gamma_{\\Lambda^*} m_{\\Lambda^*} - m_{01}^{2} + \\left(m_{\\Lambda^*}\\right)^{2}} \n", + "\\end{eqnarray}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\dots$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import sympy as sp\n", + "from IPython.display import Math, display\n", + "\n", + "for i, (symbol, expr) in enumerate(standard_model.amplitudes.items()):\n", + " if i == 3:\n", + " display(Math(R\"\\dots\"))\n", + " break\n", + " latex = sp.multiline_latex(symbol, expr, environment=\"eqnarray\")\n", + " display(Math(latex))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Importing the parameter values given by [Table 1](https://downloads.hindawi.com/journals/ahep/2020/6674595.pdf#page=13):" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "jupyter": { + "source_hidden": true + }, + "tags": [ + "full-width", + "hide-cell" + ] + }, + "outputs": [], + "source": [ + "from ampform.helicity import HelicityModel\n", + "\n", + "# fmt: off\n", + "parameter_table = {\n", + " # K*\n", + " R\"C_{\\Lambda_{c}^{+} \\to K^*_{0} p_{+1/2}; K^* \\to K^{-}_{0} \\pi^{+}_{0}}\": 1,\n", + " R\"C_{\\Lambda_{c}^{+} \\to K^*_{+1} p_{+1/2}; K^* \\to K^{-}_{0} \\pi^{+}_{0}}\": 0.5 + 0.5j,\n", + " R\"C_{\\Lambda_{c}^{+} \\to K^*_{-1} p_{-1/2}; K^* \\to K^{-}_{0} \\pi^{+}_{0}}\": 1j,\n", + " R\"C_{\\Lambda_{c}^{+} \\to K^*_{0} p_{-1/2}; K^* \\to K^{-}_{0} \\pi^{+}_{0}}\": -0.5 - 0.5j,\n", + " \"m_{K^*}\": 0.9, # GeV\n", + " R\"\\Gamma_{K^*}\": 0.2, # GeV\n", + " # Λ*\n", + " R\"C_{\\Lambda_{c}^{+} \\to \\Lambda^*_{-1/2} \\pi^{+}_{0}; \\Lambda^* \\to K^{-}_{0} p_{+1/2}}\": 1j,\n", + " R\"C_{\\Lambda_{c}^{+} \\to \\Lambda^*_{+1/2} \\pi^{+}_{0}; \\Lambda^* \\to K^{-}_{0} p_{+1/2}}\": 0.8 - 0.4j,\n", + " R\"m_{\\Lambda^*}\": 1.6, # GeV\n", + " R\"\\Gamma_{\\Lambda^*}\": 0.2, # GeV\n", + " # Δ*\n", + " R\"C_{\\Lambda_{c}^{+} \\to \\Delta^*_{+1/2} K^{-}_{0}; \\Delta^* \\to p_{+1/2} \\pi^{+}_{0}}\": 0.6 - 0.4j,\n", + " R\"C_{\\Lambda_{c}^{+} \\to \\Delta^*_{-1/2} K^{-}_{0}; \\Delta^* \\to p_{+1/2} \\pi^{+}_{0}}\": 0.1j,\n", + " R\"m_{\\Delta^*}\": 1.4, # GeV\n", + " R\"\\Gamma_{\\Delta^*}\": 0.2, # GeV\n", + "}\n", + "# fmt: on\n", + "\n", + "\n", + "def set_coefficients(model: HelicityModel) -> None:\n", + " for name, value in parameter_table.items():\n", + " model.parameter_defaults[name] = value" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "jupyter": { + "source_hidden": true + }, + "tags": [ + "hide-input" + ] + }, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{array}{lc}\n", + " C_{\\Lambda_{c}^{+} \\to K^*_{0} p_{+1/2}; K^* \\to K^{-}_{0} \\pi^{+}_{0}} & 1 \\\\\n", + " C_{\\Lambda_{c}^{+} \\to K^*_{+1} p_{+1/2}; K^* \\to K^{-}_{0} \\pi^{+}_{0}} & 0.5+0.5i \\\\\n", + " C_{\\Lambda_{c}^{+} \\to K^*_{-1} p_{-1/2}; K^* \\to K^{-}_{0} \\pi^{+}_{0}} & 1i \\\\\n", + " C_{\\Lambda_{c}^{+} \\to K^*_{0} p_{-1/2}; K^* \\to K^{-}_{0} \\pi^{+}_{0}} & -0.5-0.5i \\\\\n", + " m_{K^*} & 0.9 \\\\\n", + " \\Gamma_{K^*} & 0.2 \\\\\n", + " C_{\\Lambda_{c}^{+} \\to \\Lambda^*_{-1/2} \\pi^{+}_{0}; \\Lambda^* \\to K^{-}_{0} p_{+1/2}} & 1i \\\\\n", + " C_{\\Lambda_{c}^{+} \\to \\Lambda^*_{+1/2} \\pi^{+}_{0}; \\Lambda^* \\to K^{-}_{0} p_{+1/2}} & 0.8-0.4i \\\\\n", + " m_{\\Lambda^*} & 1.6 \\\\\n", + " \\Gamma_{\\Lambda^*} & 0.2 \\\\\n", + " C_{\\Lambda_{c}^{+} \\to \\Delta^*_{+1/2} K^{-}_{0}; \\Delta^* \\to p_{+1/2} \\pi^{+}_{0}} & 0.6-0.4i \\\\\n", + " C_{\\Lambda_{c}^{+} \\to \\Delta^*_{-1/2} K^{-}_{0}; \\Delta^* \\to p_{+1/2} \\pi^{+}_{0}} & 0.1i \\\\\n", + " m_{\\Delta^*} & 1.4 \\\\\n", + " \\Gamma_{\\Delta^*} & 0.2 \\\\\n", + "\\end{array}$" + ], + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "set_coefficients(standard_model)\n", + "\n", + "latex = R\"\\begin{array}{lc}\" + \"\\n\"\n", + "for par_name, value in parameter_table.items():\n", + " value = str(value).lstrip(\"(\").rstrip(\")\").replace(\"j\", \"i\")\n", + " symbol = sp.Symbol(par_name)\n", + " latex += Rf\" {sp.latex(symbol)} & {value} \\\\\" + \"\\n\"\n", + "latex += R\"\\end{array}\"\n", + "Math(latex)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "tags": [] + }, + "source": [ + "### Generate data" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "jupyter": { + "source_hidden": true + }, + "tags": [ + "hide-cell" + ] + }, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "from tensorwaves.data import SympyDataTransformer\n", + "from tensorwaves.function.sympy import create_function\n", + "\n", + "\n", + "def compute_sub_intensities(\n", + " model: HelicityModel, resonance_name: str, phsp, full_expression\n", + ") -> np.ndarray:\n", + " parameter_values = {}\n", + " for symbol, value in model.parameter_defaults.items():\n", + " if resonance_name not in symbol.name and symbol.name.startswith(\"C\"):\n", + " parameter_values[symbol] = 0\n", + " else:\n", + " parameter_values[symbol] = value\n", + " sub_expression = full_expression.subs(parameter_values)\n", + " sub_intensity = create_function(sub_expression, backend=\"jax\")\n", + " return np.array(sub_intensity(phsp).real)\n", + "\n", + "\n", + "def plot_distributions(model: HelicityModel) -> None:\n", + " helicity_transformer = SympyDataTransformer.from_sympy(\n", + " model.kinematic_variables, backend=\"jax\"\n", + " )\n", + " phsp = helicity_transformer(phsp_momenta)\n", + " phsp = {k: v.real for k, v in phsp.items()}\n", + "\n", + " full_expression = model.expression.doit()\n", + " substituted_expression = full_expression.xreplace(model.parameter_defaults)\n", + " intensity_func = create_function(substituted_expression, backend=\"jax\")\n", + " intensities_all = np.array(intensity_func(phsp).real)\n", + " intensities_k = compute_sub_intensities(\n", + " model, \"K^*\", phsp, full_expression\n", + " )\n", + " intensities_delta = compute_sub_intensities(\n", + " model, \"Delta^*\", phsp, full_expression\n", + " )\n", + " intensities_lambda = compute_sub_intensities(\n", + " model, \"Lambda^*\", phsp, full_expression\n", + " )\n", + "\n", + " fig, ax = plt.subplots(nrows=2, ncols=3, figsize=(8, 5))\n", + " hist_kwargs = dict(\n", + " bins=80,\n", + " histtype=\"step\",\n", + " )\n", + "\n", + " for x in ax.flatten():\n", + " x.set_yticks([])\n", + "\n", + " ax[0, 0].set_xlabel(\"$m^2(pK^-)$ [GeV$^2/c^4$]\")\n", + " ax[0, 1].set_xlabel(R\"$m^2(K^-\\pi^+)$ [GeV$^2/c^4$]\")\n", + " ax[0, 2].set_xlabel(R\"$m^2(p\\pi^+)$ [GeV$^2/c^4$]\")\n", + " ax[1, 0].set_xlabel(R\"$\\cos\\theta(p)$\")\n", + " ax[1, 1].set_xlabel(R\"$\\phi(p)$\")\n", + " ax[1, 2].set_xlabel(R\"$\\chi$\")\n", + "\n", + " for x, xticks in {\n", + " ax[0, 0]: [2, 2.5, 3, 3.5, 4, 4.5],\n", + " ax[0, 1]: [0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2],\n", + " ax[0, 2]: [1, 1.5, 2, 2.5, 3],\n", + " ax[1, 0]: [-1, -0.5, 0, 0.5, 1],\n", + " ax[1, 1]: [-3, -2, -1, 0, 1, 2, 3],\n", + " }.items():\n", + " x.set_xticks(xticks)\n", + " x.set_xticklabels(xticks)\n", + "\n", + " for weights, color, label in [\n", + " (intensities_all, \"red\", \"Model\"),\n", + " (intensities_k, \"orange\", R\"$K^*\\to\\,K^{^-}\\pi^+$\"),\n", + " (intensities_delta, \"brown\", R\"$\\Delta^{*^{++}} \\to\\,p\\pi^+$\"),\n", + " (intensities_lambda, \"purple\", R\"$\\Lambda^* \\to\\,p K^{^-}$\"),\n", + " ]:\n", + " kwargs = dict(weights=weights, color=color, **hist_kwargs)\n", + " ax[0, 0].hist(np.array(phsp[\"m_01\"] ** 2), **kwargs)\n", + " ax[0, 1].hist(np.array(phsp[\"m_12\"] ** 2), **kwargs)\n", + " ax[0, 2].hist(np.array(phsp[\"m_02\"] ** 2), **kwargs)\n", + " ax[1, 0].hist(np.array(np.cos(phsp[\"theta_01\"])), **kwargs)\n", + " ax[1, 1].hist(np.array(phsp[\"phi_01\"]), **kwargs, label=label)\n", + "\n", + " ax[1, 2].remove()\n", + " handles, labels = ax[1, 1].get_legend_handles_labels()\n", + " fig.legend(handles, labels, loc=\"lower right\")\n", + "\n", + " ax[0, 2].set_xlim(1, 3.4)\n", + " ax[1, 0].set_xlim(-1, +1)\n", + " ax[1, 1].set_xlim(-np.pi, +np.pi)\n", + "\n", + " fig.tight_layout()\n", + "\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "```{autolink-skip}\n", + "```" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "tags": [ + "hide-input" + ] + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " 2022-03-03T17:06:02.331547\n", + " image/svg+xml\n", + " \n", + " \n", + " Matplotlib v3.5.1, https://matplotlib.org/\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + 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\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + "\n" + ], + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "CPU times: user 48.5 s, sys: 3.57 s, total: 52.1 s\n", + "Wall time: 41.7 s\n" + ] + } + ], + "source": [ + "%%time\n", + "plot_distributions(standard_model)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Spin alignment sum" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now, with the spin alignment sum from [ampform#245](https://ampform--245.org.readthedocs.build/en/245/usage/helicity/spin-alignment.html) inserted:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "tags": [ + "full-width" + ] + }, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{m_{A}=-1/2}^{1/2} \\sum_{m_{0}=-1/2}^{1/2} \\sum_{m_{1}=0} \\sum_{m_{2}=0}{\\left|{\\sum_{\\lambda^{01}_{0}=-1/2}^{1/2} \\sum_{\\mu^{01}_{0}=-1/2}^{1/2} \\sum_{\\nu^{01}_{0}=-1/2}^{1/2} \\sum_{\\lambda^{01}_{1}=0} \\sum_{\\mu^{01}_{1}=0} \\sum_{\\nu^{01}_{1}=0} \\sum_{\\lambda^{01}_{2}=0}{{A^{01}}_{m_{A},\\lambda^{01}_{0},- \\lambda^{01}_{1},- \\lambda^{01}_{2}} D^{0}_{m_{1},\\nu^{01}_{1}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{0}_{m_{2},\\lambda^{01}_{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{\\mu^{01}_{1},\\lambda^{01}_{1}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{0}_{\\nu^{01}_{1},\\mu^{01}_{1}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{m_{0},\\nu^{01}_{0}}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{\\frac{1}{2}}_{\\mu^{01}_{0},\\lambda^{01}_{0}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\nu^{01}_{0},\\mu^{01}_{0}}\\left(\\phi_{01},\\theta_{01},0\\right)} + \\sum_{\\lambda^{02}_{0}=-1/2}^{1/2} \\sum_{\\mu^{02}_{0}=-1/2}^{1/2} \\sum_{\\nu^{02}_{0}=-1/2}^{1/2} \\sum_{\\lambda^{02}_{1}=0} \\sum_{\\lambda^{02}_{2}=0} \\sum_{\\mu^{02}_{2}=0} \\sum_{\\nu^{02}_{2}=0}{{A^{02}}_{m_{A},\\lambda^{02}_{0},- \\lambda^{02}_{1},- \\lambda^{02}_{2}} D^{0}_{m_{1},\\lambda^{02}_{1}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{m_{2},\\nu^{02}_{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{0}_{\\mu^{02}_{2},\\lambda^{02}_{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{0}_{\\nu^{02}_{2},\\mu^{02}_{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{m_{0},\\nu^{02}_{0}}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{\\frac{1}{2}}_{\\mu^{02}_{0},\\lambda^{02}_{0}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\nu^{02}_{0},\\mu^{02}_{0}}\\left(\\phi_{02},\\theta_{02},0\\right)} + \\sum_{\\lambda^{12}_{0}=-1/2}^{1/2} \\sum_{\\lambda^{12}_{1}=0} \\sum_{\\mu^{12}_{1}=0} \\sum_{\\nu^{12}_{1}=0} \\sum_{\\lambda^{12}_{2}=0} \\sum_{\\mu^{12}_{2}=0} \\sum_{\\nu^{12}_{2}=0}{{A^{12}}_{m_{A},\\lambda^{12}_{0},\\lambda^{12}_{1},- \\lambda^{12}_{2}} D^{0}_{m_{1},\\nu^{12}_{1}}\\left(\\alpha^{12}_{1},\\beta^{12}_{1},\\gamma^{12}_{1}\\right) D^{0}_{m_{2},\\nu^{12}_{2}}\\left(\\alpha^{12}_{2},\\beta^{12}_{2},\\gamma^{12}_{2}\\right) D^{0}_{\\mu^{12}_{1},\\lambda^{12}_{1}}\\left(\\phi^{12}_{1},\\theta^{12}_{1},0\\right) D^{0}_{\\mu^{12}_{2},\\lambda^{12}_{2}}\\left(\\phi^{12}_{1},\\theta^{12}_{1},0\\right) D^{0}_{\\nu^{12}_{1},\\mu^{12}_{1}}\\left(\\phi_{0},\\theta_{0},0\\right) D^{0}_{\\nu^{12}_{2},\\mu^{12}_{2}}\\left(\\phi_{0},\\theta_{0},0\\right) D^{\\frac{1}{2}}_{m_{0},\\lambda^{12}_{0}}\\left(\\phi_{0},\\theta_{0},0\\right)}}\\right|^{2}}$" + ], + "text/plain": [ + "PoolSum(Abs(PoolSum(A^01[m_A, lambda_0^01, -lambda_1^01, -lambda_2^01]*WignerD(0, m1, nu_1^01, alpha_1^01, beta_1^01, gamma_1^01)*WignerD(0, m2, lambda_2^01, phi_01, theta_01, 0)*WignerD(0, mu_1^01, lambda_1^01, phi_0^01, theta_0^01, 0)*WignerD(0, nu_1^01, mu_1^01, phi_01, theta_01, 0)*WignerD(1/2, m0, nu_0^01, alpha_0^01, beta_0^01, gamma_0^01)*WignerD(1/2, mu_0^01, lambda_0^01, phi_0^01, theta_0^01, 0)*WignerD(1/2, nu_0^01, mu_0^01, phi_01, theta_01, 0), (lambda_0^01, (-1/2, 1/2)), (mu_0^01, (-1/2, 1/2)), (nu_0^01, (-1/2, 1/2)), (lambda_1^01, (0,)), (mu_1^01, (0,)), (nu_1^01, (0,)), (lambda_2^01, (0,))) + PoolSum(A^02[m_A, lambda_0^02, -lambda_1^02, -lambda_2^02]*WignerD(0, m1, lambda_1^02, phi_02, theta_02, 0)*WignerD(0, m2, nu_2^02, alpha_2^02, beta_2^02, gamma_2^02)*WignerD(0, mu_2^02, lambda_2^02, phi_0^02, theta_0^02, 0)*WignerD(0, nu_2^02, mu_2^02, phi_02, theta_02, 0)*WignerD(1/2, m0, nu_0^02, alpha_0^02, beta_0^02, gamma_0^02)*WignerD(1/2, mu_0^02, lambda_0^02, phi_0^02, theta_0^02, 0)*WignerD(1/2, nu_0^02, mu_0^02, phi_02, theta_02, 0), (lambda_0^02, (-1/2, 1/2)), (mu_0^02, (-1/2, 1/2)), (nu_0^02, (-1/2, 1/2)), (lambda_1^02, (0,)), (lambda_2^02, (0,)), (mu_2^02, (0,)), (nu_2^02, (0,))) + PoolSum(A^12[m_A, lambda_0^12, lambda_1^12, -lambda_2^12]*WignerD(0, m1, nu_1^12, alpha_1^12, beta_1^12, gamma_1^12)*WignerD(0, m2, nu_2^12, alpha_2^12, beta_2^12, gamma_2^12)*WignerD(0, mu_1^12, lambda_1^12, phi_1^12, theta_1^12, 0)*WignerD(0, mu_2^12, lambda_2^12, phi_1^12, theta_1^12, 0)*WignerD(0, nu_1^12, mu_1^12, phi_0, theta_0, 0)*WignerD(0, nu_2^12, mu_2^12, phi_0, theta_0, 0)*WignerD(1/2, m0, lambda_0^12, phi_0, theta_0, 0), (lambda_0^12, (-1/2, 1/2)), (lambda_1^12, (0,)), (mu_1^12, (0,)), (nu_1^12, (0,)), (lambda_2^12, (0,)), (mu_2^12, (0,)), (nu_2^12, (0,))))**2, (m_A, (1/2, -1/2)), (m0, (1/2, -1/2)), (m1, (0,)), (m2, (0,)))" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "builder.align_spin = True\n", + "aligned_model = builder.formulate()\n", + "set_coefficients(aligned_model)\n", + "aligned_model.intensity" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + ":::{warning}\n", + "\n", + "It takes several minutes to lambdify the full expression and expressions for the Wigner rotation angles.\n", + "\n", + ":::\n", + "\n", + "```{autolink-skip}\n", + "```" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "tags": [ + "hide-input" + ] + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " 2022-03-03T17:14:02.866924\n", + " image/svg+xml\n", + " \n", + " \n", + " Matplotlib v3.5.1, https://matplotlib.org/\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " 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" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "CPU times: user 9min 14s, sys: 17.2 s, total: 9min 31s\n", + "Wall time: 8min\n" + ] + } + ], + "source": [ + "%%time\n", + "plot_distributions(aligned_model)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Compare with [Figure 2](https://downloads.hindawi.com/journals/ahep/2020/6674595.pdf#page=9). Note that the distributions differ close to threshold, because the distributions in the paper are produced [with form factors](https://ampform.readthedocs.io/en/0.12.x/api/ampform.dynamics.html#ampform.dynamics.relativistic_breit_wigner_with_ff) and an [energy-dependent width](https://ampform.readthedocs.io/en/0.12.x/api/ampform.dynamics.html#ampform.dynamics.EnergyDependentWidth)." + ] + } + ], + "metadata": { + "keep_output": true, + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "name": "python", + "version": "3.8.12" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/docs/report/014.ipynb b/docs/report/014.ipynb index 3783fefa..17500eb4 100644 --- a/docs/report/014.ipynb +++ b/docs/report/014.ipynb @@ -50,6 +50,18 @@ "```" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + ":::{seealso}\n", + "\n", + "- {doc}`/report/013`\n", + "- {doc}`/report/015`\n", + "\n", + ":::" + ] + }, { "cell_type": "code", "execution_count": null, @@ -146,7 +158,7 @@ "\n", ":::\n", "\n", - "[ampform#213](https://github.com/ComPWA/ampform/pull/213) implements spin alignment, which results in large sum combinatorics for all helicity combinations. The result is an amplitude model expression that is too large to be rendered as LaTeX.\n", + "[ampform#245](https://github.com/ComPWA/ampform/pull/245) implements spin alignment, which results in large sum combinatorics for all helicity combinations. The result is an amplitude model expression that is too large to be rendered as LaTeX.\n", "\n", "To some extend, this is already the case with the [current implementation](https://ampform.readthedocs.io/en/0.12.3/usage/formalism.html) of the 'standard' helicity formalism {cite}`jacobGeneralTheoryCollisions1959, richmanExperimenterGuideHelicity1984, kutschkeAngularDistributionCookbook1996, chungSpinFormalismsUpdated2014`: many of the terms in the total intensity expression differ only by the helicities of the final and initial state." ] @@ -205,7 +217,632 @@ "full-width" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "edge0\n", + "0: K-[0]\n", + "\n", + "\n", + "\n", + "edge1\n", + "1: p[-1/2]\n", + "\n", + "\n", + "\n", + "edge2\n", + "2: pi+[0]\n", + "\n", + "\n", + "\n", + "edge-1\n", + "Lambda(c)+[-1/2]\n", + "\n", + "\n", + "\n", + "node0\n", + "\n", + "\n", + "\n", + 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+ } + ], "source": [ "display(\n", " *map(\n", @@ -239,7 +876,26 @@ "full-width" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{align*}\n", + "I = & \\left|{D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right) + D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}\\right|^{2} \\\\\n", + "& + \\left|{D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right) + D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}\\right|^{2} \\\\\n", + "& + \\left|{D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}\\right|^{2} \\\\\n", + "& + \\left|{D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}\\right|^{2} \n", + "\\end{align*}$" + ], + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "def remove_coefficients(expr: sp.Expr) -> sp.Expr:\n", " coefficients = {s: 1 for s in expr.free_symbols if s.name.startswith(\"C_\")}\n", @@ -389,7 +1045,61 @@ "full-width" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left. \\sum_{\\lambda_{\\Lambda_c}=-1/2}^{1/2} \\sum_{\\lambda_{p}=-1/2}^{1/2} \\sum_{\\lambda_{\\pi}=0} \\sum_{\\lambda_{K}=0}{\\left|{\\sum_{\\lambda_{\\Delta}=-1/2}^{1/2}{D^{s_{\\Delta}}_{\\lambda_{\\Delta},- \\lambda_{\\pi} + \\lambda_{p}}\\left(\\phi^{12}_{1},\\theta^{12}_{1},0\\right) D^{s_{\\Lambda_c}}_{\\lambda_{\\Lambda_c},\\lambda_{K} - \\lambda_{\\Delta}}\\left(\\phi_{12},\\theta_{12},0\\right)}}\\right|^{2}} \\right|_{\\substack{ s_{\\Lambda_c}=\\frac{1}{2}\\\\ s_{\\Delta}=\\frac{3}{2} }}$" + ], + "text/plain": [ + "Subs(PoolSum(Abs(PoolSum(WignerD(s_\\Delta, \\lambda_\\Delta, -\\lambda_\\pi + \\lambda_p, phi_1^12, theta_1^12, 0)*WignerD(s_{\\Lambda_c}, \\lambda_{\\Lambda_c}, \\lambda_K - \\lambda_\\Delta, phi_12, theta_12, 0), (\\lambda_\\Delta, (-1/2, 1/2))))**2, (\\lambda_{\\Lambda_c}, (-1/2, 1/2)), (\\lambda_p, (-1/2, 1/2)), (\\lambda_\\pi, (0,)), (\\lambda_K, (0,))), (s_{\\Lambda_c}, s_\\Delta), (1/2, 3/2))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda_{\\Lambda_c}=-1/2}^{1/2} \\sum_{\\lambda_{p}=-1/2}^{1/2}{\\left|{\\sum_{\\lambda_{\\Delta}=-1/2}^{1/2}{D^{s_{\\Delta}}_{\\lambda_{\\Delta},\\lambda_{p}}\\left(\\phi^{12}_{1},\\theta^{12}_{1},0\\right) D^{s_{\\Lambda_c}}_{\\lambda_{\\Lambda_c},- \\lambda_{\\Delta}}\\left(\\phi_{12},\\theta_{12},0\\right)}}\\right|^{2}}$" + ], + "text/plain": [ + "PoolSum(Abs(PoolSum(WignerD(s_\\Delta, \\lambda_\\Delta, \\lambda_p, phi_1^12, theta_1^12, 0)*WignerD(s_{\\Lambda_c}, \\lambda_{\\Lambda_c}, -\\lambda_\\Delta, phi_12, theta_12, 0), (\\lambda_\\Delta, (-1/2, 1/2))))**2, (\\lambda_{\\Lambda_c}, (-1/2, 1/2)), (\\lambda_p, (-1/2, 1/2)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{align*}\n", + "I = & \\left|{\\sum_{\\lambda_{\\Delta}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\lambda_{\\Delta}}\\left(\\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{\\Delta},- \\frac{1}{2}}\\left(\\phi^{12}_{1},\\theta^{12}_{1},0\\right)}}\\right|^{2} \\\\\n", + "& + \\left|{\\sum_{\\lambda_{\\Delta}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\lambda_{\\Delta}}\\left(\\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{\\Delta},\\frac{1}{2}}\\left(\\phi^{12}_{1},\\theta^{12}_{1},0\\right)}}\\right|^{2} \\\\\n", + "& + \\left|{\\sum_{\\lambda_{\\Delta}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{\\frac{1}{2},- \\lambda_{\\Delta}}\\left(\\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{\\Delta},- \\frac{1}{2}}\\left(\\phi^{12}_{1},\\theta^{12}_{1},0\\right)}}\\right|^{2} \\\\\n", + "& + \\left|{\\sum_{\\lambda_{\\Delta}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{\\frac{1}{2},- \\lambda_{\\Delta}}\\left(\\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{\\Delta},\\frac{1}{2}}\\left(\\phi^{12}_{1},\\theta^{12}_{1},0\\right)}}\\right|^{2} \n", + "\\end{align*}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\frac{\\sin^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)} \\sin^{2}{\\left(\\frac{\\theta^{12}_{1}}{2} \\right)}}{8} - \\frac{3 \\sin^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)} \\sin{\\left(\\frac{\\theta^{12}_{1}}{2} \\right)} \\sin{\\left(\\frac{3 \\theta^{12}_{1}}{2} \\right)}}{4} + \\frac{9 \\sin^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)} \\sin^{2}{\\left(\\frac{3 \\theta^{12}_{1}}{2} \\right)}}{8} + \\frac{\\sin^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)} \\cos^{2}{\\left(\\frac{\\theta^{12}_{1}}{2} \\right)}}{8} + \\frac{3 \\sin^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)} \\cos{\\left(\\frac{\\theta^{12}_{1}}{2} \\right)} \\cos{\\left(\\frac{3 \\theta^{12}_{1}}{2} \\right)}}{4} + \\frac{9 \\sin^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)} \\cos^{2}{\\left(\\frac{3 \\theta^{12}_{1}}{2} \\right)}}{8} + \\frac{\\sin^{2}{\\left(\\frac{\\theta^{12}_{1}}{2} \\right)} \\cos^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)}}{8} - \\frac{3 \\sin{\\left(\\frac{\\theta^{12}_{1}}{2} \\right)} \\sin{\\left(\\frac{3 \\theta^{12}_{1}}{2} \\right)} \\cos^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)}}{4} + \\frac{9 \\sin^{2}{\\left(\\frac{3 \\theta^{12}_{1}}{2} \\right)} \\cos^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)}}{8} + \\frac{\\cos^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)} \\cos^{2}{\\left(\\frac{\\theta^{12}_{1}}{2} \\right)}}{8} + \\frac{3 \\cos^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)} \\cos{\\left(\\frac{\\theta^{12}_{1}}{2} \\right)} \\cos{\\left(\\frac{3 \\theta^{12}_{1}}{2} \\right)}}{4} + \\frac{9 \\cos^{2}{\\left(\\frac{\\theta_{12}}{2} \\right)} \\cos^{2}{\\left(\\frac{3 \\theta^{12}_{1}}{2} \\right)}}{8}$" + ], + "text/plain": [ + "sin(theta_12/2)**2*sin(theta_1^12/2)**2/8 - 3*sin(theta_12/2)**2*sin(theta_1^12/2)*sin(3*theta_1^12/2)/4 + 9*sin(theta_12/2)**2*sin(3*theta_1^12/2)**2/8 + sin(theta_12/2)**2*cos(theta_1^12/2)**2/8 + 3*sin(theta_12/2)**2*cos(theta_1^12/2)*cos(3*theta_1^12/2)/4 + 9*sin(theta_12/2)**2*cos(3*theta_1^12/2)**2/8 + sin(theta_1^12/2)**2*cos(theta_12/2)**2/8 - 3*sin(theta_1^12/2)*sin(3*theta_1^12/2)*cos(theta_12/2)**2/4 + 9*sin(3*theta_1^12/2)**2*cos(theta_12/2)**2/8 + cos(theta_12/2)**2*cos(theta_1^12/2)**2/8 + 3*cos(theta_12/2)**2*cos(theta_1^12/2)*cos(3*theta_1^12/2)/4 + 9*cos(theta_12/2)**2*cos(3*theta_1^12/2)**2/8" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "half = sp.S.Half\n", "\n", @@ -474,7 +1184,7 @@ "source": [ ":::{margin}\n", "\n", - "When introducing spin alignment ([ampform#213](https://github.com/ComPWA/ampform/pull/213)), we have to distinguish the helicity symbols between different topologies.\n", + "When introducing spin alignment ([ampform#245](https://github.com/ComPWA/ampform/pull/245)), we have to distinguish the helicity symbols between different topologies.\n", "\n", ":::\n", "\n", @@ -525,7 +1235,20 @@ "cell_type": "code", "execution_count": null, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle D^{\\frac{1}{2}}_{\\lambda,- \\lambda_{0} + \\lambda_{3}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{3},\\lambda_{1} - \\lambda_{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)$" + ], + "text/plain": [ + "WignerD(1/2, lambda, -lambda_0 + lambda_3, -phi_12, theta_12, 0)*WignerD(3/2, lambda_3, lambda_1 - lambda_2, -phi_1^12, theta_1^12, 0)" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "wigner_functions = {\n", " sp.Mul(\n", @@ -602,7 +1325,26 @@ "metadata": { "tags": [] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "{lambda_3: {-1/2, 1/2}}" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "{lambda: {-1/2, 1/2}, lambda_0: {0}, lambda_1: {-1/2, 1/2}, lambda_2: {0}}" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "inner_helicities = get_helicities(reaction, which=\"inner\")\n", "outer_helicities = get_helicities(reaction, which=\"outer\")\n", @@ -620,7 +1362,21 @@ "cell_type": "code", "execution_count": null, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda=-1/2}^{1/2} \\sum_{\\lambda_{0}=0} \\sum_{\\lambda_{1}=-1/2}^{1/2} \\sum_{\\lambda_{2}=0}{\\left|{\\sum_{\\lambda_{3}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{\\lambda,- \\lambda_{0} + \\lambda_{3}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{3},\\lambda_{1} - \\lambda_{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}}\\right|^{2}}$" + ], + "text/plain": [ + "PoolSum(Abs(PoolSum(WignerD(1/2, lambda, -lambda_0 + lambda_3, -phi_12, theta_12, 0)*WignerD(3/2, lambda_3, lambda_1 - lambda_2, -phi_1^12, theta_1^12, 0), (lambda_3, (-1/2, 1/2))))**2, (lambda, (-1/2, 1/2)), (lambda_0, (0,)), (lambda_1, (-1/2, 1/2)), (lambda_2, (0,)))" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "def formulate_intensity(reaction: ReactionInfo):\n", " wigner_functions = {\n", @@ -685,7 +1441,164 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "g0_edge0\n", + "0: K-\n", + "\n", + "\n", + "\n", + "g0_edge1\n", + "1: p\n", + "\n", + "\n", + "\n", + "g0_edge2\n", + "2: pi+\n", + "\n", + "\n", + "\n", + "g0_edge-1\n", + "Lambda(c)+\n", + "\n", + "\n", + "\n", + "g0_node0\n", + "\n", + "\n", + "\n", + "g0_edge-1->g0_node0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node0->g0_edge0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1\n", + "\n", + "\n", + "\n", + "g0_node0->g0_node1\n", + "\n", + "3: Delta(1600)++\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge1\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge2\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "g0_edge0\n", + "0: K-\n", + "\n", + "\n", + "\n", + "g0_edge1\n", + "1: p\n", + "\n", + "\n", + "\n", + "g0_edge2\n", + "2: pi+\n", + "\n", + "\n", + "\n", + "g0_edge-1\n", + "Lambda(c)+\n", + "\n", + "\n", + "\n", + "g0_node0\n", + "\n", + "\n", + "\n", + "g0_edge-1->g0_node0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node0->g0_edge2\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1\n", + "\n", + "\n", + "\n", + "g0_node0->g0_node1\n", + "\n", + "3: Lambda(1405)\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge1\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "reaction_two_resonances = qrules.generate_transitions(\n", " initial_state=\"Lambda(c)+\",\n", @@ -716,7 +1629,21 @@ "full-width" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda=-1/2}^{1/2} \\sum_{\\lambda_{1}=-1/2}^{1/2}{\\left|{\\sum_{\\lambda_{3}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{\\lambda,\\lambda_{3}}\\left(- \\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\lambda_{3},- \\lambda_{1}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) + D^{\\frac{1}{2}}_{\\lambda,\\lambda_{3}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{3},\\lambda_{1}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}}\\right|^{2}}$" + ], + "text/plain": [ + "PoolSum(Abs(PoolSum(WignerD(1/2, lambda, lambda_3, -phi_01, theta_01, 0)*WignerD(1/2, lambda_3, -lambda_1, -phi_0^01, theta_0^01, 0) + WignerD(1/2, lambda, lambda_3, -phi_12, theta_12, 