some more discussion around pseudo rapidity

prog/python/qqgg/analytical_xs.org

@@ -210,6 +210,9 @@ Define the integrand.
 #+begin_src jupyter-python :exports both :results raw drawer
   def xs_pb_int(θ):
       return 2*np.pi*gev_to_pb(np.sin(θ)*diff_xs(θ, charge=charge, esp=esp))
+
+  def xs_pb_int_η(η):
+      return 2*np.pi*gev_to_pb(diff_xs_eta(η, charge, esp))
 #+end_src
 
 #+RESULTS:
@@ -219,7 +222,7 @@ Plot the integrand. # TODO: remove duplication
 fig, ax = set_up_plot()
 ax.plot(plot_points, xs_pb_int(plot_points))
 ax.set_xlabel(r'$\theta$')
-ax.set_ylabel(r'$\sin(\theta)\cdot\frac{d\sigma}{d\theta}$ [pb]')
+ax.set_ylabel(r'$\sin(\theta)\cdot\frac{d\sigma}{d\Omega}$ [pb]')
 ax.axvline(interval[0], color='gray', linestyle='--')
 ax.axvline(interval[1], color='gray', linestyle='--', label=rf'$|\eta|={η}$')
 ax.legend()
@@ -227,7 +230,7 @@ save_fig(fig, 'xs_integrand', 'xs', size=[4, 4])
 #+end_src
 
 #+RESULTS:
-[[file:./.ob-jupyter/9e547bdeaa79bb956057b552090b4ab6305a20e6.png]]
+[[file:./.ob-jupyter/377a2432093426a961983767c2f23e1cd8b105a5.png]]
 
 Intergrate σ with the mc method.
 #+begin_src jupyter-python :exports both :results raw drawer
@@ -248,16 +251,29 @@ tex_value(xs_pb_mc, unit=r'\pico\barn', prefix=r'\sigma = ', err=xs_pb_mc_err, s
 : \(\sigma = \SI{0.05389\pm 0.00005}{\pico\barn}\)
 
 
+Plot the intgrand of the pseudo rap.
+#+begin_src jupyter-python :exports both :results raw drawer
+fig, ax = set_up_plot()
+points = np.linspace(*interval_η, 1000)
+ax.plot(points, xs_pb_int_η(points))
+ax.set_xlabel(r'$\eta$')
+ax.set_ylabel(r'$\frac{d\sigma}{d\theta}$ [pb]')
+save_fig(fig, 'xs_integrand_η', 'xs', size=[4, 4])
+#+end_src
+
+#+RESULTS:
+[[file:./.ob-jupyter/09de667c0ccb1d17fef74918e3f462a1340df113.png]]
+
 As we see, the result is much better if we use pseudo rapidity,
 because the differential cross section does not difverge anymore.
 #+begin_src jupyter-python :exports both :results raw drawer
-  xs_pb_η = monte_carlo.integrate(lambda x: 2*np.pi*gev_to_pb(diff_xs_eta(x, charge, esp)),
+  xs_pb_η = monte_carlo.integrate(xs_pb_int_η,
                         interval_η, 1000)
   xs_pb_η
 #+end_src
 
 #+RESULTS:
-| 0.05359070224781967 | 7.127342563203912e-05 |
+| 0.053758763260352345 | 7.109470263473152e-05 |
 
 And yet again export that as tex.
 #+begin_src jupyter-python :exports both :results raw drawer

prog/python/qqgg/figs/xs/xs_integrand.pgf

@@ -3545,7 +3545,7 @@
 \definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{textcolor}%
 \pgfsetfillcolor{textcolor}%
-\pgftext[x=0.466597in,y=2.197917in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle \sin(\theta)\cdot\frac{d\sigma}{d\theta}\) [pb]}%
+\pgftext[x=0.466597in,y=2.197917in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle \sin(\theta)\cdot\frac{d\sigma}{d\Omega}\) [pb]}%
 \end{pgfscope}%
 \begin{pgfscope}%
 \pgfpathrectangle{\pgfqpoint{0.866319in}{0.594444in}}{\pgfqpoint{2.935069in}{3.206944in}}%
@@ -3714,15 +3714,15 @@
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetstrokeopacity{0.800000}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{1.854752in}{3.481944in}}%
-\pgfpathlineto{\pgfqpoint{2.812956in}{3.481944in}}%
-\pgfpathquadraticcurveto{\pgfqpoint{2.840734in}{3.481944in}}{\pgfqpoint{2.840734in}{3.509722in}}%
-\pgfpathlineto{\pgfqpoint{2.840734in}{3.704167in}}%
-\pgfpathquadraticcurveto{\pgfqpoint{2.840734in}{3.731944in}}{\pgfqpoint{2.812956in}{3.731944in}}%
-\pgfpathlineto{\pgfqpoint{1.854752in}{3.731944in}}%
-\pgfpathquadraticcurveto{\pgfqpoint{1.826974in}{3.731944in}}{\pgfqpoint{1.826974in}{3.704167in}}%
-\pgfpathlineto{\pgfqpoint{1.826974in}{3.509722in}}%
-\pgfpathquadraticcurveto{\pgfqpoint{1.826974in}{3.481944in}}{\pgfqpoint{1.854752in}{3.481944in}}%
+\pgfpathmoveto{\pgfqpoint{1.018278in}{3.481944in}}%
+\pgfpathlineto{\pgfqpoint{3.704167in}{3.481944in}}%
+\pgfpathquadraticcurveto{\pgfqpoint{3.731944in}{3.481944in}}{\pgfqpoint{3.731944in}{3.509722in}}%
+\pgfpathlineto{\pgfqpoint{3.731944in}{3.704167in}}%
+\pgfpathquadraticcurveto{\pgfqpoint{3.731944in}{3.731944in}}{\pgfqpoint{3.704167in}{3.731944in}}%
+\pgfpathlineto{\pgfqpoint{1.018278in}{3.731944in}}%
+\pgfpathquadraticcurveto{\pgfqpoint{0.990500in}{3.731944in}}{\pgfqpoint{0.990500in}{3.704167in}}%
+\pgfpathlineto{\pgfqpoint{0.990500in}{3.509722in}}%
+\pgfpathquadraticcurveto{\pgfqpoint{0.990500in}{3.481944in}}{\pgfqpoint{1.018278in}{3.481944in}}%
 \pgfpathclose%
 \pgfusepath{stroke,fill}%
 \end{pgfscope}%
@@ -3733,15 +3733,15 @@
 \definecolor{currentstroke}{rgb}{0.501961,0.501961,0.501961}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{{5.550000pt}{2.400000pt}}{0.000000pt}%
-\pgfpathmoveto{\pgfqpoint{1.882530in}{3.620833in}}%
-\pgfpathlineto{\pgfqpoint{2.160308in}{3.620833in}}%
+\pgfpathmoveto{\pgfqpoint{1.046056in}{3.620833in}}%
+\pgfpathlineto{\pgfqpoint{1.323833in}{3.620833in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
 \definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{textcolor}%
 \pgfsetfillcolor{textcolor}%
-\pgftext[x=2.271419in,y=3.572222in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle |\eta|=2.5\)}%
+\pgftext[x=1.434944in,y=3.572222in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle |\eta|=<function η at 0x7f9ddc45bb80>\)}%
 \end{pgfscope}%
 \end{pgfpicture}%
 \makeatother%