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<body>
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<div id="content">
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|
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<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org255850a">1. Init</a>
|
||||
<ul>
|
||||
<li><a href="#org2f3a4e9">1.1. Required Modules</a></li>
|
||||
<li><a href="#org2e97ca2">1.2. Utilities</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org468bdb4">2. Implementation</a></li>
|
||||
<li><a href="#org9989482">3. Calculations</a>
|
||||
<ul>
|
||||
<li><a href="#org9716522">3.1. XS qq -> gamma gamma</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
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</div>
|
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</div>
|
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|
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<div id="outline-container-org255850a" class="outline-2">
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<h2 id="org255850a"><span class="section-number-2">1</span> Init</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org2f3a4e9" class="outline-3">
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<h3 id="org2f3a4e9"><span class="section-number-3">1.1</span> Required Modules</h3>
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||||
<div class="org-src-container">
|
||||
<pre class="src src-ipython" id="org91f75e0"><span style="color: #859900; font-weight: bold;">import</span> numpy <span style="color: #859900; font-weight: bold;">as</span> np
|
||||
<span style="color: #859900; font-weight: bold;">import</span> matplotlib.pyplot <span style="color: #859900; font-weight: bold;">as</span> plt
|
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</pre>
|
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</div>
|
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</div>
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</div>
|
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|
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<div id="outline-container-org2e97ca2" class="outline-3">
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<h3 id="org2e97ca2"><span class="section-number-3">1.2</span> Utilities</h3>
|
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<div class="outline-text-3" id="text-1-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-ipython" id="org41306d9">%run ../utility.py
|
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</pre>
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</div>
|
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</div>
|
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</div>
|
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</div>
|
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|
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<div id="outline-container-org468bdb4" class="outline-2">
|
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<h2 id="org468bdb4"><span class="section-number-2">2</span> Implementation</h2>
|
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<div class="outline-text-2" id="text-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-ipython" id="org30462c1"><span style="color: #35a69c; font-style: italic;">"""</span>
|
||||
<span style="color: #35a69c; font-style: italic;">Implementation of the analytical cross section for q q_bar -></span>
|
||||
<span style="color: #35a69c; font-style: italic;">gamma gamma</span>
|
||||
|
||||
<span style="color: #35a69c; font-style: italic;">Author: Valentin Boettcher <a href="mailto:hiro%40protagon.space"><hiro@protagon.space></a></span>
|
||||
<span style="color: #35a69c; font-style: italic;">"""</span>
|
||||
|
||||
<span style="color: #859900; font-weight: bold;">import</span> numpy <span style="color: #859900; font-weight: bold;">as</span> np
|
||||
<span style="color: #859900; font-weight: bold;">from</span> scipy.constants <span style="color: #859900; font-weight: bold;">import</span> alpha
|
||||
|
||||
<span style="color: #96A7A9; font-style: italic;"># </span><span style="color: #96A7A9; font-style: italic;">NOTE: a more elegant solution would be a decorator</span>
|
||||
<span style="color: #859900; font-weight: bold;">def</span> <span style="color: #b58900;">energy_factor</span>(charge, esp):
|
||||
<span style="color: #35a69c; font-style: italic;">"""</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> Calculates the factor common to all other values in this module</span>
