1. Particle Physics Chris Parkes Feynman Graphs of QFT Relativistic Quantum Mechanics QED Standard model vertices Amplitudes and Probabilities QCD Running Coupling Constants Quark confinement 2 nd Handout http://ppewww.ph.gla.ac.uk/~parkes/teaching/PP/PP.html

2. 2 Adding Relativity to QM See Advanced QM option Free particleApply QM prescription Get Schrödinger Equation Missing phenomena: Anti-particles, pair production, spin Or non relativistic Whereas relativistically Klein-Gordon Equation Applying QM prescription again gives: Quadratic equation  2 solutions One for particle, one for anti-particle Dirac Equation  4 solutions particle, anti-particle each with spin up +1/2, spin down -1/2

3. 3 Positron KG as old as QM, originally dismissed. No spin 0 particles known. Pion was only discovered in 1948. Dirac equation of 1928 described known spin ½ electron. Also described an anti-particle – Dirac boldly postulated existence of positron Discovered by Anderson in 1933 using a cloud chamber (C.Wilson) Track curves due to magnetic field F=qv×B

4. 4 Transition Probability reactions will have transition probability How likely that a particular initial state will transform to a specified final state e.g. decays Interactions We want to calculate the transition rate between initial state i and final state f, We Use Fermi’s golden rule This is what we calculate from our QFT, using Feynman graphs This tells us that  fi (transition rate) is proportional to the transition matrix element T fi squared (T fi 2 ) Transition rate  Prob y of decay/unit time  cross-section x incident flux  IV 

5. 5 Quantum ElectroDynamics (QED) Developed ~1948 Feynman,Tomonaga,Schwinger Feynman illustrated with diagrams e-e- e-e-  Time: Left to Right. Anti-particles:backwards in time. Process broken down into basic components. In this case all processes are same diagram rotated e-e- e+e+ e-e- e+e+  Photon emission Pair production annhilation We can draw lots of diagrams for electron scattering (see lecture) Compare with c.f. Dirac hole theory M&S 1.3.1,1.3.2

6. 6 Orders of  The amplitude T is the sum of all amplitudes from all possible diagrams Each vertex involves the emag coupling (  =1/137) in its amplitude Feynman graphs are calculational tools, they have terms associated with them So, we have a perturbation series – only lowest order terms needed More precision  more diagrams There can be a lot of diagrams! N photons, gives  n in amplitude c.f. anomalous magnetic moment: After 1650 two-loop Electroweak diagrams - Calculation accurate at 10 -10 level and experimental precision also!

7. 7 The main standard model vertices Strong: All quarks (and anti-quarks) No change of flavour EM: All charged particles No change of flavour Weak neutral current: All particles No change of flavour Weak charged current: All particles Flavour changes At low energy:

8. 8 Amplitude  Probability If we have several diagrams contributing to same process, we much consider interference between them e.g. e-e- e-e-  e+e+ e+e+ (a) (b) Same final state, get terms for (a+b) 2 =a 2+ b 2 +ab+ba e-e- e+e+ e+e+ e-e-   |T fi | 2 The Feynman diagrams give us the amplitude, c.f.  in QM whereas probability is |  | 2 (1) (2) So, two emag vertices: e.g. e - e +  -  + amplitude gets factor from each vertex And xsec gets amplitude squared for e - e +  qq with quarks of charge q (1/3 or 2/3) Also remember : u,d,s,c,t,b quarks and they each come in 3 colours Scattering from a nucleus would have a Z term

9. 9 Massive particle exchange Forces are due to exchange of virtual field quanta ( ,W,Z,g..) E,p conserved overall in the process but not for exchanged bosons. You can break Energy conservation - as long as you do it for a short enough time that you don’t notice! i.e. don’t break uncertainty principle. X B A Consider exchange of particle X, mass m x, in CM of A: For all p, energy not conserved Uncertainty principle Particle range R So for real photon, mass 0, range is infinite For W (80.4 GeV/c 2 ) or Z (91.2 GeV/c 2 ), range is 2x10 -3 fm

