docs: fix calculation and add qrules code

docs/usage/ls-coupling.ipynb

@@ -54,10 +54,52 @@
     "The {func}`.spin_conservation` rule is one of the more complicated checks in the {mod}`.conservation_rules` module. It provides an implementation of $LS$-couplings, which is a procedure to determine which values for **total angular momentum $L$** and **coupled spin $S$** are allowed in an interaction node. In this notebook, we illustrate this procedure with the following decay chain as an example:\n",
     "\n",
     "$$\n",
-    "J/\\psi \\to \\bar\\Sigma(1670)^ -\\Sigma^+ , \\quad \\bar\\Sigma(1670)^- \\rightarrow \\bar p K^0.\n",
+    "J/\\psi \\to \\Sigma^+ \\bar\\Sigma(1670)^-, \\quad \\bar\\Sigma(1670)^- \\rightarrow \\bar p K^0.\n",
     "$$\n",
     "\n",
-    "In this decay chain, there are two decay nodes that we investigate separately. In addition, both decays are mediated interactions by the strong force, which means there is also **parity conservation**."
+    "In this decay chain, there are two decay nodes that we investigate separately. In addition, both decays are mediated interactions by the strong force, which means there is also **parity conservation**.\n",
+    "\n",
+    "In the following derivations, the {attr}`.Particle.spin` and {attr}`.Particle.parity` values are of importance:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    },
+    "tags": [
+     "hide-input"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "from fractions import Fraction\n",
+    "\n",
+    "from IPython.display import Math\n",
+    "\n",
+    "import qrules\n",
+    "\n",
+    "PDG = qrules.load_pdg()\n",
+    "particle_names = [\n",
+    "    \"J/psi(1S)\",\n",
+    "    \"Sigma+\",\n",
+    "    \"Sigma(1670)~-\",\n",
+    "    \"p~\",\n",
+    "    \"K0\",\n",
+    "]\n",
+    "latex_expressions = []\n",
+    "for name in particle_names:\n",
+    "    particle = PDG[name]\n",
+    "    parity = \"+\" if particle.parity > 0 else \"-\"\n",
+    "    if particle.spin.is_integer():\n",
+    "        spin = int(particle.spin)\n",
+    "    else:\n",
+    "        nominator, denominator = Fraction(particle.spin).as_integer_ratio()\n",
+    "        spin = fR\"\\tfrac{{{nominator}}}{{{denominator}}}\"\n",
+    "    latex_expressions.append(f\"{particle.latex}={spin}^{parity}\")\n",
+    "Math(R\"\\qquad \".join(latex_expressions))"
    ]
   },
   {
@@ -82,25 +124,25 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## $J/\\psi\\to\\bar\\Sigma(1670)^-\\Sigma^+$"
+    "## $J/\\psi \\to \\Sigma^+\\bar\\Sigma(1670)^-$"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The spin and parity of each particle in the first transition can be summarized as $1^-\\to\\frac{3}{2}^+\\frac{1}{2}^+$. Following step 1 in the {ref}`procedure <usage/ls-coupling:procedure>`, we get:\n",
+    "The spin and parity of each particle in the first transition can be summarized as $1^-\\to\\frac{1}{2}^+\\frac{3}{2}^+$. Following step 1 in the {ref}`procedure <usage/ls-coupling:procedure>`, we get:\n",
     "\n",
     "$$\n",
