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Introduction to Quantum Electrodynamics

Introduction to Quantum Electrodynamics. Properties of Dirac Spinors Description of photons Feynman rules for QED Simple examples Spins and traces. Summary – Spin-1/2. u (1) , u (2) , v (1) , v (2) need not be pure spin states, but their sum is still “complete”.

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Introduction to Quantum Electrodynamics

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  1. Introduction to Quantum Electrodynamics Properties of Dirac Spinors Description of photons Feynman rules for QED Simple examples Spins and traces Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

  2. Summary – Spin-1/2 u(1), u(2), v(1), v(2) need not be pure spin states, but their sum is still “complete”. Brian Meadows, U. Cincinnati

  3. Summary - Photons • Photons have two spin projections (s): Brian Meadows, U. Cincinnati

  4. Feynman Rules for QED Label: • Label each external line with 4-momenta p1, … pn. Label theirspins s1, … sn. Label internal lines with 4-momenta q1, … qn Directions: Arrows on external indicate Fermion or anti-Fermion Arrows on internal lines preserve flow Arrows on internal lines preserve flow External photon arrows point in direction of motion Internal photon arrows do not matter Brian Meadows, U. Cincinnati

  5. Feynman Rules for QED • For external lines, write factor • For each vertex write a factor Always follow a Fermion line To obtain a product: (adjoint-spinor)()(spinor) E.g.: pj , sj pk , sk e - e - q u (sk)(k) ige u(s))(j) Brian Meadows, U. Cincinnati

  6. Feynman Rules for QED • Write a propagator factor for each internal line NOTE: qj2 = mj2c2 for internal lines. NOTE also: use of the ”slash – q” a 4 x 4 matrix rather than a 4-vector! Brian Meadows, U. Cincinnati

  7. Feynman Rules for QED Now conserve momentum (at each vertex) 5. Include a d function to conserve momentum at each vertex. where the k's are the 4-momenta entering the vertex 6. Integrate over all internal 4-momenta qj. I.e. write a factor For each internal line. • Cancel the d function. Result will include factor • Erase the d function and you are left with • Anti-symmetrize (“-” sign between diagrams with swapped Fermions) Brian Meadows, U. Cincinnati

  8. p1 , s1 p3 , s3 e - e - u (s3 )(3) ige u(s1 )(1) q u (s4 )(4) ige u(s2 )(2) - - p2 , s2 p4 , s4 NOTE u(k) is short for u(sk )(pk) Example – e-- e--Scattering • We already wrote down one vertex: Use index “” • The other is similar: BUT use index “” • Leads to Brian Meadows, U. Cincinnati

  9. Evaluate – e-- e--Scattering(Griffiths, problem 7:24) • Assume e - and  - move along the z-axis, each with helicity +1. After collision, they return likewise. In this case: where Brian Meadows, U. Cincinnati

  10. The electron and muon vertex functions: • So the inner product is Brian Meadows, U. Cincinnati

  11. p1 , s1 p1 , s1 p4 , s4 p3 , s3 e - e - e - e - q q e- p3 , s3 e- e- p2 , s2 e- p2 , s2 p4 , s4 Example – e-e- e-e- (Moller Scattering) • One other diagram required in which 3  4 are interchanged (not possible in e-- scattering) • Anti-symmetrization leads to: Brian Meadows, U. Cincinnati

  12. Spin Summation and Averaging • We have, so far: • Consider e- scattering. To obtain a cross-section or decay rate, we need to evaluate and (usually) sum over final spins and average over initial ones. • Each term in […] is a number, as seen above, so we can re-order them and evaluate in pairs like where1,2are 4 x 4 matrices (in this case,and, respectively) Brian Meadows, U. Cincinnati

  13. Spin Summation and Averaging • You can show that [Hints: the * is equivalent to Also, 00=1 and ] • Therefore we can re-write V with the u(1)’s together: 4 x 4 matrix Brian Meadows, U. Cincinnati

  14. Spin Summation and Averaging • To sum over spins for particle 1, use the completeness property: • Now sum over spins for particle 3: 4 x 4 matrix WHY ? scalar Scalar NO u’s !! Brian Meadows, U. Cincinnati

  15. Spin Summation and Averaging • Why is: • Expanding (i, j are summed from 1 to 4) • So 4 x 4 matrix Call it “Q” scalar ? 4 x 4 matrix (Q.E.D!) Brian Meadows, U. Cincinnati

  16. Spin Summation and Averaging • Repeat for other vertex to get: • For the specific case of e- scattering: where m is mass of electron and M is mass of . Average over 4 initial spins Brian Meadows, U. Cincinnati

  17. Spin Summation and Averaging • The result is easy to remember: • If there is a v rather than a u: • Change the + mc to a - mc Trace for Each vertex A factor like this for each external line Brian Meadows, U. Cincinnati

  18. Dirac g Matrices - Reminder • Almost done, but we need to use gm,g5, p, etc.. Brian Meadows, U. Cincinnati

  19. … About the Traces • Almost done, but the traces need some ready results: Brian Meadows, U. Cincinnati

  20. Evaluate Traces for e- Scattering • We obtained: • Expand the first factor: Spin average over initial states 4 g 0 0 Brian Meadows, U. Cincinnati

  21. Evaluate Traces for e- Scattering • Therefore • So the first factor is: • The second factor is therefore: Brian Meadows, U. Cincinnati

  22. Evaluate Traces for e- Scattering • The product is: • Contracting terms: • Done with traces! Brian Meadows, U. Cincinnati

  23. e -  e - - Evaluate Cross-Section for e- Scattering • Work in the frame where the  is at rest. Assume so that we can ignore the recoil of the, and therefore |p1| = |p3| = |p| and |p2| = |p4| = 0 • Computing terms in matrix element: Brian Meadows, U. Cincinnati

  24. Evaluate Cross-Section for e- Scattering • Insert in matrix element: • To get the cross-section: Brian Meadows, U. Cincinnati

  25. Limiting Cases: • Relativistically, we have Mott scattering (originally for e- p): • Low energy we get Rutherford scattering: • High energy limit Brian Meadows, U. Cincinnati

  26. Compton Scattering • Two diagrams in lowest order: • Apply Feynman rules (first diagram – second is similar): p2 , s2 p4 , s4 p2 , s2 p4 , s4 + Time p1 , s1 p3 , s3 p1 , s1 p3 , s3 Fermion line (backwards) g in g out Brian Meadows, U. Cincinnati

  27. Compton Scattering • Conserve energy-momentum, etc.: where q = p1-p3, etc.. • Add term for other diagram. • Write as trace • Evaluate trace • Evaluate cross-section Brian Meadows, U. Cincinnati

  28. Consider Without the loop: With the loop: In effect: Vacuum Polarization k p4 p3 q q Time p2 p1 q-k “cut-off”  1 ! “running” coupling Physics 842, U. Cincinnati, Jan, 2009 Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987 Brian Meadows, U. Cincinnati

  29. Re-normalization (very brief!) • Other 4th-order diagrams can also contribute – e.g. with a loop at either end of the propagator (rather than in the middle). • Such diagrams actually cancel each other (“Ward indentity”) • Higher order diagrams add terms in higher orders in . • In QCD, the signs change and s grows with decreasing q2 (increasing distance). • Can have Boson loops (3-gluon vertices) and Fermion loops • Also in QCD, color factors play a role. Brian Meadows, U. Cincinnati

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