0)*WignerD(3/2, lambda_3, lambda_1, -phi_1^12, theta_1^12, 0), (lambda_3, (-1/2, 1/2))))**2, (lambda, (-1/2, 1/2)), (lambda_1, (-1/2, 1/2)))" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "formulate_intensity(reaction_two_resonances).cleanup()" ] @@ -764,7 +1691,19 @@ "full-width" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "[C_{\\Lambda_{c}^{+} \\to \\Delta_{-1/2} K^{-}_{0}; \\Delta \\to p_{+1/2} \\pi^{+}_{0}},\n", + " C_{\\Lambda_{c}^{+} \\to \\Delta_{+1/2} K^{-}_{0}; \\Delta \\to p_{+1/2} \\pi^{+}_{0}}]" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "list(model.parameter_defaults)" ] @@ -782,7 +1721,21 @@ "metadata": { "tags": [] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda=-1/2}^{1/2} \\sum_{\\lambda_{0}=0} \\sum_{\\lambda_{1}=-1/2}^{1/2} \\sum_{\\lambda_{2}=0}{\\left|{\\sum_{\\lambda_{3}=-1/2}^{1/2}{{C}_{\\lambda_{0},\\lambda_{1},\\lambda_{2}} D^{\\frac{1}{2}}_{\\lambda,- \\lambda_{0} + \\lambda_{3}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{3},\\lambda_{1} - \\lambda_{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}}\\right|^{2}}$" + ], + "text/plain": [ + "PoolSum(Abs(PoolSum(C[lambda_0, lambda_1, lambda_2]*WignerD(1/2, lambda, -lambda_0 + lambda_3, -phi_12, theta_12, 0)*WignerD(3/2, lambda_3, lambda_1 - lambda_2, -phi_1^12, theta_1^12, 0), (lambda_3, (-1/2, 1/2))))**2, (lambda, (-1/2, 1/2)), (lambda_0, (0,)), (lambda_1, (-1/2, 1/2)), (lambda_2, (0,)))" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "C = sp.IndexedBase(\"C\")\n", "\n", @@ -833,7 +1786,26 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{align*}\n", + "I = & \\left|{\\sum_{\\lambda_{3}=-1/2}^{1/2}{{C}_{0,- \\frac{1}{2},0} D^{\\frac{1}{2}}_{- \\frac{1}{2},\\lambda_{3}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{3},- \\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}}\\right|^{2} \\\\\n", + "& + \\left|{\\sum_{\\lambda_{3}=-1/2}^{1/2}{{C}_{0,- \\frac{1}{2},0} D^{\\frac{1}{2}}_{\\frac{1}{2},\\lambda_{3}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{3},- \\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}}\\right|^{2} \\\\\n", + "& + \\left|{\\sum_{\\lambda_{3}=-1/2}^{1/2}{{C}_{0,\\frac{1}{2},0} D^{\\frac{1}{2}}_{- \\frac{1}{2},\\lambda_{3}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{3},\\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}}\\right|^{2} \\\\\n", + "& + \\left|{\\sum_{\\lambda_{3}=-1/2}^{1/2}{{C}_{0,\\frac{1}{2},0} D^{\\frac{1}{2}}_{\\frac{1}{2},\\lambda_{3}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{3},\\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}}\\right|^{2} \n", + "\\end{align*}$" + ], + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "latex = sp.multiline_latex(I, indexed_coefficient_expr.doit(deep=False))\n", "Math(latex)" @@ -858,7 +1830,18 @@ "metadata": { "tags": [] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "[C, C[0, -1/2, 0], C[0, 1/2, 0], phi_12, phi_1^12, theta_12, theta_1^12]" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "free_symbols = sorted(indexed_coefficient_expr.doit().free_symbols, key=str)\n", "free_symbols" @@ -877,7 +1860,24 @@ "metadata": { "tags": [] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "{C: (sympy.core.symbol.Symbol, True),\n", + " C[0, -1/2, 0]: (sympy.tensor.indexed.Indexed, False),\n", + " C[0, 1/2, 0]: (sympy.tensor.indexed.Indexed, False),\n", + " phi_12: (sympy.core.symbol.Symbol, True),\n", + " phi_1^12: (sympy.core.symbol.Symbol, True),\n", + " theta_12: (sympy.core.symbol.Symbol, True),\n", + " theta_1^12: (sympy.core.symbol.Symbol, True)}" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "{s: (type(s), isinstance(s, sp.Symbol)) for s in free_symbols}" ] @@ -897,7 +1897,18 @@ "metadata": { "tags": [] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "func = sp.lambdify(free_symbols, indexed_coefficient_expr.doit())\n", "inspect.signature(func)" @@ -916,7 +1927,18 @@ "metadata": { "tags": [] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "{C_{0,-1/2,0}, C_{0,1/2,0}, phi_12, phi_1^12, theta_12, theta_1^12}" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "indexed_coefficient_expr_symbols_only = symplot.substitute_indexed_symbols(\n", " indexed_coefficient_expr.doit()\n", @@ -930,7 +1952,18 @@ "metadata": { "tags": [] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "args = sorted(indexed_coefficient_expr_symbols_only.free_symbols, key=str)\n", "func = sp.lambdify(args, indexed_coefficient_expr_symbols_only)\n", @@ -1057,7 +2090,21 @@ "metadata": { "tags": [] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda=-1/2}^{1/2} \\sum_{\\lambda_{0}=0} \\sum_{\\lambda_{1}=-1/2}^{1/2} \\sum_{\\lambda_{2}=0}{\\left|{\\sum_{\\lambda_{3}=-1/2}^{1/2}{\\frac{\\Gamma_{\\Delta} m_{\\Delta} {C}_{\\lambda_{0},\\lambda_{1},\\lambda_{2},\\Delta} D^{\\frac{1}{2}}_{\\lambda,- \\lambda_{0} + \\lambda_{3}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{3},\\lambda_{1} - \\lambda_{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}{- i \\Gamma_{\\Delta} m_{\\Delta} - m_{12}^{2} + m_{\\Delta}^{2}}}}\\right|^{2}}$" + ], + "text/plain": [ + "PoolSum(Abs(PoolSum(\\Gamma_{\\Delta}*m_{\\Delta}*C[lambda_0, lambda_1, lambda_2, \\Delta]*WignerD(1/2, lambda, -lambda_0 + lambda_3, -phi_12, theta_12, 0)*WignerD(3/2, lambda_3, lambda_1 - lambda_2, -phi_1^12, theta_1^12, 0)/(-I*\\Gamma_{\\Delta}*m_{\\Delta} - m_12**2 + m_{\\Delta}**2), (lambda_3, (-1/2, 1/2))))**2, (lambda, (-1/2, 1/2)), (lambda_0, (0,)), (lambda_1, (-1/2, 1/2)), (lambda_2, (0,)))" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "formulate_intensity_with_dynamics(\n", " reaction,\n", @@ -1076,7 +2123,21 @@ "full-width" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda=-1/2}^{1/2} \\sum_{\\lambda_{0}=0} \\sum_{\\lambda_{1}=-1/2}^{1/2} \\sum_{\\lambda_{2}=0}{\\left|{\\sum_{\\lambda_{3}=-1/2}^{1/2}{\\frac{\\Gamma_{\\Delta} m_{\\Delta} {C}_{\\lambda_{0},\\lambda_{1},\\lambda_{2},\\Delta} D^{\\frac{1}{2}}_{\\lambda,- \\lambda_{0} + \\lambda_{3}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\lambda_{3},\\lambda_{1} - \\lambda_{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}{- i \\Gamma_{\\Delta} m_{\\Delta} - m_{12}^{2} + m_{\\Delta}^{2}} + \\frac{\\Gamma_{\\Lambda} m_{\\Lambda} {C}_{\\lambda_{0},\\lambda_{1},\\lambda_{2},\\Lambda} D^{\\frac{1}{2}}_{\\lambda,- \\lambda_{2} + \\lambda_{3}}\\left(- \\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\lambda_{3},\\lambda_{0} - \\lambda_{1}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right)}{- i \\Gamma_{\\Lambda} m_{\\Lambda} - m_{01}^{2} + m_{\\Lambda}^{2}}}}\\right|^{2}}$" + ], + "text/plain": [ + "PoolSum(Abs(PoolSum(\\Gamma_{\\Delta}*m_{\\Delta}*C[lambda_0, lambda_1, lambda_2, \\Delta]*WignerD(1/2, lambda, -lambda_0 + lambda_3, -phi_12, theta_12, 0)*WignerD(3/2, lambda_3, lambda_1 - lambda_2, -phi_1^12, theta_1^12, 0)/(-I*\\Gamma_{\\Delta}*m_{\\Delta} - m_12**2 + m_{\\Delta}**2) + \\Gamma_{\\Lambda}*m_{\\Lambda}*C[lambda_0, lambda_1, lambda_2, \\Lambda]*WignerD(1/2, lambda, -lambda_2 + lambda_3, -phi_01, theta_01, 0)*WignerD(1/2, lambda_3, lambda_0 - lambda_1, -phi_0^01, theta_0^01, 0)/(-I*\\Gamma_{\\Lambda}*m_{\\Lambda} - m_01**2 + m_{\\Lambda}**2), (lambda_3, (-1/2, 1/2))))**2, (lambda, (-1/2, 1/2)), (lambda_0, (0,)), (lambda_1, (-1/2, 1/2)), (lambda_2, (0,)))" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "formulate_intensity_with_dynamics(\n", " reaction_two_resonances,\n", @@ -1263,7 +2324,81 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{align*}\n", + "I = & \\sum_{\\lambda=-1/2}^{1/2} \\sum_{\\lambda_{0}=0} \\sum_{\\lambda_{1}=-1/2}^{1/2} \\sum_{\\lambda_{2}=0}{\\left|{\\sum_{\\lambda_{3}=-1/2}^{1/2}{{\\mathcal{A}}_{\\lambda,\\lambda_{0},\\lambda_{1},\\lambda_{2},\\lambda_{3},\\Delta} + {\\mathcal{A}}_{\\lambda,\\lambda_{0},\\lambda_{1},\\lambda_{2},\\lambda_{3},\\Lambda}}}\\right|^{2}} \n", + "\\end{align*}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{align*}\n", + "I = & \\left|{{\\mathcal{A}}_{- \\frac{1}{2},0,- \\frac{1}{2},0,- \\frac{1}{2},\\Delta} + {\\mathcal{A}}_{- \\frac{1}{2},0,- \\frac{1}{2},0,- \\frac{1}{2},\\Lambda} + {\\mathcal{A}}_{- \\frac{1}{2},0,- \\frac{1}{2},0,\\frac{1}{2},\\Delta} + {\\mathcal{A}}_{- \\frac{1}{2},0,- \\frac{1}{2},0,\\frac{1}{2},\\Lambda}}\\right|^{2} \\\\\n", + "& + \\left|{{\\mathcal{A}}_{- \\frac{1}{2},0,\\frac{1}{2},0,- \\frac{1}{2},\\Delta} + {\\mathcal{A}}_{- \\frac{1}{2},0,\\frac{1}{2},0,- \\frac{1}{2},\\Lambda} + {\\mathcal{A}}_{- \\frac{1}{2},0,\\frac{1}{2},0,\\frac{1}{2},\\Delta} + {\\mathcal{A}}_{- \\frac{1}{2},0,\\frac{1}{2},0,\\frac{1}{2},\\Lambda}}\\right|^{2} \\\\\n", + "& + \\left|{{\\mathcal{A}}_{\\frac{1}{2},0,- \\frac{1}{2},0,- \\frac{1}{2},\\Delta} + {\\mathcal{A}}_{\\frac{1}{2},0,- \\frac{1}{2},0,- \\frac{1}{2},\\Lambda} + {\\mathcal{A}}_{\\frac{1}{2},0,- \\frac{1}{2},0,\\frac{1}{2},\\Delta} + {\\mathcal{A}}_{\\frac{1}{2},0,- \\frac{1}{2},0,\\frac{1}{2},\\Lambda}}\\right|^{2} \\\\\n", + "& + \\left|{{\\mathcal{A}}_{\\frac{1}{2},0,\\frac{1}{2},0,- \\frac{1}{2},\\Delta} + {\\mathcal{A}}_{\\frac{1}{2},0,\\frac{1}{2},0,- \\frac{1}{2},\\Lambda} + {\\mathcal{A}}_{\\frac{1}{2},0,\\frac{1}{2},0,\\frac{1}{2},\\Delta} + {\\mathcal{A}}_{\\frac{1}{2},0,\\frac{1}{2},0,\\frac{1}{2},\\Lambda}}\\right|^{2} \n", + "\\end{align*}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{align*}\n", + "{\\mathcal{A}}_{- \\frac{1}{2},0,- \\frac{1}{2},0,- \\frac{1}{2},\\Delta} = & \\frac{C_{\\Lambda_{c}^{+} \\to \\Delta_{-1/2} K^{-}_{0}; \\Delta \\to p_{+1/2} \\pi^{+}_{0}} \\Gamma_{\\Delta} m_{\\Delta} D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}{- i \\Gamma_{\\Delta} m_{\\Delta} - m_{12}^{2} + m_{\\Delta}^{2}} \n", + "\\end{align*}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{align*}\n", + "{\\mathcal{A}}_{- \\frac{1}{2},0,- \\frac{1}{2},0,\\frac{1}{2},\\Delta} = & \\frac{C_{\\Lambda_{c}^{+} \\to \\Delta_{+1/2} K^{-}_{0}; \\Delta \\to p_{+1/2} \\pi^{+}_{0}} \\Gamma_{\\Delta} m_{\\Delta} D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}{- i \\Gamma_{\\Delta} m_{\\Delta} - m_{12}^{2} + m_{\\Delta}^{2}} \n", + "\\end{align*}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{align*}\n", + "{\\mathcal{A}}_{- \\frac{1}{2},0,\\frac{1}{2},0,- \\frac{1}{2},\\Delta} = & \\frac{C_{\\Lambda_{c}^{+} \\to \\Delta_{-1/2} K^{-}_{0}; \\Delta \\to p_{+1/2} \\pi^{+}_{0}} \\Gamma_{\\Delta} m_{\\Delta} D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi_{12},\\theta_{12},0\\right) D^{\\frac{3}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(- \\phi^{12}_{1},\\theta^{12}_{1},0\\right)}{- i \\Gamma_{\\Delta} m_{\\Delta} - m_{12}^{2} + m_{\\Delta}^{2}} \n", + "\\end{align*}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "display(Math(sp.multiline_latex(I, expression)))\n", "display(Math(sp.multiline_latex(I, expression.doit())))\n", @@ -1328,6 +2463,7 @@ } ], "metadata": { + "keep_output": true, "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", diff --git a/docs/report/015.ipynb b/docs/report/015.ipynb index c97570aa..cf6d9ec4 100644 --- a/docs/report/015.ipynb +++ b/docs/report/015.ipynb @@ -39,13 +39,20 @@ "tags": [] }, "source": [ - "# [TR-015] Spin alignment" + "# [TR-015] Spin alignment implementation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ + ":::{seealso}\n", + "\n", + "- {doc}`/report/013`\n", + "- {doc}`/report/014`\n", + "\n", + ":::\n", + "\n", ":::{note}\n", "\n", "This report has been implemented through [ampform#245](https://github.com/ComPWA/ampform/pull/245). For details on how to use it, see [this notebook](https://ampform--245.org.readthedocs.build/en/245/usage/helicity/spin-alignment.html).\n", @@ -75,7 +82,7 @@ }, "outputs": [], "source": [ - "%pip install -q git+https://github.com/ComPWA/ampform@98de70f qrules[viz]==0.9.7 sympy==1.9" + "%pip install -q ampform==0.13.0 qrules[viz]==0.9.7 sympy==1.9" ] }, { @@ -140,7 +147,86 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "edge0\n", + "0: K0[0]\n", + "\n", + "\n", + "\n", + "edge1\n", + "1: Sigma+[+1/2]\n", + "\n", + "\n", + "\n", + "edge2\n", + "2: p~[+1/2]\n", + "\n", + "\n", + "\n", + "edge-1\n", + "J/psi(1S)[-1]\n", + "\n", + "\n", + "\n", + "node0\n", + "\n", + "\n", + "\n", + "edge-1->node0\n", + "\n", + "\n", + "\n", + "\n", + "node0->edge1\n", + "\n", + "\n", + "\n", + "\n", + "node1\n", + "\n", + "\n", + "\n", + "node0->node1\n", + "\n", + "Sigma(1660)~-[-1/2]\n", + "\n", + "\n", + "\n", + "node1->edge0\n", + "\n", + "\n", + "\n", + "\n", + "node1->edge2\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "full_reaction = qrules.generate_transitions(\n", " initial_state=\"J/psi(1S)\",\n", @@ -177,7 +263,21 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left|{D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{-1,-1}\\left(- \\phi_{02},\\theta_{02},0\\right)}\\right|^{2}$" + ], + "text/plain": [ + "Abs(WignerD(1/2, -1/2, -1/2, -phi_0^02, theta_0^02, 0)*WignerD(1, -1, -1, -phi_02, theta_02, 0))**2" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "builder = ampform.get_builder(single_transition_reaction)\n", "model = builder.formulate()\n", @@ -195,7 +295,21 @@ "cell_type": "code", "execution_count": null, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{-1,-1}\\left(- \\phi_{02},\\theta_{02},0\\right)$" + ], + "text/plain": [ + "WignerD(1/2, -1/2, -1/2, -phi_0^02, theta_0^02, 0)*WignerD(1, -1, -1, -phi_02, theta_02, 0)" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "sp.Mul(\n", " formulate_wigner_d(transition, node_id=0),\n", @@ -223,7 +337,140 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "g0_edge0\n", + "0: K0\n", + "\n", + "\n", + "\n", + "g0_edge1\n", + "1: Sigma+\n", + "\n", + "\n", + "\n", + "g0_edge2\n", + "2: p~\n", + "\n", + "\n", + "\n", + "g0_edge-1\n", + "J/psi(1S)\n", + "\n", + "\n", + "\n", + "g0_node0\n", + "\n", + "\n", + "\n", + "g0_edge-1->g0_node0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node0->g0_edge1\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1\n", + "\n", + "\n", + "\n", + "g0_node0->g0_node1\n", + "\n", + "Sigma(1660)~-\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge0\n", + "\n", + "\n", + "\n", + "\n", + "g0_node1->g0_edge2\n", + "\n", + "\n", + "\n", + "\n", + "g1_edge0\n", + "0: K0\n", + "\n", + "\n", + "\n", + "g1_edge1\n", + "1: Sigma+\n", + "\n", + "\n", + "\n", + "g1_edge2\n", + "2: p~\n", + "\n", + "\n", + "\n", + "g1_edge-1\n", + "J/psi(1S)\n", + "\n", + "\n", + "\n", + "g1_node0\n", + "\n", + "\n", + "\n", + "g1_edge-1->g1_node0\n", + "\n", + "\n", + "\n", + "\n", + "g1_node0->g1_edge2\n", + "\n", + "\n", + "\n", + "\n", + "g1_node1\n", + "\n", + "\n", + "\n", + "g1_node0->g1_node1\n", + "\n", + "N(1650)+\n", + "\n", + "\n", + "\n", + "g1_node1->g1_edge0\n", + "\n", + "\n", + "\n", + "\n", + "g1_node1->g1_edge1\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "show_transition(full_reaction, collapse_graphs=True)" ] @@ -248,7 +495,25 @@ "full-width" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{eqnarray}\n", + "I & = & \\left|{\\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{-1,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{-1,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{-1,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{-1,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{-1,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{-1,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{-1,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{-1,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{-1,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{-1,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{-1,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{-1,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{-1,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{-1,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{-1,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{-1,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{01}}_{-1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{-1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{-1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{-1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{-1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{-1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{-1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) + {A^{01}}_{-1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{-1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{-1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{-1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{-1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} + {A^{02}}_{-1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{-1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + {A^{02}}_{-1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{-1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + {A^{02}}_{-1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{-1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{-1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{-1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} + {A^{02}}_{-1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{-1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + {A^{02}}_{-1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) + {A^{02}}_{-1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right)}\\right|^{2} \\nonumber\\\\\n", + "& & + \\left|{\\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{0,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{0,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{0,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{0,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{0,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{0,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{0,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{0,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{0,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{0,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{0,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{0,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{0,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{0,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{0,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{0,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{01}}_{0,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{0,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{0,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{0,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{0,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{0,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{0,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) + {A^{01}}_{0,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{0,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{0,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{0,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{0,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} + {A^{02}}_{0,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{0,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + {A^{02}}_{0,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{0,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + {A^{02}}_{0,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{0,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{0,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{0,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} + {A^{02}}_{0,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{0,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + {A^{02}}_{0,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) + {A^{02}}_{0,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right)}\\right|^{2} \\nonumber\\\\\n", + "& & + \\left|{\\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{1,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{1,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{1,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{1,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{1,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{1,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{1,-1}\\left(- \\phi_{01},\\theta_{01},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{1}_{1,0}\\left(- \\phi_{01},\\theta_{01},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{1,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{1,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{1,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{1,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{1,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{1,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + \\left(D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{1,-1}\\left(- \\phi_{02},\\theta_{02},0\\right) + D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(- \\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{1}_{1,0}\\left(- \\phi_{02},\\theta_{02},0\\right)\\right) D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{01}}_{1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) + {A^{01}}_{1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) + {A^{01}}_{1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right) + {A^{01}}_{1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{01},\\theta_{01},0\\right)^{2} + {A^{02}}_{1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + {A^{02}}_{1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{1,0,- \\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + {A^{02}}_{1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{1,0,- \\frac{1}{2},\\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} + {A^{02}}_{1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) + {A^{02}}_{1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) + {A^{02}}_{1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{- \\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right)^{2} D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) + {A^{02}}_{1,0,\\frac{1}{2},- \\frac{1}{2}} D^{0}_{0,0}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{0,0}\\left(\\phi_{02},\\theta_{02},0\\right) D^{0}_{0,0}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},- \\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi_{02},\\theta_{02},0\\right) D^{\\frac{1}{2}}_{\\frac{1}{2},\\frac{1}{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right)}\\right|^{2} \n", + "\\end{eqnarray}$" + ], + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "builder = ampform.get_builder(full_reaction)\n", "model = builder.formulate()\n", @@ -296,7 +561,164 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "A\n", + "A\n", + "\n", + "\n", + "\n", + "N0\n", + "\n", + "\n", + "\n", + "A->N0\n", + "\n", + "\n", + "\n", + "\n", + "0\n", + "0\n", + "\n", + "\n", + "\n", + "1\n", + "1\n", + "\n", + "\n", + "\n", + "2\n", + "2\n", + "\n", + "\n", + "\n", + "N0->2\n", + "\n", + "\n", + "\n", + "\n", + "N1\n", + "\n", + "\n", + "\n", + "N0->N1\n", + "\n", + "R = 01\n", + "\n", + "\n", + "\n", + "N1->0\n", + "\n", + "\n", + "\n", + "\n", + "N1->1\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "A\n", + "A\n", + "\n", + "\n", + "\n", + "N0\n", + "\n", + "\n", + "\n", + "A->N0\n", + "\n", + "\n", + "\n", + "\n", + "0\n", + "0\n", + "\n", + "\n", + "\n", + "1\n", + "1\n", + "\n", + "\n", + "\n", + "2\n", + "2\n", + "\n", + "\n", + "\n", + "N0->1\n", + "\n", + "\n", + "\n", + "\n", + "N1\n", + "\n", + "\n", + "\n", + "N0->N1\n", + "\n", + "S = 02\n", + "\n", + "\n", + "\n", + "N1->0\n", + "\n", + "\n", + "\n", + "\n", + "N1->2\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "dot1 = \"\"\"\n", "digraph {\n", @@ -417,7 +839,88 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "0\n", + "0\n", + "\n", + "\n", + "\n", + "1\n", + "1\n", + "\n", + "\n", + "\n", + "2\n", + "2\n", + "\n", + "\n", + "\n", + "A\n", + "A\n", + "\n", + "\n", + "\n", + "N0\n", + "\n", + "\n", + "\n", + "A->N0\n", + "\n", + "\n", + "\n", + "\n", + "N0->0\n", + "\n", + "\n", + "\n", + "\n", + "N1\n", + "\n", + "\n", + "\n", + "N0->N1\n", + "\n", + "U =\n", + "12\n", + "\n", + "\n", + "\n", + "N1->1\n", + "\n", + "\n", + "\n", + "\n", + "N1->2\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "dot3 = \"\"\"\n", "digraph {\n", @@ -552,7 +1055,158 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "edge0\n", + "0: K0[0]\n", + "\n", + "\n", + "\n", + "edge1\n", + "1: Sigma+[+1/2]\n", + "\n", + "\n", + "\n", + "edge2\n", + "2: p~[+1/2]\n", + "\n", + "\n", + "\n", + "edge-1\n", + "J/psi(1S)[+1]\n", + "\n", + "\n", + "\n", + "node0\n", + "\n", + "\n", + "\n", + "edge-1->node0\n", + "\n", + "\n", + "\n", + "\n", + "node0->edge2\n", + "\n", + "\n", + "\n", + "\n", + "node1\n", + "\n", + "\n", + "\n", + "node0->node1\n", + "\n", + "N(1650)+[+1/2]\n", + "\n", + "\n", + "\n", + "node1->edge0\n", + "\n", + "\n", + "\n", + "\n", + "node1->edge1\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle |s_0,m_0\\rangle=|0,0\\rangle \\quad (K^{0})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{0}_{\\nu^{01}_{0},0}\\left(\\phi_{01},\\theta_{01},0\\right)$" + ], + "text/plain": [ + "WignerD(0, 0, 0, phi_0^01, theta_0^01, 0)*WignerD(0, nu_0^01, 0, phi_01, theta_01, 0)" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle |s_1,m_1\\rangle=|1/2,1/2\\rangle \\quad (\\Sigma^{+})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{01}_{1}=-1/2}^{1/2} \\sum_{\\mu^{01}_{1}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{\\mu^{01}_{1},\\lambda^{01}_{1}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\nu^{01}_{1},\\mu^{01}_{1}}\\left(\\phi_{01},\\theta_{01},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(1/2, mu_1^01, lambda_1^01, phi_0^01, theta_0^01, 0)*WignerD(1/2, nu_1^01, mu_1^01, phi_01, theta_01, 0), (lambda_1^01, (-1/2, 1/2)), (mu_1^01, (-1/2, 1/2)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle |s_2,m_2\\rangle=|1/2,1/2\\rangle \\quad (\\overline{p})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{01}_{2}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{0.5,\\lambda^{01}_{2}}\\left(\\phi_{01},\\theta_{01},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(1/2, 0.5, lambda_2^01, phi_01, theta_01, 0), (lambda_2^01, (-1/2, 1/2)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "transition_r = full_reaction.transitions[-1]\n", "show_transition(transition_r)\n", @@ -579,7 +1233,80 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle |s_0,m_0\\rangle=|0,0\\rangle \\quad (K^{0})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{01}_{0}=0} \\sum_{\\mu^{01}_{0}=0} \\sum_{\\nu^{01}_{0}=0}{D^{0}_{m_{0},\\nu^{01}_{0}}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{0}_{\\mu^{01}_{0},\\lambda^{01}_{0}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{0}_{\\nu^{01}_{0},\\mu^{01}_{0}}\\left(\\phi_{01},\\theta_{01},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(0, m0, nu_0^01, alpha_0^01, beta_0^01, gamma_0^01)*WignerD(0, mu_0^01, lambda_0^01, phi_0^01, theta_0^01, 0)*WignerD(0, nu_0^01, mu_0^01, phi_01, theta_01, 0), (lambda_0^01, (0,)), (mu_0^01, (0,)), (nu_0^01, (0,)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle |s_1,m_1\\rangle=|1/2,1/2\\rangle \\quad (\\Sigma^{+})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{01}_{1}=-1/2}^{1/2} \\sum_{\\mu^{01}_{1}=-1/2}^{1/2} \\sum_{\\nu^{01}_{1}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{m_{1},\\nu^{01}_{1}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{\\mu^{01}_{1},\\lambda^{01}_{1}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\nu^{01}_{1},\\mu^{01}_{1}}\\left(\\phi_{01},\\theta_{01},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(1/2, m1, nu_1^01, alpha_1^01, beta_1^01, gamma_1^01)*WignerD(1/2, mu_1^01, lambda_1^01, phi_0^01, theta_0^01, 0)*WignerD(1/2, nu_1^01, mu_1^01, phi_01, theta_01, 0), (lambda_1^01, (-1/2, 1/2)), (mu_1^01, (-1/2, 