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prog/python/qqgg/analytical_xs.org
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@ -1,9 +1,9 @@
#+PROPERTY: header-args :exports both :output-dir results #+PROPERTY: header-args :exports both :output-dir results :session xs :kernel python3
* Init * Init
** Required Modules ** Required Modules
#+NAME: e988e3f2-ad1f-49a3-ad60-bedba3863283 #+NAME: e988e3f2-ad1f-49a3-ad60-bedba3863283
#+begin_src ipython :session :exports both :tangle tangled/xs.py #+begin_src jupyter-python :exports both :tangle tangled/xs.py
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
import monte_carlo import monte_carlo
@ -11,9 +11,10 @@
#+RESULTS: e988e3f2-ad1f-49a3-ad60-bedba3863283 #+RESULTS: e988e3f2-ad1f-49a3-ad60-bedba3863283
** Utilities ** Utilities
#+NAME: 53548778-a4c1-461a-9b1f-0f401df12b08 #+NAME: 53548778-a4c1-461a-9b1f-0f401df12b08
#+BEGIN_SRC ipython :session :exports both #+BEGIN_SRC jupyter-python :exports both
%run ../utility.py %run ../utility.py
%load_ext autoreload %load_ext autoreload
%aimport monte_carlo %aimport monte_carlo
@ -24,7 +25,7 @@
* Implementation * Implementation
#+NAME: 777a013b-6c20-44bd-b58b-6a7690c21c0e #+NAME: 777a013b-6c20-44bd-b58b-6a7690c21c0e
#+BEGIN_SRC ipython :session :exports both :results raw drawer :exports code :tangle tangled/xs.py #+BEGIN_SRC jupyter-python :exports both :results raw drawer :exports code :tangle tangled/xs.py
""" """
Implementation of the analytical cross section for q q_bar -> Implementation of the analytical cross section for q q_bar ->
gamma gamma gamma gamma
@ -131,21 +132,19 @@
#+END_SRC #+END_SRC
#+RESULTS: 777a013b-6c20-44bd-b58b-6a7690c21c0e #+RESULTS: 777a013b-6c20-44bd-b58b-6a7690c21c0e
:RESULTS:
:END:
* Calculations * Calculations
** XS qq -> gamma gamma ** XS qq -> gamma gamma
First, set up the input parameters. First, set up the input parameters.
#+NAME: 7e62918a-2935-41ac-94e0-f0e7c3af8e0d #+NAME: 7e62918a-2935-41ac-94e0-f0e7c3af8e0d
#+BEGIN_SRC ipython :session :exports both :results raw drawer #+BEGIN_SRC jupyter-python :exports both :results raw drawer
η = 2.5 η = 2.5
charge = 1/3 charge = 1/3
esp = 200 # GeV esp = 200 # GeV
#+END_SRC #+END_SRC
Set up the integration and plot intervals. Set up the integration and plot intervals.
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
interval_η = [-η, η] interval_η = [-η, η]
interval = η_to_θ([-η, η]) interval = η_to_θ([-η, η])
interval_cosθ = np.cos(interval) interval_cosθ = np.cos(interval)
@ -154,45 +153,37 @@ plot_interval = [0.1, np.pi-.1]
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS:
:END:
#+RESULTS: 7e62918a-2935-41ac-94e0-f0e7c3af8e0d #+RESULTS: 7e62918a-2935-41ac-94e0-f0e7c3af8e0d
:RESULTS:
:END:
*** Analytical Integratin *** Analytical Integratin
And now calculate the cross section in picobarn. And now calculate the cross section in picobarn.
#+NAME: cf853fb6-d338-482e-bc55-bd9f8e796495 #+NAME: cf853fb6-d338-482e-bc55-bd9f8e796495
#+BEGIN_SRC ipython :session :exports both :results drawer output file :file xs.tex #+BEGIN_SRC jupyter-python :exports both :results drawer output file :file xs.tex
xs_gev = total_xs_eta(η, charge, esp) xs_gev = total_xs_eta(η, charge, esp)
xs_pb = gev_to_pb(xs_gev) xs_pb = gev_to_pb(xs_gev)
print(tex_value(xs_pb, unit=r'\pico\barn', prefix=r'\sigma = ', prec=5)) print(tex_value(xs_pb, unit=r'\pico\barn', prefix=r'\sigma = ', prec=5))
#+END_SRC #+END_SRC
#+RESULTS: cf853fb6-d338-482e-bc55-bd9f8e796495 #+RESULTS: cf853fb6-d338-482e-bc55-bd9f8e796495
:RESULTS: : \(\sigma = \SI{0.05379}{\pico\barn}\)
[[file:results/xs.tex]]
:END:
Compared to sherpa, it's pretty close. Compared to sherpa, it's pretty close.
