final commit

literature.bib

@@ -20,4 +20,19 @@
   publisher = {Zenodo},
   year = {2020},
   copyright = {Open Access}
+}
+
+
+@article{durham,
+  doi = {10.1016/0370-2693(92)91467-n},
+  url = {https://doi.org/10.1016/0370-2693(92)91467-n},
+  year = {1992},
+  month = jul,
+  publisher = {Elsevier {BV}},
+  volume = {285},
+  number = {3},
+  pages = {291--299},
+  author = {S. Catani and Yu.L. Dokshitzer and B.R. Webber},
+  title = {The k$\bot$-clustering algorithm for jets in deep inelastic scattering and hadron collisions},
+  journal = {Physics Letters B}
 }
\ No newline at end of file

utils/crosssection.py

@@ -4,17 +4,25 @@ import numpy as np
 
 
 def chi1(s):
+    "The chi1 function from the sheet"
     out = const.kappa * s * (s - const.M_Z**2)
     out /= (s - const.M_Z**2)**2 + const.Gamma_Z**2 * const.M_Z**2
     return out
 
 
 def chi2(s):
+    "The chi2 function from the sheet"
     out = const.kappa**2 * s**2
     out /= (s - const.M_Z**2)**2 + const.Gamma_Z**2 * const.M_Z**2
     return out
 
 
+def breit_wigner_pdf(s):
+    p = const.Gamma_Z * const.M_Z
+    p /= np.pi * ((s - const.M_Z**2)**2 + const.M_Z**2 * const.Gamma_Z**2)
+    return p
+
+
 def eeqq_matrix_element(s : float, costheta : float, phi : float, q):
     """The squared ee -> qq matrix elemenent.
     
@@ -24,7 +32,8 @@ def eeqq_matrix_element(s : float, costheta : float, phi : float, q):
     costheta : The angle between the incoming e+ and the outgoing q
     phi :      The azimuthal angle of outgoing quark in the plane perpendicular
                to the beam
-    q : one of ["u", "c", "d", "s", "b"], the flavour of the outgoing quarks
+    q :        one of ["u", "c", "d", "s", "b"], the flavour of the outgoing
+               quarks
     """
     prefactor = (4 * np.pi * const.alpha)**2 * const.N_C
     term1 = 1 + costheta**2

