Interactions - Introduction to Feynman Graphs

1. Interactions - Introduction to Feynman Graphs Brian Meadows, U. Cincinnati.

2. e g Time e Quantum Electrodynamics (QED) • Basic “Vertex” • A rule: particle forward is equivalent to anti-particle backward in time • All electromagnetic processes are made up from various combinations of these. • Could have any charged particle (e§, q, §, etc.) at vertex with photon • Each vertex has “coupling” (strength) ge= \/4 (~1/137) “e” means e - Brian Meadows, U. Cincinnati

3. Time Some Examples • Two simple combinations are: Moller Scattering Bhabha Scattering e- + e- e- + e- e- + e+  e- + e+ • Both processes are of magnitude /ge2 = 4 ~ 4/137 e e e e g g e e e e Brian Meadows, U. Cincinnati

4. e e e e g Time g + e e e e Bhabha Scattering • Actually, Bhabha scattering can also be represented by another diagram. • Since it is impossible to distinguish, the two must interfere. • Only the “external lines” are observable. e- + e+  e- + e+ Brian Meadows, U. Cincinnati

5. Time More Examples • Two simple combinations are: • All these processes are of magnitude /ge2 (~ ) Pair production  +  e+ + e- Pair annihilation e+ + e-   +  Compton scattering  + e-  + e- Brian Meadows, U. Cincinnati

6. More Rules • Charge and spin must be conserved at a vertex • Energy-momentum must be conserved at each vertex At basic, single vertex this means that photon has mass! • If we imagine the vertex viewed from one e- rest frame • The CM momentum of incoming e+e- = 0 • So the momentum of recoiling = 0 • So the photon cannot be mass-less. • Internal lines are “virtual” – i.e. off the mass shell. e- e+  Brian Meadows, U. Cincinnati

7. Time Higher Order Graphs (> 2 vertices) • Should sum over all orders in number of vertices • Diagrams with 4 vertices: ~ 2 • Series can converge since  ~ 1/137 << 1 Brian Meadows, U. Cincinnati

8. - - - - + - + + - - + + Q + + - - - - + + - - - + + + + + + + Vacuum Polarization • A bare charge +Q in a medium is screened by the di-electric effect of the medium: • Effective charge is reduced by halo of opposite charge from molecules by a factor e (dielectric constant) • At distances < molecular separation ~ 5 x 10-8 cm, the “bare” charge is seen Qeff Q Inter-molecular spacing Q/e distance Brian Meadows, U. Cincinnati

9. Time Vacuum Polarization • In QED, even a vacuum can become polarized by production of e-pairs: • The e-pairs become polarized rather than molecules • They shield charge of the electron • At distances < Compton wavelength of the e-, the full “bare charge” is seen aeff lc = h/mec = 2.43 x 10-10 cm ~1/137 distance Brian Meadows, U. Cincinnati

10. Feynman Rules for Electrodynamics • More on this later ! Brian Meadows, U. Cincinnati

11. Quantum Chromo-dynamics (QCD) • Basic vertex • Simplest quark-quark interaction: • Gluons are mass-less like g’s • However: • 3 colors (rgb) replace • one electric charge • AND as > 1 (but “runs”) q (b) g (r) (rb) q q q (b) g (r) • NOTICE: • the gluon transfers color • un-like the photon (r) (rb) (b) q Brian Meadows, U. Cincinnati

12. QCD and Color • In QCD one electric charge is replaced by 3 colors (rgb) • Color is conserved so when q q’ + g the gluon transfers color from q to q’ • So gluons are bi-colored (e.q. rb) • The colors belong to the group SU(3). • Instead of 3 x 3 = 9 types of gluons, there are only {8} – an “octet”. • Since gluons are colored, they attract each other too g g g g OR etc g g g Brian Meadows, U. Cincinnati

13. Time Asymptotic Freedom • “Asymptotic freedom” - as (effective) << 1 at short distances • Vacuum polarization occurs in QCD also • Increases effective color charge at close distances • Also get polarization from gluon pairs – but with opposite effect on the effective “charge” • Decreases effective color charge at close distances Brian Meadows, U. Cincinnati