#+NAME: 81b5ed93-0312-45dc-beec-e2ba92e22626 #+NAME: 81b5ed93-0312-45dc-beec-e2ba92e22626
#+BEGIN_SRC ipython :session :exports both :results raw drawer #+BEGIN_SRC jupyter-python :exports both :results raw drawer
sherpa = 0.0538009 sherpa = 0.0538009
xs_pb/sherpa xs_pb/sherpa
#+END_SRC #+END_SRC
#+RESULTS: 81b5ed93-0312-45dc-beec-e2ba92e22626 #+RESULTS: 81b5ed93-0312-45dc-beec-e2ba92e22626
:RESULTS: : 0.9998585425137037
0.9998585425137037
:END:
I had to set the runcard option ~EW_SCHEME: alpha0~ to use the pure I had to set the runcard option ~EW_SCHEME: alpha0~ to use the pure
QED coupling constant. QED coupling constant.
*** Numerical Integration *** Numerical Integration
Plot our nice distribution: Plot our nice distribution:
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
plot_points = np.linspace(*plot_interval, 1000) plot_points = np.linspace(*plot_interval, 1000)
fig, ax = set_up_plot() fig, ax = set_up_plot()
@ -206,22 +197,18 @@ save_fig(fig, 'diff_xs', 'xs', size=[4, 4])
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS: [[file:./.ob-jupyter/d30ededeaa03958fae5b649f50f3c5c3e6ae4677.png]]
[[file:./obipy-resources/DHBTl1.png]]
:END:
Define the integrand. Define the integrand.
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
def xs_pb_int(θ): def xs_pb_int(θ):
return gev_to_pb(np.sin(θ)*diff_xs(θ, charge=charge, esp=esp)) return gev_to_pb(np.sin(θ)*diff_xs(θ, charge=charge, esp=esp))
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS:
:END:
Plot the integrand. # TODO: remove duplication Plot the integrand. # TODO: remove duplication
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
fig, ax = set_up_plot() fig, ax = set_up_plot()
ax.plot(plot_points, xs_pb_int(plot_points)) ax.plot(plot_points, xs_pb_int(plot_points))
ax.set_xlabel(r'$\theta$') ax.set_xlabel(r'$\theta$')
@ -233,59 +220,49 @@ save_fig(fig, 'xs_integrand', 'xs', size=[4, 4])
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS: [[file:./.ob-jupyter/78974a2e2315c72bd7ae8e4ac009b3d79cfe7001.png]]
[[file:./obipy-resources/4mne94.png]]
:END:
Intergrate σ with the mc method. Intergrate σ with the mc method.
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
xs_pb_mc, xs_pb_mc_err = monte_carlo.integrate(xs_pb_int, interval, 10000) xs_pb_mc, xs_pb_mc_err = monte_carlo.integrate(xs_pb_int, interval, 10000)
xs_pb_mc = xs_pb_mc*np.pi*2 xs_pb_mc = xs_pb_mc*np.pi*2
xs_pb_mc, xs_pb_mc_err xs_pb_mc, xs_pb_mc_err
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS: | 0.05365000636562272 | 4.2342293364016736e-05 |
(0.05360809379599215, 4.22681790215136e-05)
:END:
We gonna export that as tex. We gonna export that as tex.
#+begin_src ipython :session :exports both :results raw drawer output :file xs_mc.tex #+begin_src jupyter-python :exports both :results raw drawer output :file xs_mc.tex
print(tex_value(xs_pb_mc, unit=r'\pico\barn', prefix=r'\sigma = ', prec=5)) print(tex_value(xs_pb_mc, unit=r'\pico\barn', prefix=r'\sigma = ', prec=5))
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS: : \(\sigma = \SI{0.05365}{\pico\barn}\)
[[file:results/xs_mc.tex]]
:END:
*** Sampling and Analysis *** Sampling and Analysis
Define the sample number. Define the sample number.
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
sample_num = 1000 sample_num = 1000
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS:
:END:
Now we monte-carlo sample our distribution. Now we monte-carlo sample our distribution.
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
cosθ_sample = monte_carlo.sample_unweighted_array(sample_num, lambda x: cosθ_sample = monte_carlo.sample_unweighted_array(sample_num, lambda x:
diff_xs_cosθ(x, charge, esp), diff_xs_cosθ(x, charge, esp),
interval_cosθ) interval_cosθ)
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS:
:END:
Nice! And now draw some histograms. Nice! And now draw some histograms.
We define an auxilliary method for convenience. We define an auxilliary method for convenience.
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
def draw_histo(points, xlabel, bins=20): def draw_histo(points, xlabel, bins=20):
fig, ax = set_up_plot() fig, ax = set_up_plot()
ax.hist(points, bins, histtype='step') ax.hist(points, bins, histtype='step')
@ -295,22 +272,18 @@ We define an auxilliary method for convenience.
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS:
:END:
The histogram for cosθ. The histogram for cosθ.