utils/exercise1.py

+#!/usr/bin/env python
+# coding: utf-8
+
+# # Exercise 1
+# 
+# **a)** (Code: 7 points) Write a program to integrate the cross section numerically using
+#   Monte-Carlo event generation, for the case that the beam energies are fixed such that
+#   $s = M_Z^2$ , i.e. $f(s) = \delta(s − M_Z^2)$. Note that you can apply the averaging over
+#   outgoing quark flavours q by just picking a random flavour for each Monte-Carlo point.
+
+# The leading-order calculation for the e + e − → qq̄ matrix element yields the following
+# result:
+# 
+# $$
+# \begin{align*}
+# |M_{q\overline{q}} (s, \cos \theta, \phi)|^2
+# = 
+# (4\pi \alpha)^2 N_C  \left[
+#     \left(1 + \cos^2 \theta\right)
+#         \left\{
+#             Q^2_e Q^2_q + 2 Q_e Q_q V_e V_q \chi_1 (s) \\
+#             + \left(A^2_e + V_e^2\right) \left(A_q^2 + V_q^2\right) \chi_2 (s)
+#         \right\}\\
+#     + \cos \theta
+#         \left\{
+#             4 Q_e Q_q A_e A_q \chi_1 (s) + 8A_e V_e A_q V_q \chi_2 (s)
+#         \right\}
+#     \right]
+# \end{align*}
+# $$
+
+# where
+#  - $s = E_{CMS}^2$ is the centre-of-mass energy
+#  - $\theta$ the angle between the incoming positron and outgoing quark
+#  - $\alpha$ the QED coupling strength (fine structure constant)
+#  - $N_C$ the number of colour charges the quarks can carry
+#  - $Q_f$ the (electric?) charge of the particle $f$
+#  - $V_f = T_f^3 - 2 Q_f \sin^2(\theta_W)$ with $\theta_W$ the Weinberg angle is the vector coupling with 
+#  - $A_f = T_f^3$ is the axial coupling
+#  - $\chi_1(s) = \kappa \frac{s(s-M_Z^2)}{(s-M_Z^2)^2 + \Gamma_Z^2 M_Z^2}$
+#      - with $\kappa = \frac{1}{4 \sin^2(\theta_W) (1-\sin^2(\theta_W))}$
+#  - $\chi_2(s) = \kappa^2 \frac{s^2}{(s-M_Z^2)^2 + \Gamma_Z^2 M_Z^2}$
+#      - with $\kappa = \frac{1}{4 \sin^2(\theta_W) (1-\sin^2(\theta_W))}$
+
+# In[1]:
+
+
+from integrator import *
+from crosssection import *
+from plotsettings import *
+
+import numpy as np 
+import matplotlib.pyplot as plt
+import uncertainties.unumpy as unp
+import vegas
+
+
+# In[2]:
+
+
+N_try = 100000
+
+
+# ## Visualizing the integrand
+
+# In[3]:
+
+
+from mpl_toolkits.mplot3d import Axes3D  # noqa: F401 unused import
+
+
+# In[4]:
+
+
+def func(x, q):
+    "throwaway function to plot the integrand"
+    mel = eeqq_matrix_element(x[0], x[1], 0., q)
+    return 0.5 * mel / x[0]
+
+
+# In[6]:
+
+
+# generating the data for the plot
+s = np.linspace((const.M_Z - 5*const.Gamma_Z)**2, (const.M_Z + 5*const.Gamma_Z)**2, 50)
+costheta = np.linspace(-1, 1, 50)
+x, y = np.meshgrid(s, costheta)
+fu = func([x, y], "u") 
+fc = func([x, y], "c") 
+fd = func([x, y], "d") 
+fs = func([x, y], "s") 
+fb = func([x, y], "b") 
+
+
+# In[10]:
+
+
+# and generating the plot
+fig = plt.figure(figsize=figsize(0.9))
+ax = fig.add_subplot(111, projection='3d')
+
+for (f, q) in zip((fu, fc, fd, fs, fb), ("u", "c", "d", "s", "b")):
+    surf = ax.plot_surface(x, y, f, alpha=0.4,
+                       label=r"$q=" + q + "$")
+    surf._facecolors2d=surf._facecolors3d
+    surf._edgecolors2d=surf._edgecolors3d
+    
+ax.set_xlabel(r"$s~[\mathrm{GeV}]$")
+ax.set_ylabel(r"$\cos \theta$")
+ax.set_zlabel(r"$f_{q\overline{q}}(s, \cos \theta, 0)$")
+ax.set_xlim((const.M_Z-5*const.Gamma_Z)**2, (const.M_Z+5*const.Gamma_Z)**2)
+ax.plot((const.M_Z**2, const.M_Z**2), (-1.1, 1.1), (0, 0), color="k", label=r"$s=M_Z^2$")
+ax.set_ylim(-1.1, 1.1)
+ax.legend(loc=2)
+
+fig.tight_layout()
+fig.savefig("integrand.pgf")
+
+
+# # Exercise 1.a
+
+# In[27]:
+
+
+domain = [(-1,1)]  # only costheta integration
+codomain = (0, 30) # using integrator.samples this seemed to be a good
+                   # choice
+integrator = MCIntegrator.uniform(
+    lambda x: eeqq_mel_random_q(const.M_Z**2, x, 0.), domain, codomain)
+
+
+# In[28]:
+
+
+I = integrator(N_try)
+
+
+# In[29]:
+
+
+mel = (I * const.N_q * 2 * np.pi * const.f_conv
+        / (64 * np.pi**2 * const.M_Z**2))