14. Asymptotic Freedom (cont’d) • In field theory, the “renormalisation” procedure leads to: where (for nf Fermion loops and nb Boson loops): In QED (nf = 3, nb = 0) b0 = p. Therefore: In QCD (nf = 6, nb = 3) b0 = p. Therefore: aem ~1/137 Distance / 1/s Brian Meadows, U. Cincinnati

15. Confinement • As as becomes larger as distance grows, it is difficult to separate a quark from its hadron • Difficult to prove, but plausible • No free quarks have yet been observed • If a quark is pulled from its hadron, the energy generated is sufficient to produce a new q-q pair. S c+ u d c u s s d c X c0 K+ u s s d c Brian Meadows, U. Cincinnati

16. Feynman Rules for QCD Brian Meadows, U. Cincinnati

17. Weak Interactions • Basic charged current lepton vertex • Examples: ne,m,t • No current necessary • In the standard model, strength • of vertex related to aem W§ e, m,t ne nm m e nm nt W§ W§ m t m e + ne + nm t m + nm + nt Brian Meadows, U. Cincinnati

18. Neutral Current Vertices • “Neutral current lepton vertex • Interference observed between g and Z0 at higher energies e, m, t, ne,m,t • Similar to e/m vertex • In the standard model, • Z 0 is related to the g Z0 e, m, t, ne,m,t e e e e e e e e g Z0 g + + + Z0 e e e e e e e e Brian Meadows, U. Cincinnati

19. q+2/3 W§ Quarks in Weak Interactions • Basic charged current lepton vertex • So the W§ can connect quarks and leptons: • Couples quarks with different charges • NOTE – quark flavor can change • e.g. c  s, etc.. q-1/3 u d u ne nt e W§ W§ d t b-decay: d e + ne + u t+ p+ + nt Brian Meadows, U. Cincinnati

20. Quarks in Weak Interactions • Basic neutral current lepton vertex • Dominant in ne proton scattering: q • Couples quarks with same charges • NOTE – quark flavor CANNOT change! Z0 q u u ne e BUT NOT Why? Z 0 Z 0 ne u e u Brian Meadows, U. Cincinnati

21. q+2/3 W§ q-1/3 p+ K - u u d s W§ c u D 0 Flavor Changing Weak Interactions • Basic charged current lepton vertex • Provides mechanism for flavor changing: • Recall - in strong interactions, flavor is strictly conserved • Couples quarks with different charges • NOTE – quark flavor can change • e.g. c  s, etc.. etc. Brian Meadows, U. Cincinnati

22. Flavor Changing Weak Interactions • SM explains non-conservation of flavor in terms of quark doublets mixing: • The transformation is through the Cabbibo-Kobayashi-Maskawa (CKM) unitary matrix U: • Experimentally, the matrix is found to be approximately Brian Meadows, U. Cincinnati

23. Feynman Rules for GWS Model Brian Meadows, U. Cincinnati

24. The Force Scales • Particles that leave “tracks” in detectors either • are stable OR • decay by weak interaction Brian Meadows, U. Cincinnati

25. The Force Scales • Partial decay rates (Gf) to modes (f) may vary greatly because of • Conservation principles (embedded in matrix element Mif) e.g. G(p+ m+ nm) G(p+ e+ ne) or G(J/y  p-p+p0only 91.0 § 3.2 keV - “OZI” rule OR • Phase space: e.g. G (K+ p+ + p+ + p-) G (K+ p+ + p0) ~ 104 due to spin (helicity) conservation ~ 3.762 Brian Meadows, U. Cincinnati

26. Conserved Quantities Brian Meadows, U. Cincinnati

27. u d K - s or c s d for J/y f s or c d s u K+ d OZI suppressed Not OZI suppressed What is the OZI Rule? • Named after its inventors (Okubo, Zweig and Iizuka) based on observation rather than sound logic: If you can cut a diagram in half by cutting only gluon lines, then it is suppressed (but NOT forbidden). e.g. f p-p+p0 is OZI suppressed relative to K-K+ that is kinematically suppressed Brian Meadows, U. Cincinnati