Compton Scattering Sum Rules for Massive Vector Bosons

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Chapter 3 Compton Scattering and Sum Rules Laboratory system (LS) In the LS, the target particle is initially at rest. Thus we have the following initial and final momenta, where ν is the photon energy: initial final p = (m, 0) , p ′ = (E, k sin φ, 0, k cos φ) , (3.28a) q = (ν, 0, 0, ν) , q ′ = (ν ′ , ν ′ sin θ, 0, ν ′ cos θ) . (3.28b) We consider only real Compton scattering, which implies the on-shell conditions for initial and final state, p 2 = M 2 = p ′ 2 and q 2 = 0 = q ′2 . (3.29) The Mandelstam variables are related to the LS by s = M 2 + 2νM = M 2 + 2ν ′ (E − k cos (θ + φ)) , t = 2M 2 − 2EM = −2νν ′ (1 − cos θ) , u = M 2 − 2ν ′ M = M 2 − 2ν (E − k cos φ) . (3.30) From these relations we can deduce Compton’s formula for the shift in the photon wavelength: ν ′ = ν 1 + ν (3.31) (1 − cos θ). M Note that in this frame, the photon energy ν is related to s by ν(s) = s−M 2 . Hence, 2M the LS prefactor is given by 1 ϕ (ν) = 64πν 2 M . (3.32) 2 The angles θ and φ in the LS are related in a complicated manner. Deriving the amplitude in this frame requires involved calculations. It is much more convenient if we choose the CMS for our calculations. Center-of-Momentum System (CMS) In the CMS, both particles are initially travelling towards each other with equal absolute 3-momenta. After the collision, both will deviate from their former course by a scattering angle ϑ, while maintaining their velocity. 30
3.2 Scattering Kinematics ϑ Figure 3.3: Compton scattering in the center-of-momentum frame. Expressed as initial and final 4-momenta, this is written as initial final p = (E cm , 0, 0, −ω) , p ′ = (E ′ cm, −ω sin ϑ, 0, −ω cos ϑ) , (3.33a) q = (ω, 0, 0, ω) , q ′ = (ω, ω sin ϑ, 0, ω cos ϑ) . (3.33b) Note that in the CMS, the photon energy is ω ≡ ω cm (s) = s−M 2 2 √ . The CMS prefactor s ϕ(s) in terms of ω is ϕ (ω) = 1 64π (2Eω 3 + 2ω 4 + ω 2 M 2 ) . (3.34) With regard to the CMS, we can write the differential cross section as dσ dΩ cm 1 = 1 4π = 1 64π 2 s 1 8π 2 λ 1 2 (s, M 2 , 0) |p ′ cm| |p cm | |M fi| 2 . ∫ d 3 p cm 2E cm d 3 q ′ cm 2ω δ(4) (p + q − p ′ − q ′ )|M fi | 2 (3.35) However, we want to express the differential cross section in an invariant manner. Therefore, we change the integration variable to the Mandelstam variable t. Its differential is related to the CMS differential solid angle via the substitution dt = 2 |p cm b | |p cm = 2ω 2 dcos ϑ, 2 | dcos ϑ = 1 π |pcm b | |p cm 2 | dΩ cm 2 (3.36) so that we finally obtain the invariant cross section dσ dt = 1 |M 16πλ(s, M 2 fi (s, t)| 2 . (3.37) , 0) } {{ } =ϕ(s) 31
Page 1 and 2: Compton Scattering Sum Rules for Ma
Page 3 and 4: Contents Introduction vii 1 Physica
Page 5: Contents Acknowledgements 85 v
Page 8 and 9: Introduction Purpose of this Study
Page 10 and 11: Introduction Euclidian metric and s
Page 12 and 13: Chapter 1 Physical method the term
Page 14 and 15: Chapter 1 Physical method where the
Page 16 and 17: Chapter 1 Physical method Lagrangia
Page 18 and 19: Chapter 1 Physical method where F (
Page 20 and 21: Chapter 1 Physical method F (ν) (n
Page 22 and 23: Chapter 2 Lagrangian Description of
Page 34 and 35: Chapter 3 Compton Scattering and Su
Page 50 and 51: Chapter 4 Testing the Sum Rules in
Page 60 and 61: Chapter 5 Quantum One-Loop Correcti
Page 73 and 74: Chapter 6 Conclusion and Outlook We
Page 75 and 76: Appendix A Feynman Rules A.1 Feynma
Page 77 and 78: A.2 Plane Wave Expansions A.2 Plane
Page 79: A.2 Plane Wave Expansions Contracti
Page 82 and 83: Appendix B Loop Integration Im l 0
Page 84 and 85: Appendix B Loop Integration It is u
Page 86 and 87: Appendix C Diagrams µ q β p k k
Page 89 and 90: Bibliography [A + 06] J. Alcaraz et
Page 91 and 92:
Bibliography [DHK + 04] [DKT01] [DP
Page 93 and 94:
Bibliography [Nus72] [Pai67] [Pai68
Page 95:
Acknowledgements First and foremost
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Compton Scattering Sum Rules for Massive Vector Bosons

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