diff --git a/.cspell.json b/.cspell.json index fa5a946c..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", @@ -174,6 +176,8 @@ "kutschke", "lambdifier", "lambdifygenerated", + "linecap", + "linejoin", "linestyle", "linewidth", "linkcheck", @@ -197,6 +201,7 @@ "ncols", "ndarray", "nonlocal", + "nonumber", "noqa", "noreply", "nrows", @@ -241,6 +246,7 @@ "spflueger", "startswith", "subslide", + "substack", "suptitle", "symplot", "theano", @@ -256,6 +262,7 @@ "xlim", "xlink", "xreplace", + "xtick", "xticklabels", "xticks", "ylabel", 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 index ace0b98e..87a8e204 100644 --- a/docs/report/013.ipynb +++ b/docs/report/013.ipynb @@ -74,7 +74,7 @@ }, "outputs": [], "source": [ - "%pip install -q git+https://github.com/ComPWA/ampform@98de70f qrules[viz]==0.9.7 tensorwaves[jax,pwa]==0.4.2" + "%pip install -q ampform==0.13.0 qrules[viz]==0.9.7 tensorwaves[jax,pwa]==0.4.2" ] }, { @@ -156,7 +156,7 @@ " },\n", ")\n", "rng = TFUniformRealNumberGenerator(seed=0)\n", - "phsp_momenta = phsp_generator.generate(400_000, rng)" + "phsp_momenta = phsp_generator.generate(1_000_000, rng)" ] }, { @@ -238,7 +238,242 @@ "full-width" ] }, - "outputs": [], + "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", + 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"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", @@ -269,7 +504,21 @@ "metadata": { "tags": [] }, - "outputs": [], + "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", @@ -297,7 +546,65 @@ "full-width" ] }, - "outputs": [], + "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", @@ -372,7 +679,36 @@ "hide-input" ] }, - "outputs": [], + "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" + } + ], 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" + ] + }, + "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)" @@ -544,7 +5415,21 @@ "full-width" ] }, - "outputs": [], + "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", @@ -574,7 +5459,4538 @@ "hide-input" ] }, - "outputs": [], + "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", + " 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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)" @@ -589,6 +10005,7 @@ } ], "metadata": { + "keep_output": true, "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", diff --git a/docs/report/014.ipynb b/docs/report/014.ipynb index c36faacd..17500eb4 100644 --- a/docs/report/014.ipynb +++ b/docs/report/014.ipynb @@ -217,7 +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", + 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"metadata": {}, + "output_type": "display_data" + } + ], "source": [ "display(\n", " *map(\n", @@ -251,7 +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", @@ -401,7 +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", @@ -537,7 +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", @@ -614,7 +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", @@ -632,7 +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", @@ -697,7 +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", @@ -728,7 +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()" ] @@ -776,7 +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)" ] @@ -794,7 +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", @@ -845,7 +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)" @@ -870,7 +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" @@ -889,7 +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}" ] @@ -909,7 +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)" @@ -928,7 +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", @@ -942,7 +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", @@ -1069,7 +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", @@ -1088,7 +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", @@ -1275,7 +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", @@ -1340,6 +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 960bff40..cf6d9ec4 100644 --- a/docs/report/015.ipynb +++ b/docs/report/015.ipynb @@ -82,7 +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" ] }, { @@ -147,7 +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", @@ -184,7 +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", @@ -202,7 +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", @@ -230,7 +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)" ] @@ -255,7 +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", @@ -303,7 +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", @@ -424,7 +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", @@ -559,7 +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", @@ -586,7 +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)" ] @@ -608,7 +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()" @@ -632,7 +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", @@ -664,7 +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", @@ -698,7 +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", @@ -725,7 +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", @@ -768,7 +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", @@ -813,7 +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", @@ -896,7 +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" @@ -906,7 +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", @@ -924,7 +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", @@ -949,7 +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)" @@ -984,7 +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", @@ -997,7 +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)" @@ -1005,6 +2494,7 @@ } ], "metadata": { + "keep_output": true, "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python",