Download High Energy Cross Sections by Monte Carlo

Topic 6
Quantum Field Theory
Lecture 2
High Energy Cross Sections by Monte Carlo Quadrature
Thomson Scattering in Electrodynamics
Jackson, Classical Electrodynamics, Section 14.8 Thomson Scattering of Radiation
Jackson Figures 14.17 and 14.18: Differential scattering cross section of unpolarized radiation by a point
charged particle initially at rest in the laboratory. The solid curve is the classical Thomson result. The
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dashed curves are the quantum-mechanical results for a spinless particle, with the numbers giving the values
of ~ω/mc2. For ~ω/mc2 = 0.25, 1.0 the dotted curves show the results for spin 12 point particles (electrons).
The differential scattering cross section is defined by
dσ Energy radiated/unit time/unit solid angle
=
dΩ
Incident energy flux/unit area/second
Number of photons detected/unit time/unit solid angle
=
Number of photons incident/unit area/unit time
The differential cross section for a monochromatic plane wave with wavevector k0 = kez and polarization
0 incident on a free particle of charge e and mass m is
2 2
dσ
e
2
∗
|
·
|
=
0
dΩ
mc2
where k = kn, are the wavevector and polarization of the outgoing wave. The polarization vectors
1 = cos θ(ex cos φ + ey sin φ) − ez sin θ ,
2 = −ex sin φ + ey cos φ
represent polarization in and perpendicular to the scattering plane defined by the initial and final wavevectors
k0, n. For plane polarized incident radiation and summing over final polarizations
(
X
cos2 θ cos2 φ + sin2 φ for 0 = ex
2
∗
| · 0| =
cos2 θ sin2 φ + cos2 φ for 0 = ey
1,2
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Lecture 2
which give the unpolarized differential and total cross sections
2
2 2
8π
dσ 1 e2
e
2
(1
+
cos
θ)
,
σ
=
.
=
T
dΩ 2 mc2
3 mc2
The Compton Scattering Cross Sections
The Tompson cross sections are valid only at low frequencies ~ω/c when the momentum of the incident
photon is much smaller than mc. In this limit the charged particle remains at rest and the energy of the
photon does not change.
When ~k ≥ mc, the charged particle will recoil and the photon will change in magnitude and direction,
giving rise to Compton scattering. The Compton formula for the change in frequency of the photon is
ω
~ω
=
1
+
(1 − cos θ)
ω0
mc2
where θ is scattering angle in the rest frame of the particle before the collision.
To derive this result, let kµ, kµ0 be the 4-momenta of the photon before and after the scattering, and pµ, p0mu
the initial and final 4-momenta of the target particle. In the rest frame of the target particle (laboratory
frame)
µ
p = (mc, 0, 0, 0) ,
PHY 411-506 Computational Physics 2
~ω 0
k = (1, sin θ cos φ, sin θ sin φ, cos θ)
c
~ω
k = (1, 0, 0, 1)
,
c
0µ
µ
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Lecture 2
Using conservation of 4-momentum
p0 · p0 = m2c2 = (p + k − k 0) · (p0 + k 0 − k) = p2 + 2p · (k − k 0) + (k − k 0)2
= m2c2 + 2m~(ω − ω 0) − 2~2ωω 0(1 − cos θ)
The cross section is
2 2 0 2
e
ω
dσ
2
∗
=
|
·
|
.
0
dΩ
mc2
ω
Compton derived these formulas simply by using relativistic kinematics and conservation of energy and
momentum. They are valid for scattering of a photon from a spinless charged particle.
The Klein-Nishina Formula in Quantum Electrodynamics
The quantum field theory which describes the interaction of charged electrons and positrons with photons
is Quantum Electrodynamics or QED.
Relativistic quantum field theory predicts that virtual electron-positron pairs will modify scattering cross
section. In QED, these corrections are computed as a perturbation series in the the Fine-structure constant
α = e2/~c ' 1/137.036.
The cross section for Compton scattering in to lowest order in α is determined by the two Feynman diagrams
from Section 5.5 of Peskin and Schroeder
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Lecture 2
The probability amplitude for the process is given in Eq. (5.74)
µ
ν
µ ν
ν
µ
ν µ
γ
6
kγ
+
2γ
p
−γ
6
kγ
+
2γ
p
+
u(p) ,
iM = −ie2∗µ(k 0)ν (k)ū(p0)
2p · k
−2p · k 0
where u(p) is a 4-component positive energy spinor wavefunction solution of the Dirac equation and γ µ are
Dirac gamma matrices. This amplitude defines the scattering of a photon from an electron for arbitrary
photon polarizations and electron spin. To compare with the Compton cross section for spinless particles,
the amplitude must be squared to obtain the scattering probability, which is then averaged over the two
initial spin states of the electron and summed over the two final spin states. This is a straightforward
calculation done in detail in Peskin and Schroeder. The final result in the rest frame of the initial electron
is (Jackson footnote on page 697)
2 2 0 2 0 2
dσ
e
ω
(ω
−
ω
)
2
∗
=
|
·
|
+
{1 + (∗×) · (0×∗0 )} .
0
2
0
dΩ
mc
ω
4ωω
The second dot product can be simplified
(∗×) · (0×∗0 ) = (∗·0)(·∗0 ) − (∗·∗0 )(·0) = |∗·0|2 − |·0|2
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Lecture 2
It vanishes for pure linear or circular polarization but can contribute for elliptical polarization.
This is the Klein-Nishina formula for Compton scattering.
