Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Compton-scatteringof the cosmic background radiation off a ultrarelativsitic cosmic proton
andpair production
by a (back-scattered) photon
Manfred Hanke, August 2005:QED-Project (guided by Prof. A. Schäfer)
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
0. Introduction- Cosmic background radiation- Cosmic rays- Compton-scattering
1. Energy-loss of a cosmic proton due to Compton-scattering- Cross-section- Kinematics- Differential probabilities- Mean energy-loss- Result
2. Mean free path of a back-scattered photon- Cross-section- Differential probabilities and mean free path- Result
3. Summary
Contents of this talk:
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Cosmic background radiation
- predicted by G. Gamow and R. Alpher in the 1940s
- discovered by A. Penzias and R. W. Wilson in 1964 (Nobelprize in 1978)
- follows Planck‘s formula for black-body-radiation with T = 2,725 K:
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Cosmic background radiation
- predicted by G. Gamow and R. Alpher in the 1940s
- discovered by A. Penzias and R. W. Wilson in 1964 (Nobelprize in 1978)
- follows Planck‘s formula for black-body-radiation with T = 2,725 K:
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Cosmic rays
- high-energy particles (up to 1020 eV)
- mostly (97%) nucleons, especially protons, -particles
- discovered in 1912 by V. Hess (Nobelprize 1936)
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Fluxes of Cosmic Rays
(1 particle per m²·s)
Knee(1 particle per m²·year)
(1 particle per km²·year)Ankle
Flux
Energy
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Cosmic rays
- high-energy particles (up to 1020 eV)
- mostly (97%) nucleons, especially protons, -particles
- origin: solar eruptions, supernovae, cosmic jets (from black holes / pulsars), ..., ?
- Nucleons with energies higher than 5·1019 eV loose their energy by the GZK-effect: (Greisen-Zatsepin-Kuzmin)
+ p + N + What is the energy-loss through Compton-scattering?
- discovered in 1912 by V. Hess (Nobelprize 1936)
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
QED -Compton-scattering
e + e +
(Klein-Nishina)
Easy calculation of the cross-section in the Dirac-theory:
How to calculate Compton-scattering off a proton?
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
0. Introduction- Cosmic background radiation- Cosmic rays- Compton-scattering
1. Energy-loss of a cosmic proton due to Compton-scattering- Cross-section- Kinematics- Differential probabilities- Mean energy-loss- Result
2. Mean free path of a back-scattered photon- Cross-section- Differential probabilities and mean free path- Result
3. Summary
Contents of this talk:
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
The cross-sectionTo calculate the energy-loss through Compton-scattering,one needs...
for Compton-scattering off a proton
, withone finds:
In Hildebrandt, Griesshammer, Hemmert, Pasquini: “Signatures of Chiral Dynamics in Low Energy Compton Scattering off the Nucleon“ (nucl-th/0307070)
the Ai‘s defined as page-long integrals over two Feynman parameters (!)
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
,forexample,is givenby:
A1
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Where do these expressions come from? EFT (Chiral Effective Field Theory)
„The Heavy Baryon Chiral Perturbation Theory only involves explicit πN degrees of freedom.“
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Where do these expressions come from? EFT (Chiral Effective Field Theory)
„The Heavy Baryon Chiral Perturbation Theory only involves explicit πN degrees of freedom, whereas the Small Scale Expansion formalism includes explicit spin 3/2 nucleon resonance degrees of freedom.“
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Where do these expressions come from? EFT (Chiral Effective Field Theory)
„The Heavy Baryon Chiral Perturbation Theory only involves explicit πN degrees of freedom, whereas the Small Scale Expansion formalism includes explicit spin 3/2 nucleon resonance degrees of freedom (and within that – in my opinion – very exotic couplings, like N or N N, for which the parameters have been fitted from experimental cross section data).“
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Here, the following abbreviations and constants are used:
for
> 0 - m 130
MeV,the values get imaginarydue to the resonance,and zero-values cause numerical divergenciesby the denominators!
Problem:
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
numericalresults
for < 130 MeV
20 nbarn
The cross-section
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
- for the energy-loss of the proton:
, z := cos (proton, scattered photon)cm
To calculate the energy-loss through Compton-scattering,one needs...
In the relativistic limit, one gets
- for the photon-energy in the center-of-mass-frame:
Here isk := energy of the cosmic background photon, := cos (proton, photon)lab
Kinematics
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Now, one can calculate...
k := energy of the cosmic background photon, := cos (proton, photon)lab , z := cos (proton, scattered photon)c
Differential probabilities
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
one can look at the spectrum of interacting photons:
Now, as one has calculated
the differential probability
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Do you see any differenceto the Planck-spectrum?
(Ep = 1019 eV)Spectrum of interacting photons
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Now, as one has calculated
the differential probability,
one can look at the
For the numerical simulation,the -function is realized by a histogram.
spectrum of energy-loss:
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Spectrum of energy-loss(Ep = 1019 eV)
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
One can rewrite the -function and perform the integral over z
to get an analytic expression for
that is only an integral over k and , which can more easily be numerically determined.
Spectrum of energy-loss
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Spectrum of energy-loss(Ep = 1019 eV)
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
The mean energy-loss
5.3 MeV / lyfor proton withEp = 1019 eV
~ Ep2
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Result- The low energy-loss is due to the small cross-section for Compton-scattering.