     "\\begin{eqnarray}\n",
-    "\\left|s_{\\bar\\Sigma(1670)^-} - s_{\\Sigma^+}\\right| & \\le S & \\le s_{\\bar\\Sigma(1670)^-} + s_{\\Sigma^+} \\\\\n",
-    "\\left|\\tfrac{3}{2}-\\tfrac{1}{2}\\right| & \\le S & \\le \\tfrac{3}{2} + \\tfrac{1}{2} \\\\\n",
+    "\\left|s_{\\Sigma^+} - s_{\\bar\\Sigma(1670)^-}\\right| & \\le S & \\le s_{\\Sigma^+} + s_{\\bar\\Sigma(1670)^-} \\\\\n",
+    "\\left|\\tfrac{1}{2}-\\tfrac{3}{2}\\right| & \\le S & \\le \\tfrac{1}{2} + \\tfrac{3}{2} \\\\\n",
     "1 & \\le S & \\le 2\n",
     "\\end{eqnarray}\n",
     "$$\n",
     "\n",
     "$$\n",
-    "\\Rightarrow S=1 \\; \\text{or} \\;  S=2.\n",
+    "\\Rightarrow S=1 \\quad \\text{or} \\quad S=2\n",
     "$$"
    ]
   },
@@ -112,11 +154,8 @@
     "\n",
     "$$\n",
     "\\begin{eqnarray}\n",
-    "& |L-S| \\le s_{J/\\psi} \\le L+S \\\\\n",
-    "& \\begin{cases}\n",
-    "|L-1| \\le 1 \\le L+1 \\\\\n",
-    "|L-2| \\le 1 \\le L+2\n",
-    "\\end{cases}\n",
+    "|L-S| & \\le s_{J/\\psi} & \\le L+S \\\\\n",
+    "|L-S| & \\le 1 & \\le L+S\n",
     "\\end{eqnarray}\n",
     "$$\n",
     "\n",
@@ -147,9 +186,9 @@
     "\n",
     "$$\n",
     "\\begin{eqnarray}\n",
-    "\\eta_{J/\\psi} & =&\\eta_{\\bar\\Sigma(1670)^-}\\cdot\\eta_{\\Sigma^+}\\cdot(-1)^L \\\\\n",
-    "(-1) & = & (+1)\\cdot(-1)\\cdot(-1)^L \\\\\n",
-    "(+1) & = & (-1)^{L}.\n",
+    "\\eta_{J/\\psi} & = & \\eta_{\\Sigma^+} \\cdot \\eta_{\\bar\\Sigma(1670)^-} \\cdot(-1)^L \\\\\n",
+    "(-1) & = & (+1)\\cdot(+1)\\cdot(-1)^L \\\\\n",
+    "(-1) & = & (-1)^{L}.\n",
     "\\end{eqnarray}\n",
     "$$"
    ]
@@ -158,10 +197,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "From this, we can easily see that only even $L$ values are possible, which leaves us with **3 $LS$-combinations**:\n",
+    "From this, we can easily see that only odd $L$ values are possible, which leaves us with **3 $LS$-combinations**:\n",
     "\n",
     "$$\n",
-    "(L,S)=(0,1), (2,1), (2,2).\n",
+    "(L,S) = (1,1), (1,2), (3,2).\n",
     "$$"
    ]
   },
@@ -234,6 +273,46 @@
     "(L,S)=\\left(2,\\tfrac{1}{2}\\right)\n",
     "$$"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Check with QRules"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Finally, let's use {func}`.generate_transitions` to check whether the allowed $LS$-couplings are found by {mod}`qrules` as well. Note that we have to increase the maximum angular momentum to find the $(L,S)=(3,2)$ combination as well."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    },
+    "tags": [
+     "hide-input"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "import graphviz\n",
+    "\n",
+    "reaction = qrules.generate_transitions(\n",
+    "    initial_state=\"J/psi(1S)\",\n",
+    "    final_state=[\"K0\", \"Sigma+\", \"p~\"],\n",
+    "    allowed_intermediate_particles=[\"Sigma(1670)\"],\n",
+    "    allowed_interaction_types=\"strong\",\n",
+    "    max_angular_momentum=3,\n",
+    ")\n",
+    "dot = qrules.io.asdot(reaction, render_node=True, strip_spin=True)\n",
+    "graphviz.Source(dot)"
+   ]
   }
  ],
  "metadata": {