+print(f"σ(s=M_z²) = {mel} pb")
+
+
+# In[30]:
+
+
+print(f"sampling efficiency: {integrator.N_acc / integrator.N_try}")
+
+
+# In[12]:
+
+
+# generating data for the log-log plot of the monte carlo error estimates
+N_trys = np.logspace(1, 4, 100)
+N_trys = np.floor(N_trys)
+N_trys = N_trys.astype(int)
+N_trys
+variances_s_const = np.empty(100)
+integrator.reset()
+for (i, N_try) in enumerate(N_trys):
+    integrator(N_try)
+    s = (integrator.I * const.N_q * 2 * np.pi * const.f_conv
+         / (64 * np.pi**2 * const.M_Z**2))
+    variances_s_const[i] = s.std_dev
+
+
+# ## Sampling uniformly in $s$
+
+# In[59]:
+
+
+domain = [((const.M_Z-3*const.Gamma_Z)**2, (const.M_Z+3*const.Gamma_Z)**2),
+          (-1,1)]
+codomain = (0, 5e-7)
+def func(x):
+    "throwaway function that we will integrate"
+    mel = eeqq_mel_random_q(x[0], x[1], 0.)
+    mel /= domain[0][1] - domain[0][0]
+    return 0.5 * mel / x[0]
+
+integrator = MCIntegrator.uniform(func, domain, codomain, histograms=True, histbins=50)
+
+
+# In[60]:
+
+
+I = integrator(10 * N_try)
+
+
+# In[61]:
+
+
+mel = I * const.N_q * 2 * np.pi * const.f_conv / (32 * np.pi**2)
+print(f"σ(s∈[(M_Z-3Γ_z)^2, (M_Z+3Γ_z)^2]) = {mel} pb")
+
+
+# In[62]:
+
+
+print(f"sampling efficiency: {integrator.N_acc / integrator.N_try}")
+
+
+# In[63]:
+
+
+# generating data for the log-log plot of the monte carlo error estimates
+N_trys = np.logspace(1, 4, 100)
+N_trys = np.floor(N_trys)
+N_trys = N_trys.astype(int)
+N_trys
+variances_s_uniform = np.empty(100)
+integrator.reset()
+for (i, N_try) in enumerate(N_trys):
+    integrator(N_try)
+    s = integrator.I * const.N_q * 2 * np.pi * const.f_conv / (32 * np.pi**2)
+    variances_s_uniform[i] = s.std_dev
+
+
+# ## $s$-Scanning
+
+# In[17]:
+
+
+n_slices = 50
+s_domain = np.linspace((const.M_Z-3*const.Gamma_Z)**2, (const.M_Z+3*const.Gamma_Z)**2,
+                       n_slices)
+slices = unp.uarray([0.] * n_slices, [0.] * n_slices)
+
+
+# In[18]:
+
+
+domain = [(-1,1)]
+codomain = (0, 30)
+for (i, s) in enumerate(s_domain):
+    integ = MCIntegrator.uniform(
+                    lambda x : eeqq_mel_random_q(s, x, 0.),
+                    domain, codomain)
+    I = integ(10000)
+    slices[i] = (I * const.N_q * 2 * np.pi * const.f_conv
+                 / (64 * np.pi**2 * s))
+
+
+# In[19]:
+
+
+slices /= np.sum(unp.nominal_values(slices)) * (s_domain[-1] - s_domain[1]) / (n_slices)
+
+
+# In[20]:
+
+
+# and plotting acceptance probabilities together with the integral slices
+x = np.linspace(7000, 9700, 100)
+y = breit_wigner_pdf(x)
+y /= np.sum(y) * (s_domain[-1] - s_domain[1]) / 99
+
+fig, ax = plt.subplots(figsize=figsize(0.8))
+integrator.histograms[0].Plot(ax=ax, c="C0", label=r"$p_{acc}(s)$")
+ax.step(s_domain, unp.nominal_values(slices), c="C1",
+        label=r"$\propto \int \, f(s, \cos \, \theta, 0) \, \mathrm{d}\cos \, \theta$")
+
+
+ax.plot(x, 1.3 * y, label=r"$\propto \left[(s-M_Z^2)^2 + M_Z^2 \Gamma_Z^2\right]^{-1}$",
+        color="gray")
+ax.legend(loc=1)
+ax.set_xlabel(r"$s ~[\mathrm{GeV}]$")
+fig.tight_layout()
+fig.savefig("histograms.pgf")
+
+
+# ## Monte Carlo Error plots
+
+# In[21]:
+
+
+fig, ax = plt.subplots(1, 2, figsize=figsize(1., 0.4))
+ax[0].plot(np.cumsum(N_trys), 6e4 / np.sqrt(np.cumsum(N_trys)),
+        c="C1", label=r"$\propto N_{try}^{- \frac{1}{2}}$")
+ax[0].scatter(np.cumsum(N_trys), variances_s_const,
+              label=r"$\sigma_I$", marker="+")
+ax[0].legend(loc=3)
+ax[0].set_xlabel(r"Number of samples $N_{try}$")
+ax[0].set_xscale("log")
+ax[0].set_yscale("log")
+ax[0].annotate(r"$\mathbf{a)}$", (50000, 10000))
+ax[0].set_title(r"$s = M_Z^2$")
+
+ax[1].plot(np.cumsum(N_trys), 3e4/np.sqrt(np.cumsum(N_trys)),
+        c="C1", label=r"$\propto N_{try}^{- \frac{1}{2}}$")
+ax[1].scatter(np.cumsum(N_trys), variances_s_uniform,
+              label=r"$\sigma_I$", marker="+")
+ax[1].legend(loc=3)
+ax[1].set_xlabel(r"Number of samples $N_{try}$")
