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Compton Scattering of Cosmic Radiation and Proton Energy Loss

This talk, guided by Prof. A. Schäfer, explores the mechanics of Compton scattering involving cosmic background radiation and ultrarelativistic protons. It delves into the energy losses experienced by these protons as they interact with photons, including calculations of cross-sections, kinematics, and mean energy loss. The presentation also details the mean free path of back-scattered photons and summarizes the findings in the context of cosmic rays, their origins, and implications related to high-energy physics.

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Compton Scattering of Cosmic Radiation and Proton Energy Loss

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  1. QED-Project Manfred Hanke, August 2005: (guided by Prof. A. Schäfer) Compton-scatteringof the cosmic background radiation off a ultrarelativsitic cosmic proton andpair productionby a (back-scattered) photon

  2. 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

  3. Cosmic background radiation • follows Planck‘s formula for black-body-radiation with T = 2,725 K: • predicted by G. Gamow and R. Alpher in the 1940s • discovered by A. Penzias and R. W. Wilson in 1964 (Nobelprize in 1978)

  4. Cosmic background radiation • follows Planck‘s formula for black-body-radiation with T = 2,725 K: • predicted by G. Gamow and R. Alpher in the 1940s • discovered by A. Penzias and R. W. Wilson in 1964 (Nobelprize in 1978)

  5. Cosmic rays - discovered in 1912 by V. Hess (Nobelprize 1936) - high-energy particles (up to 1020 eV) - mostly (97%) nucleons, especially protons, -particles

  6. Fluxes of Cosmic Rays Flux (1 particle per m²·s) Knee(1 particle per m²·year) Ankle (1 particle per km²·year) Energy

  7. Cosmic rays - discovered in 1912 by V. Hess (Nobelprize 1936) - 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?

  8. QED -Compton-scattering e +  e +  Easy calculation of the cross-section in the Dirac-theory: (Klein-Nishina) How to calculate Compton-scattering off a proton?

  9. 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

  10. The cross-section In Hildebrandt, Griesshammer, Hemmert, Pasquini: “Signatures of Chiral Dynamics in Low Energy Compton Scattering off the Nucleon“ (nucl-th/0307070) one finds: , with To calculate the energy-loss through Compton-scattering,one needs... for Compton-scattering off a proton the Ai‘s defined as page-long integrals over two Feynman parameters (!)

  11. A1 , forexample, is given by:

  12. Where do these expressions come from? • EFT (Chiral Effective Field Theory) „The Heavy Baryon Chiral Perturbation Theory only involves explicit πN degrees of freedom.“

  13. 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.“

  14. 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).“

  15. Problem: for > 0 - m  130 MeV, the values get imaginarydue to the resonance,and zero-values cause numerical divergencies by the denominators! Here, the following abbreviations and constants are used:

  16. The cross-section 20 nbarn  numericalresults for < 130 MeV

  17. Kinematics 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 is k := energy of the cosmic background photon,  := cos (proton, photon)lab - for the energy-loss of the proton: , z := cos (proton, scattered photon)cm

  18. Differential probabilities  := cos (proton, photon)lab , z := cos (proton, scattered photon)c k := energy of the cosmic background photon, Now, one can calculate...

  19. the differential probability one can look at the spectrum of interacting photons: Now, as one has calculated

  20. Spectrum of interacting photons (Ep = 1019 eV) Do you see any difference to the Planck-spectrum?

  21. spectrum of energy-loss: Now, as one has calculated the differential probability, one can look at the For the numerical simulation, the -function is realized by a histogram.

  22. Spectrum of energy-loss (Ep = 1019 eV)

  23. Spectrum of energy-loss  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.

  24. Spectrum of energy-loss (Ep = 1019 eV)

  25. The mean energy-loss  5.3 MeV / ly for proton with Ep = 1019 eV ~ Ep2

  26. Result 1. Energy-loss of a cosmic proton • The low energy-loss is due to the small cross-section for Compton-scattering. • 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)

  27. 2. Mean free path of a back-scattered photon 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.

  28. 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

  29. The total cross-section (Breit-Wheeler) It is . As a result from kinematics:  , for e+ / e– - pair production from two photons

  30. The total cross-section

  31. Differential probabilities k0 = 3.21 · 109 MeV kmax(CMB) kmax()  maximum

  32. Differential probabilities k0 = 5 · 107 MeV kmax(CMB) <kmax()  suppression by the exp-factor

  33. Differential probabilities k0 = 1011 MeV kmax() < kmax(CMB) suppression by the k²-factor

  34. Mean free path rapid decrease of probability for k0 < 5 · 108 MeV 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

  35. Mean free path (slow) decrease of probability for k0 > 1011 MeV dW/dL(k0 = 1010 MeV) = 3.0 · 10-5/lydW/dL(k0 = 1011 MeV) = 9.4 · 10-6/ly dW/dL(k0 = 1012 MeV) = 2.0 · 10-6/ly

  36. Mean free path minimal probability at k0 = 3.21 · 109 MeV • dW/dL = 3.8 · 10-5/ly • maximal mean free path<L> = 26 · 103 ly

  37. Result 2. 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 eV should 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.)

  38. 3. Summary 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

  39. 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 from EFT:  20 nbarn : spectrum of energy-loss

  40. Spectrum of a protons energy-loss due to Compton-scattering

  41. 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 - Result 2. 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

  42. 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: ~ Ep2, but only 5.3 MeV / ly for Ep = 1019 eV - Result 2. Mean free path of a back-scattered photon- Cross-section- Differential probabilities and mean free path- Result 3. Summary :  26 · 103 ly (k0,min = 3.2·1015 eV) : no -rays with 2·1014 eV < k0 < 1019 eV

  43. That‘s it! Thank you very muchfor your attention!

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