1/2)), (nu_1^01, (-1/2, 1/2)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle |s_2,m_2\\rangle=|1/2,1/2\\rangle \\quad (\\overline{p})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{01}_{2}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{m_{2},\\lambda^{01}_{2}}\\left(\\phi_{01},\\theta_{01},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(1/2, m2, lambda_2^01, phi_01, theta_01, 0), (lambda_2^01, (-1/2, 1/2)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "show_all_spin_matrices(transition_r, formulate_rotation_chain, cleanup=False)" ] @@ -601,7 +1328,21 @@ "full-width" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{01}_{1}=-1/2}^{1/2} \\sum_{\\mu^{01}_{1}=-1/2}^{1/2} \\sum_{\\nu^{01}_{1}=-1/2}^{1/2} \\sum_{\\lambda^{01}_{2}=-1/2}^{1/2}{D^{0}_{0,0}\\left(\\phi_{01},\\theta_{01},0\\right) D^{0}_{0,0}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{0}_{m_{0},0}\\left(\\alpha^{01}_{0},\\beta^{01}_{0},\\gamma^{01}_{0}\\right) D^{\\frac{1}{2}}_{m_{1},\\nu^{01}_{1}}\\left(\\alpha^{01}_{1},\\beta^{01}_{1},\\gamma^{01}_{1}\\right) D^{\\frac{1}{2}}_{m_{2},\\lambda^{01}_{2}}\\left(\\phi_{01},\\theta_{01},0\\right) D^{\\frac{1}{2}}_{\\mu^{01}_{1},\\lambda^{01}_{1}}\\left(\\phi^{01}_{0},\\theta^{01}_{0},0\\right) D^{\\frac{1}{2}}_{\\nu^{01}_{1},\\mu^{01}_{1}}\\left(\\phi_{01},\\theta_{01},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(0, 0, 0, phi_01, theta_01, 0)*WignerD(0, 0, 0, phi_0^01, theta_0^01, 0)*WignerD(0, m0, 0, alpha_0^01, beta_0^01, gamma_0^01)*WignerD(1/2, m1, nu_1^01, alpha_1^01, beta_1^01, gamma_1^01)*WignerD(1/2, m2, lambda_2^01, phi_01, theta_01, 0)*WignerD(1/2, mu_1^01, lambda_1^01, phi_0^01, theta_0^01, 0)*WignerD(1/2, nu_1^01, mu_1^01, phi_01, theta_01, 0), (lambda_1^01, (-1/2, 1/2)), (mu_1^01, (-1/2, 1/2)), (nu_1^01, (-1/2, 1/2)), (lambda_2^01, (-1/2, 1/2)))" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "alignment_summation = formulate_spin_alignment(transition_r)\n", "alignment_summation.cleanup()" @@ -625,7 +1366,158 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "edge0\n", + "0: K0[0]\n", + "\n", + "\n", + "\n", + "edge1\n", + "1: Sigma+[+1/2]\n", + "\n", + "\n", + "\n", + "edge2\n", + "2: p~[+1/2]\n", + "\n", + "\n", + "\n", + "edge-1\n", + "J/psi(1S)[-1]\n", + "\n", + "\n", + "\n", + "node0\n", + "\n", + "\n", + "\n", + "edge-1->node0\n", + "\n", + "\n", + "\n", + "\n", + "node0->edge1\n", + "\n", + "\n", + "\n", + "\n", + "node1\n", + "\n", + "\n", + "\n", + "node0->node1\n", + "\n", + "N(1650)+[-1/2]\n", + "\n", + "\n", + "\n", + "node1->edge0\n", + "\n", + "\n", + "\n", + "\n", + "node1->edge2\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle |s_0,m_0\\rangle=|0,0\\rangle \\quad (K^{0})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{02}_{0}=0} \\sum_{\\mu^{02}_{0}=0} \\sum_{\\nu^{02}_{0}=0}{D^{0}_{m_{0},\\nu^{02}_{0}}\\left(\\alpha^{02}_{0},\\beta^{02}_{0},\\gamma^{02}_{0}\\right) D^{0}_{\\mu^{02}_{0},\\lambda^{02}_{0}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{0}_{\\nu^{02}_{0},\\mu^{02}_{0}}\\left(\\phi_{02},\\theta_{02},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(0, m0, nu_0^02, alpha_0^02, beta_0^02, gamma_0^02)*WignerD(0, mu_0^02, lambda_0^02, phi_0^02, theta_0^02, 0)*WignerD(0, nu_0^02, mu_0^02, phi_02, theta_02, 0), (lambda_0^02, (0,)), (mu_0^02, (0,)), (nu_0^02, (0,)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle |s_1,m_1\\rangle=|1/2,1/2\\rangle \\quad (\\Sigma^{+})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{02}_{1}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{m_{1},\\lambda^{02}_{1}}\\left(\\phi_{02},\\theta_{02},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(1/2, m1, lambda_1^02, phi_02, theta_02, 0), (lambda_1^02, (-1/2, 1/2)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle |s_2,m_2\\rangle=|1/2,1/2\\rangle \\quad (\\overline{p})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{02}_{2}=-1/2}^{1/2} \\sum_{\\mu^{02}_{2}=-1/2}^{1/2} \\sum_{\\nu^{02}_{2}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{m_{2},\\nu^{02}_{2}}\\left(\\alpha^{02}_{2},\\beta^{02}_{2},\\gamma^{02}_{2}\\right) D^{\\frac{1}{2}}_{\\mu^{02}_{2},\\lambda^{02}_{2}}\\left(\\phi^{02}_{0},\\theta^{02}_{0},0\\right) D^{\\frac{1}{2}}_{\\nu^{02}_{2},\\mu^{02}_{2}}\\left(\\phi_{02},\\theta_{02},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(1/2, m2, nu_2^02, alpha_2^02, beta_2^02, gamma_2^02)*WignerD(1/2, mu_2^02, lambda_2^02, phi_0^02, theta_0^02, 0)*WignerD(1/2, nu_2^02, mu_2^02, phi_02, theta_02, 0), (lambda_2^02, (-1/2, 1/2)), (mu_2^02, (-1/2, 1/2)), (nu_2^02, (-1/2, 1/2)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "reaction_s = qrules.generate_transitions(\n", " initial_state=\"J/psi(1S)\",\n", @@ -657,7 +1549,158 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "edge0\n", + "0: K0[0]\n", + "\n", + "\n", + "\n", + "edge1\n", + "1: Sigma+[+1/2]\n", + "\n", + "\n", + "\n", + "edge2\n", + "2: p~[+1/2]\n", + "\n", + "\n", + "\n", + "edge-1\n", + "J/psi(1S)[-1]\n", + "\n", + "\n", + "\n", + "node0\n", + "\n", + "\n", + "\n", + "edge-1->node0\n", + "\n", + "\n", + "\n", + "\n", + "node0->edge0\n", + "\n", + "\n", + "\n", + "\n", + "node1\n", + "\n", + "\n", + "\n", + "node0->node1\n", + "\n", + "K*(1680)~0[-1]\n", + "\n", + "\n", + "\n", + "node1->edge1\n", + "\n", + "\n", + "\n", + "\n", + "node1->edge2\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle |s_0,m_0\\rangle=|0,0\\rangle \\quad (K^{0})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{12}_{0}=0}{D^{0}_{m_{0},\\lambda^{12}_{0}}\\left(\\phi_{0},\\theta_{0},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(0, m0, lambda_0^12, phi_0, theta_0, 0), (lambda_0^12, (0,)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle |s_1,m_1\\rangle=|1/2,1/2\\rangle \\quad (\\Sigma^{+})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{12}_{1}=-1/2}^{1/2} \\sum_{\\mu^{12}_{1}=-1/2}^{1/2} \\sum_{\\nu^{12}_{1}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{m_{1},\\nu^{12}_{1}}\\left(\\alpha^{12}_{1},\\beta^{12}_{1},\\gamma^{12}_{1}\\right) D^{\\frac{1}{2}}_{\\mu^{12}_{1},\\lambda^{12}_{1}}\\left(\\phi^{12}_{1},\\theta^{12}_{1},0\\right) D^{\\frac{1}{2}}_{\\nu^{12}_{1},\\mu^{12}_{1}}\\left(\\phi_{0},\\theta_{0},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(1/2, m1, nu_1^12, alpha_1^12, beta_1^12, gamma_1^12)*WignerD(1/2, mu_1^12, lambda_1^12, phi_1^12, theta_1^12, 0)*WignerD(1/2, nu_1^12, mu_1^12, phi_0, theta_0, 0), (lambda_1^12, (-1/2, 1/2)), (mu_1^12, (-1/2, 1/2)), (nu_1^12, (-1/2, 1/2)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle |s_2,m_2\\rangle=|1/2,1/2\\rangle \\quad (\\overline{p})$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\sum_{\\lambda^{12}_{2}=-1/2}^{1/2} \\sum_{\\mu^{12}_{2}=-1/2}^{1/2} \\sum_{\\nu^{12}_{2}=-1/2}^{1/2}{D^{\\frac{1}{2}}_{m_{2},\\nu^{12}_{2}}\\left(\\alpha^{12}_{2},\\beta^{12}_{2},\\gamma^{12}_{2}\\right) D^{\\frac{1}{2}}_{\\mu^{12}_{2},\\lambda^{12}_{2}}\\left(\\phi^{12}_{1},\\theta^{12}_{1},0\\right) D^{\\frac{1}{2}}_{\\nu^{12}_{2},\\mu^{12}_{2}}\\left(\\phi_{0},\\theta_{0},0\\right)}$" + ], + "text/plain": [ + "PoolSum(WignerD(1/2, m2, nu_2^12, alpha_2^12, beta_2^12, gamma_2^12)*WignerD(1/2, mu_2^12, lambda_2^12, phi_1^12, theta_1^12, 0)*WignerD(1/2, nu_2^12, mu_2^12, phi_0, theta_0, 0), (lambda_2^12, (-1/2, 1/2)), (mu_2^12, (-1/2, 1/2)), (nu_2^12, (-1/2, 1/2)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "reaction_u = qrules.generate_transitions(\n", " initial_state=\"J/psi(1S)\",\n", @@ -691,7 +1734,122 @@ "cell_type": "code", "execution_count": null, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "edge0\n", + "0: K0[0]\n", + "\n", + "\n", + "\n", + "edge1\n", + "1: Sigma+[+1/2]\n", + "\n", + "\n", + "\n", + "edge2\n", + "2: p~[+1/2]\n", + "\n", + "\n", + "\n", + "edge-1\n", + "J/psi(1S)[-1]\n", + "\n", + "\n", + "\n", + "node0\n", + "\n", + "\n", + "\n", + "edge-1->node0\n", + "\n", + "\n", + "\n", + "\n", + "node0->edge0\n", + "\n", + "\n", + "\n", + "\n", + "node1\n", + "\n", + "\n", + "\n", + "node0->node1\n", + "\n", + "K*(1680)~0[-1]\n", + "\n", + "\n", + "\n", + "node1->edge1\n", + "\n", + "\n", + "\n", + "\n", + "node1->edge2\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\boldsymbol{B}\\left(p_{0}\\right)\\end{matrix}\\right]$" + ], + "text/plain": [ + "[BoostMatrix(p0)]" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\boldsymbol{B}\\left({p}_{12}\\right) & \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{1}\\right)\\end{matrix}\\right]$" + ], + "text/plain": [ + "[BoostMatrix(p1 + p2), BoostMatrix(ArrayMultiplication(BoostMatrix(p1 + p2), p1))]" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\boldsymbol{B}\\left({p}_{12}\\right) & \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{2}\\right)\\end{matrix}\\right]$" + ], + "text/plain": [ + "[BoostMatrix(p1 + p2), BoostMatrix(ArrayMultiplication(BoostMatrix(p1 + p2), p2))]" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "from ampform.kinematics import (\n", " compute_boost_chain,\n", @@ -718,7 +1876,44 @@ "cell_type": "code", "execution_count": null, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\boldsymbol{B}\\left(-\\left(p_{0}\\right)\\right) \\boldsymbol{B}\\left(p_{0}\\right)$" + ], + "text/plain": [ + "MatrixMultiplication(BoostMatrix(NegativeMomentum(p0)), BoostMatrix(p0))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\boldsymbol{B}\\left(-\\left(p_{1}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{1}\\right)$" + ], + "text/plain": [ + "MatrixMultiplication(BoostMatrix(NegativeMomentum(p1)), BoostMatrix(p1 + p2), BoostMatrix(ArrayMultiplication(BoostMatrix(p1 + p2), p1)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\boldsymbol{B}\\left(-\\left(p_{2}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{2}\\right)$" + ], + "text/plain": [ + "MatrixMultiplication(BoostMatrix(NegativeMomentum(p2)), BoostMatrix(p1 + p2), BoostMatrix(ArrayMultiplication(BoostMatrix(p1 + p2), p2)))" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "from ampform.kinematics import compute_wigner_rotation_matrix\n", "\n", @@ -761,7 +1956,31 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{eqnarray}\n", + "\\alpha^{12}_{0}&=&\\operatorname{atan_{2}}{\\left(\\boldsymbol{B}\\left(-\\left(p_{0}\\right)\\right) \\boldsymbol{B}\\left(p_{0}\\right)\\left[:, 3, 2\\right],\\boldsymbol{B}\\left(-\\left(p_{0}\\right)\\right) \\boldsymbol{B}\\left(p_{0}\\right)\\left[:, 3, 1\\right] \\right)}\\\\\n", + "\\beta^{12}_{0}&=&\\operatorname{acos}{\\left(\\boldsymbol{B}\\left(-\\left(p_{0}\\right)\\right) \\boldsymbol{B}\\left(p_{0}\\right)\\left[:, 3, 3\\right] \\right)}\\\\\n", + "\\gamma^{12}_{0}&=&\\operatorname{atan_{2}}{\\left(\\boldsymbol{B}\\left(-\\left(p_{0}\\right)\\right) \\boldsymbol{B}\\left(p_{0}\\right)\\left[:, 2, 3\\right],- \\boldsymbol{B}\\left(-\\left(p_{0}\\right)\\right) \\boldsymbol{B}\\left(p_{0}\\right)\\left[:, 1, 3\\right] \\right)}\\\\\n", + "\\alpha^{12}_{1}&=&\\operatorname{atan_{2}}{\\left(\\boldsymbol{B}\\left(-\\left(p_{1}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{1}\\right)\\left[:, 3, 2\\right],\\boldsymbol{B}\\left(-\\left(p_{1}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{1}\\right)\\left[:, 3, 1\\right] \\right)}\\\\\n", + "\\beta^{12}_{1}&=&\\operatorname{acos}{\\left(\\boldsymbol{B}\\left(-\\left(p_{1}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{1}\\right)\\left[:, 3, 3\\right] \\right)}\\\\\n", + "\\gamma^{12}_{1}&=&\\operatorname{atan_{2}}{\\left(\\boldsymbol{B}\\left(-\\left(p_{1}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{1}\\right)\\left[:, 2, 3\\right],- \\boldsymbol{B}\\left(-\\left(p_{1}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{1}\\right)\\left[:, 1, 3\\right] \\right)}\\\\\n", + "\\alpha^{12}_{2}&=&\\operatorname{atan_{2}}{\\left(\\boldsymbol{B}\\left(-\\left(p_{2}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{2}\\right)\\left[:, 3, 2\\right],\\boldsymbol{B}\\left(-\\left(p_{2}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{2}\\right)\\left[:, 3, 1\\right] \\right)}\\\\\n", + "\\beta^{12}_{2}&=&\\operatorname{acos}{\\left(\\boldsymbol{B}\\left(-\\left(p_{2}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{2}\\right)\\left[:, 3, 3\\right] \\right)}\\\\\n", + "\\gamma^{12}_{2}&=&\\operatorname{atan_{2}}{\\left(\\boldsymbol{B}\\left(-\\left(p_{2}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{2}\\right)\\left[:, 2, 3\\right],- \\boldsymbol{B}\\left(-\\left(p_{2}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{2}\\right)\\left[:, 1, 3\\right] \\right)}\\\\\n", + "\\end{eqnarray}$" + ], + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "latex_lines = [R\"\\begin{eqnarray}\"]\n", "for symbol, expr in angles.items():\n", @@ -806,7 +2025,21 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\beta^{12}_{1}:\\quad\\text{2,147 characters in generated code}$" + ], + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "import inspect\n", "\n", @@ -889,7 +2122,21 @@ "cell_type": "code", "execution_count": null, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\boldsymbol{B}\\left(-\\left(p_{2}\\right)\\right) \\boldsymbol{B}\\left({p}_{12}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{12}\\right) p_{2}\\right)$" + ], + "text/plain": [ + "MatrixMultiplication(BoostMatrix(NegativeMomentum(p2)), BoostMatrix(p1 + p2), BoostMatrix(ArrayMultiplication(BoostMatrix(p1 + p2), p2)))" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "matrix_expr = compute_wigner_rotation_matrix(topology, momenta, state_id=2)\n", "matrix_expr" @@ -899,7 +2146,41 @@ "cell_type": "code", "execution_count": null, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "array([[[ 1. , -0. , -0. , -0. ],\n", + " [-0. , 1. , 0.02, -0.02],\n", + " [-0. , -0.02, 1. , 0.03],\n", + " [ 0. , 0.02, -0.03, 1. ]],\n", + "\n", + " [[ 1. , 0. , -0. , 0. ],\n", + " [ 0. , 1. , -0.02, -0.04],\n", + " [ 0. , 0.02, 1. , -0. ],\n", + " [ 0. , 0.04, 0. , 1. ]],\n", + "\n", + " [[ 1. , 0. , 0. , 0. ],\n", + " [-0. , 1. , 0.02, -0.01],\n", + " [ 0. , -0.02, 1. , 0.02],\n", + " [ 0. , 0.01, -0.02, 1. ]],\n", + "\n", + " [[ 1. , 0. , -0. , 0. ],\n", + " [ 0. , 1. , -0.01, 0.02],\n", + " [ 0. , 0.01, 1. , 0.02],\n", + " [ 0. , -0.02, -0.02, 1. ]],\n", + "\n", + " [[ 1. , -0. , 0. , -0. ],\n", + " [-0. , 1. , -0.01, -0.01],\n", + " [ 0. , 0.01, 1. , 0.01],\n", + " [-0. , 0.01, -0.01, 1. ]]])" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "matrix_func = sp.lambdify(momenta.values(), matrix_expr.doit(), cse=True)\n", "matrix_array = matrix_func(*phsp.values())\n", @@ -917,7 +2198,31 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\begin{eqnarray}\n", + "\\alpha^{12}_{0}&=&\\left[\\begin{matrix}-1.25 & 0.0 & 0.79 & 1.33 & 2.36\\end{matrix}\\right]\\\\\n", + "\\beta^{12}_{0}&=&\\left[\\begin{matrix}0.0 & \\text{NaN} & 0.0 & 0.0 & 0.0\\end{matrix}\\right]\\\\\n", + "\\gamma^{12}_{0}&=&\\left[\\begin{matrix}-1.89 & 3.14 & 2.36 & 1.82 & 0.79\\end{matrix}\\right]\\\\\n", + "\\alpha^{12}_{1}&=&\\left[\\begin{matrix}2.03 & -3.04 & 1.9 & 0.74 & 2.14\\end{matrix}\\right]\\\\\n", + "\\beta^{12}_{1}&=&\\left[\\begin{matrix}0.03 & 0.03 & 0.01 & 0.02 & 0.01\\end{matrix}\\right]\\\\\n", + "\\gamma^{12}_{1}&=&\\left[\\begin{matrix}-2.05 & 3.06 & -1.92 & -0.73 & -2.13\\end{matrix}\\right]\\\\\n", + "\\alpha^{12}_{2}&=&\\left[\\begin{matrix}-1.09 & 0.08 & -1.22 & -2.41 & -1.01\\end{matrix}\\right]\\\\\n", + "\\beta^{12}_{2}&=&\\left[\\begin{matrix}0.04 & 0.04 & 0.02 & 0.03 & 0.01\\end{matrix}\\right]\\\\\n", + "\\gamma^{12}_{2}&=&\\left[\\begin{matrix}1.11 & -0.1 & 1.25 & 2.4 & 1.0\\end{matrix}\\right]\\\\\n", + "\\end{eqnarray}$" + ], + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "latex_lines = [R\"\\begin{eqnarray}\"]\n", "for angle_symbol, angle_expr in angles.items():\n", @@ -942,7 +2247,87 @@ "hide-input" ] }, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "edge0\n", + "0: K0[0]\n", + "\n", + "\n", + "\n", + "edge1\n", + "1: Sigma+[+1/2]\n", + "\n", + "\n", + "\n", + "edge2\n", + "2: p~[+1/2]\n", + "\n", + "\n", + "\n", + "edge-1\n", + "J/psi(1S)[-1]\n", + "\n", + "\n", + "\n", + "node0\n", + "\n", + "\n", + "\n", + "edge-1->node0\n", + "\n", + "\n", + "\n", + "\n", + "node0->edge0\n", + "\n", + "\n", + "\n", + "\n", + "node1\n", + "\n", + "\n", + "\n", + "node0->node1\n", + "\n", + "K*(1680)~0[-1]\n", + "\n", + "\n", + "\n", + "node1->edge1\n", + "\n", + "\n", + "\n", + "\n", + "node1->edge2\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "dot = qrules.io.asdot(transition_u, collapse_graphs=True)\n", "graphviz.Source(dot)" @@ -977,7 +2362,104 @@ "cell_type": "code", "execution_count": null, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "%3\n", + "\n", + "\n", + "\n", + "edge0\n", + "0\n", + "\n", + "\n", + "\n", + "edge1\n", + "1\n", + "\n", + "\n", + "\n", + "edge2\n", + "2\n", + "\n", + "\n", + "\n", + "edge3\n", + "3\n", + "\n", + "\n", + "\n", + "edge-1\n", + "\n", + "\n", + "\n", + "node0\n", + "\n", + "\n", + "\n", + "edge-1->node0\n", + "\n", + "\n", + "\n", + "\n", + "node0->edge0\n", + "\n", + "\n", + "\n", + "\n", + "node1\n", + "\n", + "\n", + "\n", + "node0->node1\n", + "\n", + "\n", + "\n", + "\n", + "node1->edge1\n", + "\n", + "\n", + "\n", + "\n", + "node2\n", + "\n", + "\n", + "\n", + "node1->node2\n", + "\n", + "\n", + "\n", + "\n", + "node2->edge2\n", + "\n", + "\n", + "\n", + "\n", + "node2->edge3\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "from qrules.topology import create_isobar_topologies\n", "\n", @@ -990,7 +2472,21 @@ "cell_type": "code", "execution_count": null, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\boldsymbol{B}\\left(-\\left(p_{3}\\right)\\right) \\boldsymbol{B}\\left({p}_{123}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{123}\\right) {p}_{23}\\right) \\boldsymbol{B}\\left(\\boldsymbol{B}\\left(\\boldsymbol{B}\\left({p}_{123}\\right) {p}_{23}\\right) \\boldsymbol{B}\\left({p}_{123}\\right) p_{3}\\right)$" + ], + "text/plain": [ + "MatrixMultiplication(BoostMatrix(NegativeMomentum(p3)), BoostMatrix(p1 + p2 + p3), BoostMatrix(ArrayMultiplication(BoostMatrix(p1 + p2 + p3), p2 + p3)), BoostMatrix(ArrayMultiplication(BoostMatrix(ArrayMultiplication(BoostMatrix(p1 + p2 + p3), p2 + p3)), ArrayMultiplication(BoostMatrix(p1 + p2 + p3), p3))))" + ] + }, + "execution_count": null, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "momenta_4body = create_four_momentum_symbols(topology_4body)\n", "compute_wigner_rotation_matrix(topology_4body, momenta_4body, state_id=3)" @@ -998,6 +2494,7 @@ } ], "metadata": { + "keep_output": true, "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python",