+ax[1].set_xscale("log")
+ax[1].set_yscale("log")
+ax[1].annotate(r"$\mathbf{b)}$", (50000, 5000))
+ax[1].set_title(r"$s \in \left[(M_Z - 3 \Gamma_Z)^2, (M_Z + 3 \Gamma_Z)^2\right]$")
+fig.tight_layout()
+fig.savefig("monte_carlo_errors.pgf")
+
+
+# ## What happens in Vegas, stays in Vegas
+
+# In[64]:
+
+
+domain = [((const.M_Z-3*const.Gamma_Z)**2, (const.M_Z+3*const.Gamma_Z)**2), (-1,1)]
+codomain = (0, 5e-7)
+
+def func(x):
+    "yet another throwaway function to integrate over"
+    mel = eeqq_mel_random_q(x[0], x[1], 0.)
+    mel /= domain[0][1] - domain[0][0]
+    return 0.5 * mel / x[0]
+
+
+# In[65]:
+
+
+integ = vegas.Integrator(domain)
+I = integ(func, nitn=10, neval=1000)
+
+
+# In[66]:
+
+
+mel = I * const.N_q * 2 * np.pi * const.f_conv / (32 * np.pi**2)
+print(f"with vegas: σ(s∈[(M_Z-3Γ_z)^2, (M_Z+3Γ_z)^2]) = {mel} pb")
+
+
+# In[67]:
+
+
+print(I.summary())
+
+
+# In[58]:
+
+
+# Plotting the vegas grid
+vlines, hlines = integ.map.grid.base[0], integ.map.grid.base[1]
+fig, ax = plt.subplots(figsize=figsize(0.7))
+ax.hlines(hlines[::2], min(vlines), max(vlines))
+ax.vlines(vlines[::3], min(hlines), max(hlines))
+ax.set_xlabel(r"$s$")
+ax.set_ylabel(r"$\cos (\theta)$")
+for spine in ax.spines.values():
+    spine.set_visible(False)
+    
+ax.axvline(const.M_Z**2, -1, 1, color="C1", lw=2)    
+
+ax.set_ylim(min(hlines), max(hlines))
+ax.set_xlim(min(vlines), max(vlines))
+ax.tick_params(axis=u"both", which=u"both", length=0)
+fig.tight_layout()
+fig.savefig("vegas_grid.pgf")
+
+
+# ## Importance sampling
+
+# In[72]:
+
+
+def func(x):
+    "you know what this is..."
+    mel = eeqq_mel_random_q(x[0], x[1], 0.)
+    mel /= domain[0][1] - domain[0][0]
+    return 0.5 * mel / x[0]
+
+
+# In[73]:
+
+
+integrator = MCIntegrator(func, sampler=BreitWignerSampler(prefactor=0.0004))
+
+
+# In[74]:
+
+
+I = integrator(10 * N_try)
+
+
+# In[75]:
+
+
+mel = I * const.N_q * 2 * np.pi * const.f_conv / (32 * np.pi**2)
+print(f"with importance sampling: σ(s∈[-∞, ∞]) = {mel} pb")
+
+
+# In[76]:
+
+
+print(f"sampling efficiency: {integrator.N_acc / integrator.N_try}")
+
+
+# ## Checking importance sampling
+
+# In[31]:
+
+
+sampler = BreitWignerSampler(0.0004)
+
+X = np.empty(shape=(1000, 2))
+Y = np.empty(1000)
+mask = np.empty(1000, dtype=bool)
+for i in range(1000):
+    j, k = sampler()
+    X[i,:] = j
+    Y[i] = k
+    mask[i] = func(j) > k
+
+
+# In[32]:
+
+
+from mpl_toolkits.mplot3d import Axes3D  # noqa: F401 unused import
+
+
+# In[33]:
+
+
+s = np.linspace((const.M_Z-5*const.Gamma_Z)**2, (const.M_Z+5*const.Gamma_Z)**2, 50)
+costheta = np.linspace(-1, 1, 50)
+x, y = np.meshgrid(s, costheta)
+f = func([x, y]) 
+
+
+# In[34]:
+
+
+fig = plt.figure(figsize=figsize(1))
+ax = fig.add_subplot(111, projection='3d')
+ax.plot(X[mask,0], X[mask,1], Y[mask], label="accepted samples",
+        marker=".", c="C0", ls="", zorder=1, markersize=3)
+surf = ax.plot_surface(x, y, f, color="C1", alpha=0.8, zorder=2,
+                       label=r"$\mathrm{d}\sigma_{q\overline{q}}(s, \cos \theta, 0)$")
+surf._facecolors2d=surf._facecolors3d
+surf._edgecolors2d=surf._edgecolors3d
+ax.plot(X[np.invert(mask),0], X[np.invert(mask),1], Y[np.invert(mask)],
+           marker=".", c="C3", zorder=3, ls="", markersize=3, label="rejected samples")
+
+ax.set_xlabel(r"$s~[\mathrm{GeV}]$")
+ax.set_ylabel(r"$\cos \theta$")
+#ax.set_zlabel()
+ax.set_xlim((const.M_Z-5*const.Gamma_Z)**2, (const.M_Z+5*const.Gamma_Z)**2)
+ax.legend(loc=2)
+
+fig.tight_layout()
+fig.savefig("3dplot.pgf")
+
+
+# In[11]:
+
+
+# just to show, that our importance sampling works for other distributions and integrands
+def func(x):
+    return np.exp(-0.5*x**2) / np.sqrt(2 * np.pi)
+
+sampler = CauchySampler(2.0)
+integrator = MCIntegrator(func, sampler)
+I = integrator(10000)
+print(f"Integrating a normal distribution with the Cauchy Sampler works: {I}")
\ No newline at end of file