|
||||
|
||||
<span style="color: #35a69c; font-style: italic;"> Arguments:</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> esp -- center of momentum energy in GeV</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> charge -- charge of the particle in units of the elementary charge</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> """</span>
|
||||
|
||||
<span style="color: #859900; font-weight: bold;">return</span> charge**4*(alpha/esp)**2/6
|
||||
|
||||
|
||||
<span style="color: #859900; font-weight: bold;">def</span> <span style="color: #b58900;">diff_xs</span>(theta, charge, esp):
|
||||
<span style="color: #35a69c; font-style: italic;">"""</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> Calculates the differential cross section as a function of the</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> azimuth angle theta in units of 1/GeV^2.</span>
|
||||
|
||||
<span style="color: #35a69c; font-style: italic;"> Arguments:</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> theta -- azimuth angle</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> esp -- center of momentum energy in GeV</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> charge -- charge of the particle in units of the elementary charge</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> """</span>
|
||||
|
||||
<span style="color: #268bd2;">f</span> = energy_factor(charge, esp)
|
||||
<span style="color: #859900; font-weight: bold;">return</span> f*((np.cos(theta)**+1)/np.sin(theta)**2)
|
||||
|
||||
<span style="color: #859900; font-weight: bold;">def</span> <span style="color: #b58900;">diff_xs_eta</span>(eta, charge, esp):
|
||||
<span style="color: #35a69c; font-style: italic;">"""</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> Calculates the differential cross section as a function of the</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> pseudo rapidity of the photons in units of 1/GeV^2.</span>
|
||||
|
||||
<span style="color: #35a69c; font-style: italic;"> Arguments:</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> eta -- pseudo rapidity</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> esp -- center of momentum energy in GeV</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> charge -- charge of the particle in units of the elementary charge</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> """</span>
|
||||
|
||||
<span style="color: #268bd2;">f</span> = energy_factor(charge, esp)
|
||||
<span style="color: #859900; font-weight: bold;">return</span> f*(2*np.cosh(eta)**2 - 1)
|
||||
|
||||
<span style="color: #859900; font-weight: bold;">def</span> <span style="color: #b58900;">total_xs_eta</span>(eta, charge, esp):
|
||||
<span style="color: #35a69c; font-style: italic;">"""</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> Calculates the total cross section as a function of the pseudo</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> rapidity of the photons in units of 1/GeV^2. If the rapditiy is</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> specified as a tuple, it is interpreted as an interval. Otherwise</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> the interval [-eta, eta] will be used.</span>
|
||||
|
||||
<span style="color: #35a69c; font-style: italic;"> Arguments:</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> eta -- pseudo rapidity (tuple or number)</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> esp -- center of momentum energy in GeV</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> charge -- charge of the particle in units of the elementar charge</span>
|
||||
<span style="color: #35a69c; font-style: italic;"> """</span>
|
||||
|
||||
<span style="color: #268bd2;">f</span> = energy_factor(charge, esp)
|
||||
<span style="color: #859900; font-weight: bold;">if</span> <span style="color: #859900; font-weight: bold;">not</span> <span style="color: #d33682; font-style: italic;">isinstance</span>(eta, <span style="color: #d33682; font-style: italic;">tuple</span>):
|
||||
<span style="color: #268bd2;">eta</span> = (-eta, eta)
|
||||
|
||||
<span style="color: #859900; font-weight: bold;">if</span> <span style="color: #d33682; font-style: italic;">len</span>(eta) != 2:
|
||||
<span style="color: #859900; font-weight: bold;">raise</span> <span style="color: #b58900; font-style: italic;">ValueError</span>(<span style="color: #2aa198;">'Invalid eta cut.'</span>)
|
||||
|
