bachelor_thesis/prog/python/qqgg/analytical_xs.org

• Init
  • Required Modules
  • Utilities
• Implementation
• Calculations
  • XS qq -> gamma gamma

Init

Required Modules

import numpy as np
import matplotlib.pyplot as plt

Utilities

%run ../utility.py

Implementation

  """
  Implementation of the analytical cross section for q q_bar ->
  gamma gamma

  Author: Valentin Boettcher <hiro@protagon.space>
  """

  import numpy as np
  from scipy.constants import alpha

  # NOTE: a more elegant solution would be a decorator
  def energy_factor(charge, esp):
      """
      Calculates the factor common to all other values in this module

      Arguments:
      esp -- center of momentum energy in GeV
      charge -- charge of the particle in units of the elementary charge
      """

      return charge**4*(alpha/esp)**2/6


  def diff_xs(theta, charge, esp):
      """
      Calculates the differential cross section as a function of the
      azimuth angle theta in units of 1/GeV^2.

      Arguments:
      theta -- azimuth angle
      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*((np.cos(theta)**+1)/np.sin(theta)**2)

  def diff_xs_eta(eta, charge, esp):
      """
      Calculates the differential cross section as a function of the
      pseudo rapidity of the photons in units of 1/GeV^2.

      Arguments:
      eta -- pseudo rapidity
      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*(2*np.cosh(eta)**2 - 1)

  def total_xs_eta(eta, charge, esp):
      """
      Calculates the total cross section as a function of the pseudo
      rapidity of the photons in units of 1/GeV^2.  If the rapditiy is
      specified as a tuple, it is interpreted as an interval.  Otherwise
      the interval [-eta, eta] will be used.

      Arguments:
      eta -- pseudo rapidity (tuple or number)
      esp -- center of momentum energy in GeV
      charge -- charge of the particle in units of the elementar charge
      """

      f = energy_factor(charge, esp)
      if not isinstance(eta, tuple):
          eta = (-eta, eta)

      if len(eta) != 2:
          raise ValueError('Invalid eta cut.')

      def F(x):
          return np.tanh(x) - 2*x

      return 2*np.pi*f*(F(eta[0]) - F(eta[1]))

Calculations

XS qq -> gamma gamma

First, set up the input parameters.

eta = 2.5
charge = 1/3
esp = 200  # GeV

And now calculate the cross section in picobarn.

xs_gev = total_xs_eta(eta, charge, esp)
xs_pb = gev_to_pb(xs_gev)
xs_pb

0.053793289459925515

Compared to sherpa, it's pretty close.

  sherpa = 0.0538009
  xs_pb/sherpa

0.9998585425137037

I had to set the runcard option EW_SCHEME: alpha0 to use the pure QED coupling constant.