/* Copyright 2008 Mike Williams (mwill@jlab.org)
*
* This file is part of qft++.
*
* qft++ is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* qft++ is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with qft++. If not, see <http://www.gnu.org/licenses/>.
*/
//_____________________________________________________________________________
/** @file compton_scattering.cxx
* @author Mike Williams
*
* @brief Calculates compton scattering \f$\frac{d\sigma}{dcos(\theta)}\f$.
*
* This tutorial shows how to use qft++ classes to calculate the compton
* scattering differential cross section. Both the s- and u-channel
* diagrams are calculated using the qft++ classes with no algebra needed.
*
* The amplitudes calculated are:
*
* \f[A_{s-channel} = \bar{u}(p',m_p')(\gamma^{\mu}
* \epsilon^*_{\mu}(k',m_{\gamma}'))
* \frac{(p+k)^{\beta}\gamma_{\beta} + m}{(p+k)^2 -m^2}
* (\gamma^{\nu}\epsilon_{\nu}(k,m_{\gamma})) u(p,m_p)\f]
*
* \f[A_{u-channel} = \bar{u}(p',m_p')(\gamma^{\nu}
* \epsilon_{\nu}(k,m_{\gamma}))
* \frac{(p-k')^{\beta}\gamma_{\beta} + m}{(p-k')^2 -m^2}
* (\gamma^{\mu}\epsilon^*_{\mu}(k',m_{\gamma}')) u(p,m_p)\f]
*
* where p(p') denotes the initial(final) electron momentum and k(k') denotes
* the initial(final) photon momentum.
*
* The user when running bin/compton_scattering is asked to provide the
* kinematic info for the event, the incident photon energy along with the
* final photon angle. The qft++ answer is produced along with the famous
* Klein-Nishina equation result (which should give the same answer...if you
* find some set of kinematics where they don't email me).
*
* The user can then look at this source code to see how it calculates the
* amplitudes...once the 4-momenta are all set, it's 2 lines of code!
*/
//_____________________________________________________________________________
#include <getopt.h>
#include "qft++/topincludes/relativistic-quantum-mechanics.hh"
using namespace std;
//_____________________________________________________________________________
/// Prints program usage to the screen
void PrintUsage();
/// Prints a line accross the screen
void PrintLine(char __c){
for(int i = 0; i < 80; i++) cout << __c;
cout << endl;
}
//_____________________________________________________________________________
int main(int __argc,char *__argv[]){
/*__________________________Parse the Command Line_________________________*/
int c;
while((c = getopt(__argc,__argv,"h")) != -1){
switch(c){
case 'h': // help option
PrintUsage();
return EXIT_SUCCESS;
break;
default:
break;
}
}
double m_e = 0.511; // election mass
double theta_f; // polar angle of final photon
double e_gi,e_gf; // inital, final photon energy
double PI = 4*atan(1.);
double dsigma_dcos;
/*__________________________________Set Up_________________________________*/
Vector4<double> pi,pf,ki,kf; // inital,final e-,inital, final gamma
DiracGamma gamma; // gamma^{mu}
DiracSpinor ui,uf; // initial, final e- spinors
DiracSpinor us,uu; // s-,u-channel exchange spinors
PolVector epsi,epsf; // inital, final gamma polarization 4-vectors
Spin mz_ei,mz_ef,mz_gi,mz_gf; // z spin projections
Matrix<complex<double> > amp_s(1,1); // s-channel amplitude
Matrix<complex<double> > amp_u(1,1); // u-channel amplitude
complex<double> amp; // total amplitude
/*______________________________Set 4-Momenta______________________________*/
PrintLine(':');
cout << "Enter incident photon energy (MeV): ";
cin >> e_gi;
cout << "Enter final photon angle (degrees): ";
cin >> theta_f;
theta_f *= PI/180.;
e_gf = e_gi/(1 + e_gi*(1 - cos(theta_f))/m_e);
ki.SetP4(e_gi,0,0,e_gi);
pi.SetP4(m_e,0,0,0);
kf.SetP4(e_gf,e_gf*sin(theta_f),0,e_gf*cos(theta_f));
pf = pi + ki - kf;
PrintLine(':');
cout << "dsigma_dcos(theta) calculated by:" << endl;
cout << "->Klein-Nishina equation: ";
dsigma_dcos = (1/(16*m_e*m_e*PI))*pow(e_gf/e_gi,2)
*(e_gf/e_gi + e_gi/e_gf - pow(sin(theta_f),2));
cout << dsigma_dcos << endl;
/*_________________Initialize Spinors/Polarization Vectors_________________*/
ui.SetP4(pi,m_e);
uf.SetP4(pf,m_e);
epsi.SetP4(ki,0.);
epsf.SetP4(kf,0.);
us.SetP4(pi+ki,m_e);
uu.SetP4(pi-kf,m_e);
/*___________________________Calculate Intensity___________________________*/
double intensity = 0.;
for(mz_ei = -1/2.; mz_ei <= 1/2.; mz_ei++){ // sum over initial e- spins
for(mz_gi = -1; mz_gi <= 1; mz_gi+=2){ // sum over initial gamma spins
for(mz_ef = -1/2.; mz_ef <= 1/2.; mz_ef++){ // sum over final e- spins
for(mz_gf = -1; mz_gf <= 1; mz_gf+=2){ // sum over final gamma spins
// s-channel piece
amp_s = Bar(uf(mz_ef))*(gamma*(epsf(mz_gf).Conjugate()))
*us.Propagator()*(gamma*epsi(mz_gi))*ui(mz_ei);
// u-channel piece
amp_u = Bar(uf(mz_ef))*(gamma*epsi(mz_gi))*uu.Propagator()
*(gamma*(epsf(mz_gf).Conjugate()))*ui(mz_ei);
amp = (amp_s + amp_u)(0,0); // total amp
intensity += norm(amp);
}
}
}
}
// average over initial spins
intensity /= 4.;
double dsigma_dcos_factor = (1./(2*m_e*2*e_gi))*(e_gf*e_gf/(8*PI*e_gi*m_e));
dsigma_dcos = dsigma_dcos_factor*intensity;
cout << "->qft++: " << dsigma_dcos << endl;
PrintLine(':');
return EXIT_SUCCESS;
}
//_____________________________________________________________________________
void PrintUsage(){
cout << "Usage: compton_scattering " << endl;
cout << "This executable calculates the compton scattering differential "
<< "cross section\nfor a given kinematics." << endl;
}
//_____________________________________________________________________________