||||
<span style="color: #859900; font-weight: bold;">def</span> <span style="color: #b58900;">F</span>(x):
|
||||
<span style="color: #859900; font-weight: bold;">return</span> np.tanh(x) - 2*x
|
||||
|
||||
<span style="color: #859900; font-weight: bold;">return</span> 2*np.pi*f*(F(eta[0]) - F(eta[1]))
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9989482" class="outline-2">
|
||||
<h2 id="org9989482"><span class="section-number-2">3</span> Calculations</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
</div>
|
||||
<div id="outline-container-org9716522" class="outline-3">
|
||||
<h3 id="org9716522"><span class="section-number-3">3.1</span> XS qq -> gamma gamma</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
First, set up the input parameters.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-ipython" id="org9135f0f"><span style="color: #268bd2;">eta</span> = 2.5
|
||||
<span style="color: #268bd2;">charge</span> = 1/3
|
||||
<span style="color: #268bd2;">esp</span> = 200 <span style="color: #96A7A9; font-style: italic;"># </span><span style="color: #96A7A9; font-style: italic;">GeV</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And now calculate the cross section in picobarn.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-ipython" id="org1b205cb"><span style="color: #268bd2;">xs_gev</span> = total_xs_eta(eta, charge, esp)
|
||||
<span style="color: #268bd2;">xs_pb</span> = gev_to_pb(xs_gev)
|
||||
<span style="color: #859900; font-weight: bold;">print</span>(tex_value(xs_pb, unit=r<span style="color: #2aa198;">'\pico\barn'</span>, prefix=r<span style="color: #2aa198;">'\eta = '</span>))
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
<a href="../../../results/xs/xs.tex">../../../results/xs/xs.tex</a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Compared to sherpa, it's pretty close.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-ipython" id="org2bcb666"><span style="color: #268bd2;">sherpa</span> = 0.0538009
|
||||
xs_pb/sherpa
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
0.9998585425137037
|
||||
</p>
|
||||
|
||||
|
||||
|
||||
<p>
|
||||
I had to set the runcard option <code>EW_SCHEME: alpha0</code> to use the pure
|
||||
QED coupling constant.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="date">Created: 2020-03-28 Sat 11:30</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -66,7 +66,7 @@
|
|||
cosine of the azimuth angle θ in units of 1/GeV².
|
||||
|
||||
Arguments:
|
||||
θ -- azimuth angle
|
||||
cosθ -- cosine of the azimuth angle
|
||||
esp -- center of momentum energy in GeV
|
||||
charge -- charge of the particle in units of the elementary charge
|
||||
"""
|
||||
|
|
@ -88,6 +88,20 @@
|
|||
f = energy_factor(charge, esp)
|
||||
return f*(2*np.cosh(η)**2 - 1)
|
||||
|
||||
def diff_xs_pt(pt, charge, esp):
|
||||
"""
|
||||
Calculates the differential cross section as a function of the
|
||||
transversal impulse of the photons in units of 1/GeV^2.
|
||||
|
||||
Arguments:
|
||||
η -- transversal impulse
|
||||
esp -- center of momentum energy in GeV
|
||||
charge -- charge of the particle in units of the elementary charge
|
||||
"""
|
||||
|
||||
f = energy_factor(charge, esp)
|
||||
return f*((esp/pt)**2/2 - 1)
|
||||
|
||||
def total_xs_eta(η, charge, esp):
|
||||
"""
|
||||
Calculates the total cross section as a function of the pseudo
|
||||
|
|
@ -161,12 +175,13 @@ Compared to sherpa, it's pretty close.
|
|||
|
||||
I had to set the runcard option ~EW_SCHEME: alpha0~ to use the pure
|
||||
QED coupling constant.
|
||||
*** Numerical Integration
|
||||
*** Numerical Integration and Sampling
|
||||
Set up the integration and plot intervals.
|
||||
#+begin_src ipython :session :exports both :results raw drawer
|
||||
interval_η = [-η, η]
|
||||
interval = η_to_θ([-η, η])
|
||||
interval_cosθ = np.cos(interval)
|
||||
interval_pt = η_to_pt([0, η], esp/2)
|
||||
plot_interval = [0.1, np.pi-.1]
|
||||
#+end_src
|
||||
|
||||
|
|
@ -190,7 +205,7 @@ save_fig(fig, 'diff_xs', 'xs', size=[4, 4])
|
|||
|
||||
#+RESULTS:
|
||||
:RESULTS:
|
||||
[[file:./obipy-resources/VutHnr.png]]
|
||||
[[file:./obipy-resources/EvlB5m.png]]
|
||||
:END:
|
||||
|
||||
Define the integrand.
|
||||
|
|
@ -217,7 +232,7 @@ save_fig(fig, 'xs_integrand', 'xs', size=[4, 4])
|
|||
|
||||
#+RESULTS:
|
||||
:RESULTS:
|
||||
[[file:./obipy-resources/vkG9zr.png]]
|
||||
[[file:./obipy-resources/lOkEKe.png]]
|
||||
:END:
|
||||
|
||||
|
||||
|
|
@ -230,7 +245,7 @@ Intergrate σ with the mc method.
