qed- project

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)


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:


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:


Cosmic rays

- high-energy particles (up to 1020 eV)

- mostly (97%) nucleons, especially protons, -particles

- discovered in 1912 by V. Hess (Nobelprize 1936)


Fluxes of Cosmic Rays

(1 particle per m²·s)

Knee(1 particle per m²·year)

(1 particle per km²·year)Ankle

Flux

Energy


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)


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?





3. Summary



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 (!)


,forexample,is givenby:

A1


Where do these expressions come from? EFT (Chiral Effective Field Theory)

„The Heavy Baryon Chiral Perturbation Theory only involves explicit πN degrees of freedom.“



„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.“



„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).“


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:


numericalresults

for < 130 MeV

20 nbarn

The cross-section


- 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


Now, one can calculate...

k := energy of the cosmic background photon, := cos (proton, photon)lab , z := cos (proton, scattered photon)c

Differential probabilities


one can look at the spectrum of interacting photons:

Now, as one has calculated

the differential probability


Do you see any differenceto the Planck-spectrum?

(Ep = 1019 eV)Spectrum of interacting photons


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:


Spectrum of energy-loss(Ep = 1019 eV)


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


Spectrum of energy-loss(Ep = 1019 eV)


The mean energy-loss

5.3 MeV / lyfor proton withEp = 1019 eV

~ Ep2


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)


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





3. Summary



The total cross-sectionfor e+ / e– - pair production from two photons

(Breit-Wheeler)

It is .

As a result from kinematics:

,


The total cross-section



k0 = 3.21 · 109 MeV

kmax(CMB) kmax()

maximum



k0 = 5 · 107 MeV

kmax(CMB) < kmax()

suppression by the exp-factor



k0 = 1011 MeV

kmax() < kmax(CMB)

suppression by the k²-factor


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


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


Mean free path

minimal probabilityat k0 = 3.21 · 109 MeVdW/dL = 3.8 · 10-5/ly

maximal mean free path<L> = 26 · 103 ly


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.)






3. Summary


from EFT: 20 nbarn

Contents of this talk:0. Introduction

- Cosmic background radiation- Cosmic rays- Compton-scattering



3. Summary

: spectrum of energy-loss


Spectrum of a protons energy-loss due to 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




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)


Thank you very muchfor your attention!

That‘s it!

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