utils/exercise2.py

+#!/usr/bin/env python
+
+# In[1]:
+
+from integrator import *
+from crosssection import *
+from plotsettings import *
+
+import numpy as np 
+import matplotlib.pyplot as plt
+import uncertainties.unumpy as unp
+import vegas
+
+
+# In[2]:
+
+
+from alphas import *
+from particle import *
+from shower import *
+from vector import *
+from analysis import *
+from yoda import *
+
+
+# In[3]:
+
+
+alpha_s = AlphaS(const.M_Z, const.alpha_M_Z)
+
+
+# In[4]:
+
+
+ts = np.logspace(0, 4)
+couplings = np.empty_like(ts)
+for (i, t) in enumerate(ts):
+    couplings[i] = alpha_s(t)
+
+
+# In[5]:
+
+
+fig, ax = plt.subplots(figsize=figsize(0.7))
+ax.plot(ts, couplings)
+ax.set_xscale("log")
+ax.set_xlabel(r"$t~[\mathrm{GeV}^2]$")
+ax.set_ylabel(r"$\alpha_s(t)$")
+
+fig.tight_layout()
+fig.savefig("running_coupling.pgf")
+
+
+# In[6]:
+
+
+def f(x, q):
+    "The weight function"
+    mel = eeqq_matrix_element(x[0], x[1], x[2], q)
+    return mel / x[0]
+
+
+# In[7]:
+
+
+def sampler():
+    "Returns some uniformly distributed ee->qq events together with their weight"
+    s = np.random.uniform((const.M_Z - 3*const.Gamma_Z)**2,
+                          (const.M_Z + 3*const.Gamma_Z)**2)
+    costheta = np.random.uniform(-1, 1)
+    phi = np.random.uniform(0, 2 * np.pi)
+    pid = np.random.randint(1, 5)
+    color = np.random.randint(1, 3)
+    
+    x = np.array([s, costheta, phi]) 
+    weight = f(x, const.out_flavours[pid])
+    return np.array([s, costheta, phi]), pid, color, weight
+
+
+# In[8]:
+
+
+# with uniformly sampled events and weights
+def runshower(t0=1.0):
+    """"Create a ee -> qq event distributed according to the results of 1
+    and run the shower algorithm on it.
+    
+    Returns
+    -------
+