1. Energy-loss of a cosmic proton
- A mean energy-loss of 5.3 MeV / ly for 1019 eV- protons corresponds to a mean free path of 1.9 · 1012 ly. (The mean distance between galaxies is of order 106 ly.)
- Compton-scattering of the cosmic background radiation off such a ultra-high-energy cosmic proton therefore does not lead to a noticeable decceleration of cosmic rays.
The result is, that there is no result.(what concerns the decceleration of cosmic protons)
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
But:The proton‘s energy-loss (up to 1018 eV for Ep = 1019 eV)is added to the photon‘s energy. (This is known as Compton-back-scattering / inverse Compton-scattering,which is one way to produce ultra-high-energy cosmic -rays.)
What happens with these high-energetic photons?
e+ / e– - pair production from single photons is not allowed,but they can interact with the cosmic background radiation.
2. Mean free path of a back-scattered photon
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
0. Introduction- Cosmic background radiation- Cosmic rays- Compton-scattering
1. Energy-loss of a cosmic proton due to Compton-scattering- Cross-section- Kinematics- Differential probabilities- Mean energy-loss- Result
2. Mean free path of a back-scattered photon- Cross-section- Differential probabilities and mean free path- Result
3. Summary
Contents of this talk:
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
The total cross-sectionfor e+ / e– - pair production from two photons
(Breit-Wheeler)
It is .
As a result from kinematics:
,
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
The total cross-section
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Differential probabilities
k0 = 3.21 · 109 MeV
kmax(CMB) kmax()
maximum
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Differential probabilities
k0 = 5 · 107 MeV
kmax(CMB) < kmax()
suppression by the exp-factor
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Differential probabilities
k0 = 1011 MeV
kmax() < kmax(CMB)
suppression by the k²-factor
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
rapid decrease of probabilityfor k0 < 5 · 108 MeV
Mean free path
dW/dL(k0 = 109 MeV) = 2.52 · 10-5/ly
dW/dL(k0 = 108 MeV) = 2.28 · 10-9/ly
dW/dL(k0 = 107 MeV) = 1.60 · 10-54/ly
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Mean free path
(slow) decrease of probabilityfor k0 > 1011 MeVdW/dL(k0 = 1010 MeV) = 3.0 · 10-5/ly
dW/dL(k0 = 1011 MeV) = 9.4 · 10-6/ly
dW/dL(k0 = 1012 MeV) = 2.0 · 10-6/ly
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Mean free path
minimal probabilityat k0 = 3.21 · 109 MeVdW/dL = 3.8 · 10-5/ly
maximal mean free path<L> = 26 · 103 ly
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Result2. Mean free path of a back-scattered photon
The universe should be almost transparent for very-high-energy -rays with k0 < 1014 eV(at least what concerns e+/e–-pair production)– the mean free paths are billions of lightyears!
Photons with ultra-high energies 2·1014 eV < k0 < 1019 eVshould interact with the cosmic background radiationand create e+/e–-pairs within less than 3 million ly,what is approximately the mean distance of galaxies.There should be no ultra-high-energy extragalactic -rays!(Back-scattered photons with these energies can‘t be observed.)
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
0. Introduction- Cosmic background radiation- Cosmic rays- Compton-scattering
1. Energy-loss of a cosmic proton due to Compton-scattering- Cross-section- Kinematics- Differential probabilities- Mean energy-loss- Result
2. Mean free path of a back-scattered photon- Cross-section- Differential probabilities and mean free path- Result
Contents of this talk:
3. Summary
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
from EFT: 20 nbarn
Contents of this talk:0. Introduction
- Cosmic background radiation- Cosmic rays- Compton-scattering
1. Energy-loss of a cosmic proton due to Compton-scattering- Cross-section- Kinematics- Differential probabilities- Mean energy-loss- Result
2. Mean free path of a back-scattered photon- Cross-section- Differential probabilities and mean free path- Result
3. Summary
: spectrum of energy-loss
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Spectrum of a protons energy-loss due to Compton-scattering
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Contents of this talk:0. Introduction
- Cosmic background radiation- Cosmic rays- Compton-scattering
1. Energy-loss of a cosmic proton due to Compton-scattering- Cross-section from EFT: 20 nbarn - Kinematics- Differential probabilities: spectrum of energy-loss- Mean energy-loss
- Result2. Mean free path of a back-scattered photon
- Cross-section- Differential probabilities and mean free path- Result
3. Summary
: ~ Ep2, but only 5.3 MeV / ly for Ep = 1019 eV
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Contents of this talk:0. Introduction
- Cosmic background radiation- Cosmic rays- Compton-scattering
1. Energy-loss of a cosmic proton due to Compton-scattering- Cross-section from EFT: 20 nbarn - Kinematics- Differential probabilities: spectrum of energy-loss- Mean energy-loss: ~ Ep
2, but only 5.3 MeV / ly for Ep = 1019 eV - Result2. Mean free path of a back-scattered photon
- Cross-section- Differential probabilities and mean free path- Result
3. Summary
: 26 · 103 ly: no -rays with 2·1014 eV < k0 < 1019 eV (k0,min = 3.2·1015 eV)
Manfred Hanke / Prof. Schäfer, Institut für theoretische Kern- und Teilchenphysik
Thank you very muchfor your attention!
That‘s it!