10. 10 Virtual particles This particle exchanged is virtual (off mass shell) e-e- e+e+ ++ -- e.g.  (E,p) (E,-p) (E , p  ) symmetric Electron-positron collider Yukawa Potential Strong Force was explained in previous course as neutral pion exchange Consider again: Spin-0 boson exchanged, so obeys Klein-Gordon equation See M&S 1.4.2, can show solution is R is range For m x  0, get coulomb potential Can rewrite in terms of dimensionless strength parameter

11. 11 Quantum Chromodynamics (QCD) QED – mediated by spin 1 bosons (photons) coupling to conserved electric charge QCD – mediated by spin 1 bosons (gluons) coupling to conserved colour charge u,d,c,s,t,b have same 3 colours (red,green,blue), so identical strong interactions [c.f. isospin symmetry for u,d], leptons are colourless so don’t feel strong force Significant difference from QED: photons have no electric charge But gluons do have colour charge – eight different colour mixtures. 7.1 M&S Hence, gluons interact with each other. Additional Feynman graph vertices: 3-gluon 4-gluon These diagrams and the difference in size of the coupling constants are responsible for the difference between EM and QCD Self-interaction

12. 12 Running Coupling Constants - QED +Q + - - + + - Charge +Q in dielectric medium Molecules nearby screened, At large distances don’t see full charge Only at small distances see +Q Also happens in vacuum – due to spontaneous production of virtual e + e - pairs And diagrams with two loops,three loops…. each with smaller effect: ,  2 …. e+e+ e-e-    e + e - As a result coupling strength grows with |q 2 | of photon, higher energy  smaller wavelength gets closer to bare charge |q 2 | 1/137 1/128 0 (90GeV) 2 QED – small variation 

13. 13 Coupling constant in QCD Exactly same replacing photons with gluons and electrons with quarks But also have gluon splitting diagrams g g g g This gives anti-screening effect. Coupling strength falls as |q 2 | increases Grand Unification ? Strong variation in strong coupling From  s  1 at |q 2 | of 1 GeV 2 To  s at |q 2 | of 10 4 GeV 2 LEP data Hence: Quarks scatter freely at high energy Perturbation theory converges very Slowly as  s  0.1 at current expts And lots of gluon self interaction diagrams

14. 14 Range of Strong Force Gluons are massless, hence expect a QED like long range force But potential is changed by gluon self coupling NB Hadrons are colourless, Force between hadrons due to pion exchange. 140MeV  1.4fm QED QCD - + Standard EM field Field lines pulled into strings By gluon self interaction Qualitatively: QCD – energy/unit length stored in field ~ constant. Need infinite energy to separate qqbar pair. Instead energy in colour field exceeds 2m q and new q qbar pair created in vacuum This explains absence of free quarks in nature. Instead jets (fragmentation) of mesons/baryons Form of QCD potential: Coulomb like to start with, but on ~1 fermi scale energy sufficient for fragmentation q q

15. 15 Formation of jets 1.Quantum Field Theory – calculation 2.Parton shower development 3.Hadronisation

16. 16 Summary 1.Add Relativity to QM  anti-particles,spin 2.Quantum Field Theory of Emag – QED Feynman graphs represent terms in perturbation series in powers of α Couples to electric charge 3.Standard Model vertices for Emag, Weak,Strong Diagrams only exist if coupling exists e.g. neutrino no electric charge, so no emag diagram 4. QCD – like QED but.. Gluon self-coupling diagrams α strong larger than α emag 5.Running Coupling Constants α strong varies, perturbation series approach breaks down 6.QCD potential – differ from QED due to gluon interactions Absence of free quarks, fragmentation into colourless hadrons Now, consider evidence for quarks, gluons….