Lecture 8 Feynman diagramms. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2

Transcription

1 Lecture 8 Feynman diagramms SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1

2 Photon propagator Electron-proton scattering by an exchange of virtual photons ( Dirac-photons ) (1) e - virtual photon p The photon vector field A μ follows the wave equation: e - p (2) where J μ is the proton 4-current and in Lorentz gauge. Solve the inhomogenuous wave equation (2) using the Green function: (3) The inhomogenuous solution of equation (2) can be written as (4) Indeed, using (4) and (3) one obtaines eq. (2) again: (5) 2

3 Photon propagator The Green s function can be written as a Fourier transform (6) (7) From (7) we obtain for the photon propagator: (8) Consequences: A wave function of positive energy will spread only forward in time t > t and of negative energy - backward in time t < t 3

4 Photon propagator The proton charge 4-current J μ can be written as: (9) where ψ i and ψ f are the spinors of the proton in the initial and final states: (10) From (10) and (9) we get u u + 0 γ (11) The photon vector field A μ : substitute (8) into (4) (12) (13) 4

5 Feynmann diagramms For the free electron: (14) S-matrix element (15) (16) The matrix element (16) can be presented as a Feynman diagramm: electron virtual photon proton 5

6 Feynmann diagramms For the electron: For the photon: For the incoming and outgoing photon we have: 6

7 Feynmann diagramms Electron propagator: S-matrix element: Electron current : Proton current: 7

8 Feynmann diagramms emission absorption of photon 1) Scattering of charged particles absorption emission of virtual photon 2) Compton scattering emission of real photon virtual electron absorption of real photon 8

9 Feynmann diagramms 3) Bremsstrahlung emission of real photon virtual electron Ze absorption of virtual photon 4) Pair creation and annihilation virtual photon (time-like) 9

10 Scattering of electrons on an external potential Consider the scattering of electrons on an external Coulomb potential photon vector field A μ : In first order perturbation theory : particles = plane waves S-matrix element: (16a) E i, E f are total initial and final energy of the system energy conservation to t use that 10

11 Scattering of electrons on an external potential S-matrix element is (17) matrix element Potential acts during the time period T (i.e. T is the interaction time) : -T/2 < t < T/2 (18) (19) Off-shell function f(ω): maximum at ω=0 (E f =E i ), the amplitude ~T 2 ; width ~1/T 11

12 Feynmann diagramms: scattering of electrons Uncertainty relation: (20) If t >>T, f(ω) δ-function During the time -T/2 < t < T/2 the system can be in a state within the energy interval E f, E f +δe f ρ(e f )δe f number of energy levels in this interval ρ(e f ) level density = number of states per energy interval Differential transition probability dw is (21) Thus, transition probability W is obtained by an integration of dw over de f : (22) 12

13 Feynmann diagramms: scattering of electrons For t >>T, f(ω) δ-function (22) Gold rule from Fermi: (23) w is the transition probability per unit time: w=w/t Total cross section: = transition probability per unit time over j ein - initial current: (24) 13

14 Feynmann diagramms: scattering of electrons The scattering probability (17) is then (25) T 2πδ ( ω ) Use that S-matrix element (26) Furthermore, perfom the integration over the phase space: i.e. over the number of levels in the energy interval E f, E f +δe f and the direction of the particle in the solid angle element dω f (27) factor 1/2E f due to the normalization of 2E f particles per volume V 14

15 Feynmann diagramms: scattering of electrons Then, the differential transition probability is (28) matrix element Initial current: (29) Differential cross section: (30) Using the explicit form for the matrix element (17), we obtain: (31) where since 15

16 Feynmann diagramms: non-relativistic case Consider the non-relativistic limit: (32) From (31) Rutherford formula (33) 16

17 Feynmann diagramms: relativistic case Consider the scattering of relativistic electrons (34) For fermions: 1) avaraging over the spin of initial fermions 2) summation over the spin of the final fermions 1 2 S i S f 1 2 Si, S f Consider (35) Spin avaraging: (36) For different components α,β: (37) 17

18 Dirac spinors (cf. Lecture N 7) Free (anti-)fermions are fully defined by the spinors specified above (with normalization (L7.24)): 1) Spinors with positive energy (fermions): (L7.25) spin up : spin down 2) Spinors with negative energy (anti-fermions): (L7.26) Wave vector ψ : fermions anti-fermions (L7.27) 8

19 Spinor tensors Summing over spin gives: (37) Notation: u u + γ 0 19

20 Feynmann diagramms Summing over spins of the initial and final fermions we may write in matrix form: (38) (39) (40) with Result: (41) Notation: Spur=Sp=Tr 20

21 Feynmann diagramms 2) Spur (even number of γ μ ) =0 where I is the 4x4 unitary matrix (42) Use that μ 0 μ + 0 γ = γ γ γ = γ μ Notation: â (42) Using (42), averaging over the spin of the initial fermions and summation over the spin of final fermions leads to: (43) mass of electron 21

22 Feynmann diagramms Evaluate the scalar product in the cms of the process i+f: (44) β p E (45) Note: non-relativisticely: Thus, substitute (45) in (31) : (46) eq. (46) describes the scattering of a relativistic spin ½ particle on a spin 0 target (e.g. nucleus) with large mass M! 22

23 Feynmann diagramms For the high energy electrons, i.e. E>>m, β 1 we obtain the Mott cross section including the backscattering of the target of finite mass M: (47) eq. (47) describes the scattering of a relativistic spin ½ electron on a spin 0 target with large mass M (e.g. nuclei, pion) 23