Plots_of_q2_dependent_observables.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Flavio examples: plots of $q^2$-dependent observables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When dealing with $q^2$-dependent observables, like differential branching ratios or angular observables, flavio can produce three different kinds of plots:\n",
    "\n",
    "- the differential, i.e. smooth, theory prediction (via `flavio.plots.q2_plot_th_ff`)\n",
    "- binned experimental measurements (via `flavio.plots.q2_plot_exp`)\n",
    "- the binned theory prediction (via `flavio.plots.q2_plot_th_bin`)\n",
    "\n",
    "This notebook illustrates the usage of these functions. For details, see the [API docs](https://flav-io.github.io/apidoc/flavio/plots/plotfunctions.m.html).\n",
    "\n",
    "For illustration, we will focus on the branching ratio of the $B^0\\to \\rho^+l^-\\bar\\nu$ decay that is used to measure $V_{ub}$ at $B$ factories."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Preliminaries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0.24'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import flavio\n",
    "import flavio.plots\n",
    "flavio.__version__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "%config InlineBackend.figure_format = 'svg'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Differential theory prediction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As simple as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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    "Note the discontinuity at $q^2$ close to 0. This can be avoided by increasing the number of $q^2$ steps from the default (100):"
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    "The spectrum we just plotted is actually the average of the electron and muon modes. Since the two have a different low-$q^2$ end point ($m_e^2$ vs. $m_\\mu^2$), the average shows a plateau close to 0:"
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   "source": [
    "flavio.plots.q2_plot_th_diff('dBR/dq2(B0->rholnu)', q2min=0, q2max=0.05, label=r'$\\ell$')\n",
    "flavio.plots.q2_plot_th_diff('dBR/dq2(B0->rhoenu)', q2min=0, q2max=0.05, c='C1', label=r'$e$')\n",
    "flavio.plots.q2_plot_th_diff('dBR/dq2(B0->rhomunu)', q2min=0, q2max=0.05, c='C0', label=r'$\\mu$')\n",
    "plt.legend();"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This plot also demonstrates how easy it is to add a plot legend: just use the `label` keyword argument on the plot functions and add `plt.legend()` at the end. More generally, all additional keyword arguments added to the polot functions will be passed on to the actual `matplotlib` functions (`pyplot.plot`, in this case)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### New physics\n",
    "\n",
    "New physics can be added easily by supplying a `WilsonCoefficients` instance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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    "wc_np = flavio.WilsonCoefficients()\n",
    "wc_np.set_initial({'CV_buenu': 0.2, 'CV_bumunu': 0.2}, scale=160 ) # increasing the b->ulnu amplitude by 20%\n",
    "flavio.plots.q2_plot_th_diff('dBR/dq2(B0->rholnu)', q2min=0, q2max=21, label='Standard Model')\n",
    "flavio.plots.q2_plot_th_diff('dBR/dq2(B0->rholnu)', q2min=0, q2max=21, wc=wc_np, label='new physics', c='C1')\n",
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   "source": [
    "## Experimental measurements\n",
    "\n",
    "The measuremens defined in `measurements.yaml` can be used to automatically plot data points. The labels for the legend are set automatically."
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  },
  {
   "cell_type": "code",
   "execution_count": 7,
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    "However, this plot is actually **highly misleading**. This is because the measurements refer to *integrated branching ratios*, so we shouldn't be comparing bins with different sizes. To remedy this problem, we have to use the option `divide_binwidth`, which just divides each measurement by the bin width. As a result, we get a *binned differential branching ratio* that we can now compare to the theory prediction as well:"
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    "flavio.plots.q2_plot_exp('<BR>(B0->rholnu)', divide_binwidth=True)\n",
    "flavio.plots.q2_plot_th_diff('dBR/dq2(B0->rholnu)', 0, 21, label='theory')\n",
    "plt.xlabel(r'$q^2$ [GeV$^2$]')\n",
    "plt.ylabel(flavio.Observable.get_instance('dBR/dq2(B0->rholnu)').tex)\n",
    "plt.legend()"
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   "metadata": {},
   "source": [
    "The above plot also demonstrates the use of $\\LaTeX$ in axis labels. For the y axis, we have even been to lazy to type the name of the observable, but simple used the `tex` attribute of the `Observable` object that is already predefined in flavio."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Binned theory prediction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Instead of plotting the differential theory prediction, we can also directly compare the measurements to the *binned* theory prediction. \n",
    "\n",
    "Also for `q2_plot_th_bin`, we are using the `divide_binwidth` option."
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   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
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    "bin_list = [(q2, q2+4) for q2 in range(0, 20, 4)]\n",
    "flavio.plots.q2_plot_th_bin('<BR>(B0->rholnu)', bin_list=bin_list, divide_binwidth=True, label='binned theory')\n",
    "flavio.plots.q2_plot_exp('<BR>(B0->rholnu)', divide_binwidth=True)\n",
    "plt.xlabel(r'$q^2$ [GeV$^2$]')\n",
    "plt.ylabel(flavio.Observable.get_instance('dBR/dq2(B0->rholnu)').tex)\n",
    "plt.legend()"
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