Numerical Evaluation of the Klein-Nishina Formula
To compare this theoretical prediction with experimental measurements in the laboratory frame we need to
express the result in Barns. The Classical electron radius is defined to be
e2
−13
2
=
2.81794033×10
cm
,
r
= 0.07940787703 b ' 79.4 mb .
re =
e
mc2
Using dΩ = sin θ dθ dφ the following dimensionless cross sections can be defined:
Z +1
Z 2π
1
1 dσ
σ
=
d(cos
θ)
dφ
T
re2
re2 dΩ
−1
0
Z 2π
1 dσ
1 dσ
=
sin
θ
dφ
re2 dθ
re2 dΩ
0
0 2 0 2
1 dσ
ω
(ω − ω )
2
∗
1 + |∗ · 0|2 − | · 0|2
=
|
·
|
+
0
2
0
re dΩ
ω
4ωω
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Lecture 2
Polarization Vectors
Polarization vectors are defined in Jackson: Figure 7.2 on the left shows the complex electric field vector
for linear and Figure 7.3 on the right shows the field for circular polarization. The most general polarization
is a linear superposition of two independent linear or helicity eigenstates
E(x, t) = (1E1 + 2E2)eik·x−iωt ,
E(x, t) = (+E+ + −E−)eik·x−iωt ,
1
± = √ (1 ± i2) ,
2
where + represents photons with positive helicity (left-circular polarization in optics) and − represents
negative helicity (right-circular polarization).
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Lecture 2
Specifying Initial and Final State Polarization
The Klein-Nishina formulas given above have been averaged over initial electron spin states and summed
over final electron spin states. The photon polarizations 0, can be specified arbitrarily.
The incident photon has wavevector k̂0 = ez in the positive direction and is scattered in the direction
k̂ = n = (sin θ cos φ, sin θ sin φ, cos θ).
The simplest and physically most meaningful way to specify high energy photon polarizations is to use the
helicity basis. The incident polarization is specified by two complex numbers (E0+, E0−) and the scattered
state polarization by (E+, E−). Linear polarizations are more natural at lower frequencies.
To evaluate the scalar products of the polarization vectors note that the unit vectors perpendicular and
parallel to the scattering plane are
2 =
n̂×k0
= (− sin φ, cos φ, 0) ,
|n̂×k0|
1 =
2×n
= (cos θ cos φ, cos θ sin φ, − sin θ) .
|2×n|
The polarization vectors are 0 = (E10, E20, 0) and
= E11 + E22 = (E1 cos θ cos φ − E2 sin φ, E1 cos θ sin φ + E2 cos φ, −E1 sin θ) ,
which can be expressed in terms photon helicity components
1
E1 = √ (E+ + E−) ,
2
PHY 411-506 Computational Physics 2
i
E2 = √ (E+ − E−) .
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Quantum Field Theory
Lecture 2
C++ Code to Evaluate the Cross Sections
compton.cpp
#include <cmath>
#include <complex>
#include <cstdlib>
#include <iostream>
using namespace std;
const double pi = 4 * atan(1.0);
#include "../tools/quadrature.hpp"
double omega;
complex<double>
complex<double>
complex<double>
complex<double>
ep_10;
ep_20;
ep_1;
ep_2;
//
//
//
//
//
incident photon angular frequency
x-component of incident photon polarization
y-component
in-plane scattered photon polarization
perpendicular component
double f(const Vector& x, const double wgt)
{
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Lecture 2
double theta = x[0];
double phi = x[1];
double omega_prime = omega / (1 + omega * (1 - cos(theta)));
// compute polarization dot products
complex<double> ep_star_dot_ep0 =
conj(ep_1 * cos(theta) * cos(phi) - ep_2 * sin(phi)) * ep_10 +
conj(ep_1 * cos(theta) * sin(phi) + ep_2 * cos(phi)) * ep_20;
complex<double> ep_dot_ep0 =
(ep_1 * cos(theta) * cos(phi) - ep_2 * sin(phi)) * ep_10 +
(ep_1 * cos(theta) * sin(phi) + ep_2 * cos(phi)) * ep_20;
return pow(omega_prime / omega, 2.0) *
(norm(ep_star_dot_ep0) + pow(omega - omega_prime, 2.0) /
(4 * omega * omega_prime) * (1 + norm(ep_star_dot_ep0) norm(ep_dot_ep0)));
}
int main()
{
cout << " Relativistic Compton Scattering Cross Sections using VEGAS\n";
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Quantum Field Theory
double tgral, sd, chi2a;
Vector regn(4);
regn[0] = 0.0; regn[2] = pi;
regn[1] = 0.0; regn[3] = 2 * pi;
Lecture 2
// 0 < theta < pi
// 0 < phi < 2 pi
// set global parameters
omega = 0.0025;
// incident angular frequency
cout << " omega = " << omega << endl;
// define and normalize polarization vectors
ep_10 = 1.0;
ep_20 = 0.0;
ep_1 = 1.0;
ep_2 = 0.0;
double n0 = sqrt(norm(ep_10) + norm(ep_20)),
n = sqrt(norm(ep_1) + norm(ep_2));
ep_10 /= n0;
ep_20 /= n0;
ep_1 /= n;
ep_2 /= n;
int init = 0;
PHY 411-506 Computational Physics 2
// initialize grid
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int ncall = 50000;
int itmx = 10;
int nprn = 0;
Quantum Field Theory
Lecture 2
// number of function evaluations
// number of iterations
// print some info after each iteration
vegas(regn, f, init, ncall, itmx, nprn, tgral, sd, chi2a);
cout << " sigma_tot = " << tgral << " +/- " << sd << endl;
}
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