DOC: add explanatory text

docs/report/draft.ipynb

@@ -82,7 +82,7 @@
    "source": [
     "Follow-up to {need}`TR-004`, where we investigate and reproduce the Riemann sheets shown in [Fig. 50.1](https://pdg.lbl.gov/2023/reviews/rpp2022-rev-resonances.pdf#page=2) and [50.2](https://pdg.lbl.gov/2023/reviews/rpp2022-rev-resonances.pdf#page=4) of the PDG.\n",
     "\n",
-    "First, we formulate the $T$ matrix in terms of a $K$ matrix."
+    "First, we formulate the $T$ matrix in terms of a $K$ matrix. There are two ways to do this and we associate the one with the $+$ with **Sheet I** and the one with $-$ with **Sheet II**."
    ]
   },
   {
@@ -161,13 +161,20 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Note that this is equivalent to [Eq. (50.31)](https://pdg.lbl.gov/2023/reviews/rpp2022-rev-resonances.pdf#page=4) (omitting the $n$):\n",
+    "Note that this the above inversion is equivalent to [Eq. (50.31)](https://pdg.lbl.gov/2023/reviews/rpp2022-rev-resonances.pdf#page=4) (omitting the form factors&nsbp;$n$):\n",
     "\n",
     "$$\n",
     "T = (1 \\pm iK\\rho)^{-1}K.\n",
     "$$"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "As an aside, we also define a special expression class for a square root where you can choose the sign for negative arguments. This can be used later for the phase space factor $\\rho$."
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
@@ -186,6 +193,7 @@
    "source": [
     "class SignedSqrt(NumPyPrintable):\n",
     "    is_commutative = True\n",
+    "    is_real = False\n",
     "\n",
     "    def __new__(cls, x, sign, *args, **kwargs):\n",
     "        x = sp.sympify(x)\n",
@@ -239,8 +247,15 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "s_real = sp.Symbol(\"x\")\n",
-    "SignedSqrt(s_real, -1) + SignedSqrt(s_real, +1)"
+    "x = sp.Symbol(\"x\")\n",
+    "SignedSqrt(x, -1) + SignedSqrt(x, +1)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This gives us all the ingredients to formulate expressions for the parametrization of the matrix elements."
    ]
   },
   {
@@ -262,6 +277,7 @@
     "@implement_doit_method\n",
     "class EqualMassPhspFactor(UnevaluatedExpression):\n",
     "    is_commutative = True\n",
+    "    is_real = False\n",
     "\n",
     "    def __new__(cls, s, m, sign, **hints) -> EqualMassPhspFactor:\n",
     "        return create_expression(cls, s, m, sign, **hints)\n",
@@ -282,7 +298,9 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "s, m0, m1, Γ, sign = sp.symbols(R\"s m0 m1 Gamma \\pm\")\n",
+    "s = sp.Symbol(\"s\", complex=True)\n",
+    "m0, m1, Γ = sp.symbols(\"m0 m1 Gamma\", real=True, nonnegative=True)\n",
+    "sign = sp.symbols(R\"\\pm\", integer=True)\n",
     "K_expr = (m0**2 - s) / (m0 * Γ)\n",
     "ρ_expr = EqualMassPhspFactor(s, m1, sign)"
    ]
@@ -338,6 +356,13 @@
     "Math(src)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Finally, we convert these expressions to numerical functions and use these to plot the Riemann sheets over a complex plane for the Mandelstam variable $s$."
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
@@ -355,6 +380,9 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    },
     "mystnb": {
      "code_prompt_show": "Define mesh grid for plotting"
     },
@@ -537,6 +565,20 @@
     "fig.tight_layout()\n",
     "display(w.VBox([output, UI]))"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Note that we matched the colors [of the PDG](https://pdg.lbl.gov/2023/reviews/rpp2022-rev-resonances.pdf#page=4), but that we had to remap the sheet associations. As can be seen in the plotting code, we have the following associations:\n",
+    "\n",
+    "|                      | Sheet I  | Sheet II |\n",
+    "|----------------------|----------|----------|\n",
+    "| $\\mathrm{Im}(s) < 0$ | $T^{I}$  | $T^{II}$ |\n",
+    "| $\\mathrm{Im}(s) > 0$ | $T^{II}$ | $T^{I}$  |\n",
+    "\n",
+    "So the sheet numbers 'flip' for $\\mathrm{Im}(s) > 0$ and what we see in the third figure is just $T^{II}$."
+   ]
   }
  ],
  "metadata": {