|
|||
|
||||
#+RESULTS:
|
||||
:RESULTS:
|
||||
(0.05364189417398581, 4.241792187836808e-05)
|
||||
(0.05382327328187836, 4.2568280254619665e-05)
|
||||
:END:
|
||||
|
||||
#+begin_src ipython :session :exports both :results raw drawer output :file xs_mc.tex
|
||||
|
|
@ -246,6 +261,8 @@ Now we monte-carlo sample our distribution.
|
|||
#+begin_src ipython :session :exports both :results raw drawer
|
||||
cosθ_sample = sample(lambda x: diff_xs_cosθ(x, charge, esp), interval_cosθ)
|
||||
η_sample = sample(lambda x: diff_xs_eta(x, charge, esp), interval_η)
|
||||
pt_sample = sample(lambda x: diff_xs_pt(x, charge, esp), interval_pt)
|
||||
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
|
|
@ -276,7 +293,7 @@ save_fig(fig, 'histo_cos_theta', 'xs', size=(4,2))
|
|||
|
||||
#+RESULTS:
|
||||
:RESULTS:
|
||||
[[file:./obipy-resources/XUeHmM.png]]
|
||||
[[file:./obipy-resources/UtLSDE.png]]
|
||||
:END:
|
||||
|
||||
And the histogram for η.
|
||||
|
|
@ -287,5 +304,16 @@ save_fig(fig, 'histo_eta', 'xs', size=(4,2))
|
|||
|
||||
#+RESULTS:
|
||||
:RESULTS:
|
||||
[[file:./obipy-resources/lxq7dj.png]]
|
||||
[[file:./obipy-resources/I7AUEF.png]]
|
||||
:END:
|
||||
|
||||
And the same for pt.
|
||||
#+begin_src ipython :session :exports both :results raw drawer
|
||||
draw_histo(pt_sample, r'$p_{T}$ [GeV]')
|
||||
save_fig(fig, 'histo_pt', 'xs', size=(4,2))
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
:RESULTS:
|
||||
[[file:./obipy-resources/Ix0X0o.png]]
|
||||
:END:
|
||||
|
|
|
|||
|
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|
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|
|
@ -3811,7 +3811,7 @@
|
|||
\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 \(\displaystyle \frac{d\sigma}{d\omega}\) [pb]}%
|
||||
\pgftext[x=0.466597in,y=2.197917in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle \frac{d\sigma}{d\Omega}\) [pb]}%
|
||||
\end{pgfscope}%
|
||||
\begin{pgfscope}%
|
||||
\pgfpathrectangle{\pgfqpoint{0.866319in}{0.594444in}}{\pgfqpoint{2.935069in}{3.206944in}}%
|
||||
|
|
|
|||
BIN
prog/python/qqgg/figs/xs/histo_pt.pdf
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3955
prog/python/qqgg/figs/xs/histo_pt.pgf
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1
prog/python/qqgg/figs/xs/histo_pt.tex
Normal file
|
|
@ -0,0 +1 @@
|
|||
\input{figs/xs/histo_pt.pgf}
|
||||
|
|
@ -42,6 +42,7 @@ def sample(f, interval, point_density=10000,
|
|||
a, b = interval
|
||||
if b < a:
|
||||
a, b = b, a
|
||||
interval = [a, b]
|
||||
|
||||
np.random.seed(seed)
|
||||
|
||||
|
|
@ -59,4 +60,5 @@ def sample(f, interval, point_density=10000,
|
|||
points = np.random.uniform([a, 0], [b, 1], [num_points_x, 2])
|
||||
sample_points = points[:, 0] \
|
||||
[np.where(f(points[:, 0]) > points[:, 1]*upper_bound)]
|
||||
|
||||
return sample_points
|
||||
|
|
|
|||
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