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
fig, _ = draw_histo(cosθ_sample, r'$\cos\theta$') fig, _ = draw_histo(cosθ_sample, r'$\cos\theta$')
save_fig(fig, 'histo_cos_theta', 'xs', size=(4,3)) save_fig(fig, 'histo_cos_theta', 'xs', size=(4,3))
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS: [[file:./.ob-jupyter/ddc5e5b2a628d9f9add43555d7386acf4d92c6ee.png]]
[[file:./obipy-resources/ZSJaBQ.png]]
:END:
Now we define some utilities to draw real 4-impulse samples. Now we define some utilities to draw real 4-impulse samples.
#+begin_src ipython :session :exports both :tangle tangled/xs.py #+begin_src jupyter-python :exports both :tangle tangled/xs.py
def sample_impulses(sample_num, interval, charge, esp, seed=None): def sample_impulses(sample_num, interval, charge, esp, seed=None):
"""Samples `sample_num` unweighted photon 4-impulses from the cross-section. """Samples `sample_num` unweighted photon 4-impulses from the cross-section.
@ -342,8 +315,10 @@ Now we define some utilities to draw real 4-impulse samples.
#+RESULTS: #+RESULTS:
To generate histograms of other obeservables, we have to define them as functions on 4-impuleses. To generate histograms of other obeservables, we have to define them
#+begin_src ipython :session :exports both :results raw drawer :tangle tangled/observables.py as functions on 4-impuleses. Using those to transform samples is
analogous to transforming the distribution itself.
#+begin_src jupyter-python :exports both :results raw drawer :tangle tangled/observables.py
"""This module defines some observables on arrays of 4-pulses.""" """This module defines some observables on arrays of 4-pulses."""
import numpy as np import numpy as np
@ -365,56 +340,44 @@ To generate histograms of other obeservables, we have to define them as function
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS:
:END:
Lets try it out. Lets try it out.
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
impulse_sample = sample_impulses(2000, interval_cosθ, charge, esp) impulse_sample = sample_impulses(2000, interval_cosθ, charge, esp)
impulse_sample impulse_sample
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS: : array([[100. , 60.93780026, 38.29391655, 69.42737539],
#+BEGIN_EXAMPLE : [100. , 16.62473755, 5.08308744, -98.47730867],
array([[100. , 48.55787717, 64.05713855, 59.48794471], : [100. , 62.52584971, 41.05712399, 66.3688985 ],
[100. , 42.68070092, 33.17436113, -84.12977792], : ...,
[100. , 18.42611283, 27.20055109, -94.44897239], : [100. , 36.93115123, 10.77808502, -92.30342871],
..., : [100. , 34.39831699, 43.0134429 , 83.46615792],
[100. , 21.40152914, 14.7440014 , 96.56391134], : [100. , 69.87424822, 3.87926805, 71.43207063]])
[100. , 35.84656512, 5.33864248, -93.20151643],
[100. , 38.37094512, 8.92583559, 91.9130025 ]])
#+END_EXAMPLE
:END:
Now let's make a histogram of the η distribution. Now let's make a histogram of the η distribution.
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
η_sample = η(impulse_sample) η_sample = η(impulse_sample)
draw_histo(η_sample, r'$\eta$') draw_histo(η_sample, r'$\eta$')
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS: :RESULTS:
#+BEGIN_EXAMPLE | <Figure | size | 432x288 | with | 1 | Axes> | <matplotlib.axes._subplots.AxesSubplot | at | 0x7fb464af2040> |
(<Figure size 432x288 with 1 Axes>, [[file:./.ob-jupyter/347b6d473f38cf692e5614a095c9bc1a0e89c763.png]]
<matplotlib.axes._subplots.AxesSubplot at 0x7ff36151dd60>)
#+END_EXAMPLE
[[file:./obipy-resources/S2OvbR.png]]
:END: :END:
And the same for the p_t (transverse impulse) distribution. And the same for the p_t (transverse impulse) distribution.
#+begin_src ipython :session :exports both :results raw drawer #+begin_src jupyter-python :exports both :results raw drawer
p_t_sample = p_t(impulse_sample) p_t_sample = p_t(impulse_sample)
draw_histo(p_t_sample, r'$p_T$') draw_histo(p_t_sample, r'$p_T$')
#+end_src #+end_src
#+RESULTS: #+RESULTS:
:RESULTS: :RESULTS:
#+BEGIN_EXAMPLE | <Figure | size | 432x288 | with | 1 | Axes> | <matplotlib.axes._subplots.AxesSubplot | at | 0x7fb463469d60> |
(<Figure size 432x288 with 1 Axes>, [[file:./.ob-jupyter/880ac31d31bd9a537c0faacd56dc38f9eb668c7d.png]]
<matplotlib.axes._subplots.AxesSubplot at 0x7ff364951370>)
#+END_EXAMPLE
[[file:./obipy-resources/nW1TKv.png]]
:END: :END:

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