1. Particle Physics Chris Parkes 5 th Handout http://ppewww.ph.gla.ac.uk/~parkes/teaching/PP/PP.html Electroweak Theory 1.Divergences: cancellation requires introduce W, introduce Z, introduce Higgs 2.Gauge theories: gauge symmetries  bosons, introduction of W+Z; Problems with massive W+Zs  the Higgs.

2. 2 Unification e-e- e-e- Force from B-field S S`e-e- e-e- Force from E field p p (See Feynman Lect. Vol.2 13-8) Grand Unification ? Emag, weak, strong, gravity –Distinct characteristics (conservation rules of interaction, coupling strength..) Different aspects of single universal interaction at v. high energy ? –Symmetry broken at low mass or energy scales e.g. Electricty & Magnetism –Single theory, electromagnetic field –One arbitrary constant, c

3. 3 Electroweak Electromagnetic and Weak –Different aspects of single electroweak interaction –Same coupling e –Low energy broken symmetry –Massless , massive W +,W -,Z

4. 4 Divergences Predicted amplitudes for physical processes finite at all energies, orders of coupling constants –QED arbitrary parameters h,e,m(electron) Fermi Weak Theory –point-like contact interaction Elastic Scattering process –Scattered intensity cannot exceed incident intensity –Unitarity limit Cross-section exceeds wave theory limit –xsec grows as s, – i.e. at some point more particles out than you put in ! See Perkins, Intro to HEP, 3 rd edition chapter 9

5. 5 Divergences – Add W Introduce W boson –Propagator – ‘spread’ interaction over finite range For q2 large, tends to Cancels s dependence –i.e. well behaved at high energy BUT diverergences appear in other processes Need to systematically cancel them WW Coupling strength, propagator W-W- Propagator term

6. 6 Divergences with QED diagrams as well as W –Adding Neutral currents solves divergences Diagrams contribute to amplitude Total xsec well behaved Divergences – Add Z

7. 7 Divergence in Electroweak 1) Electroweak - Cancel all divergences –well behaved theory –Photon and weak couplings related – unification –Same intrinsic coupling strength 2) Works exactly if electron mass=0 –For finite electron mass need to add Higgs boson also (numerical factors neglected)

8. 8 Unification Conditions (g Z depends on particles at vertex, discuss form later) Predicts mass M W At low energy W interaction strength given by G F Fermi constant

9. 9 Higher Orders Measure neutrino – nucleon scattering NUTEV expt t WW b H Z/W Z q  W q q sin 2 θ W =0.2277  0.0013(stat)  0.0009(syst)  M W =78.1  0.2 GeV/c 2 using unification formula BUT measurements (LEP,TeVatron) give MW =80.39  0.03 GeV/c 2 Why ? Higher order diagrams e.g. Hence, M W sensitive to mass top quark, mass Higgs boson

10. 10 In Q.M. connection between Global transformations and conserved quantities, e.g. 1.Translational Invariance  Linear momentum conservation 2.Rotational Invariance  Angular momentum conservation 3.Translations in time  Energy conservation Emmy Noether Noether’s theorem – Symmetries (invariances) naturally lead to conserved quantities Gauge Transformations So, ψ’ still satisfies eq n of motion  no change in observables Physics invariant under Global transformation of this form (known as U(1)) Schrodinger or Dirac eq n of form:

11. 11 Local Gauge Transform - QED Now consider local transformation Add Electromagnetism –Can now be made invariant ! –i.e. invariance under U(1) local transformation  electromagnetic field (Conserved quantity is electric charge) Interpretation: 1.Change of phase  change in E,p Exactly compensated by changes in emag. Field 2.Emag field carries changes away Virtual photons 3.To cancel over all space-time range must be  so, photon massless Phase θ different at every point is space-time Ψ’ no longer a solution of eq n of motion for free particle

12. 12 Local Gauge Transform - QCD This time use colour state of quark  3 component vector Λ in r,g,b, space Symmetry group is SU(3) λ are matrices which transform the colour state –8 basis states i.e. SU(3) gauge symmetry  8 massless, coloured gluons

13. 13 Weak Interactions So QED local gauge QCD What about weak ? Need nature of particle also to change Transform Symmetry group SU(2) –Λ is a 2 component vector –Τ are the matrix states

14. 14 Weak Transform Generators of SU(2) T are Pauli matrices:  3 basis states W +,W -,W 0 Arrange particles in pairs in generations: Weak Isospin space – up and down components e.g. Left-handed doublet Right-handed singlet Weak force acts on LH Caveat: RH neutrinos?

15. 15 Electroweak Transform Combined Electroweak –Symmetry SU(2)xU(1) –Triplet (W ,W 0 ) and singlet (B 0 ) of massless (  range) fields –Predicts W+,W-, neutral currents, photon –Explains Fermi theory, cancels divergences Two problems remain: 1.W 0 same form and strength as W  But not true experimentally 2.W +,W -,W 0 all predicted massless But heavy, W ~ 80 GeV/c 2, Z ~91 GeV/c 2 And g Z depends on particles at vertex

16. 16 Problem 1: neutral bosons Clearly ,Z 0 related to W 0,B 0 but how ? Mixtures  W  - weak force –couples left-handed particle states discussed earlier Z – mixture weak & electromagnetic –Emag part couples to electric charge of particle Same for LH,RH parts –Weak part couples to weak isospin i.e. only to RH part of particle –e.g.ν – only weak component of coupling e - - weak part & emag part for charge 1 u - weak part & emag part for charge 2/3 Mixtures give rise to unification condition  relate ,W,Z couplings and explain g Z variation with particle type

17. 17 Problem 2: Masses for W & Z Gauge invariance leads to zero masses –Need to cancel at infinite range –QED – massless  –QCD – massless g BUT not for (Electro)Weak Overcome by introducing Higgs Field Mechanism to: give particles masses make theory gauge invariant Higgs boson is the quanta of the Higgs field. Only particle in SM not discovered

18. 18 Higgs Mechanism Cocktail party –People at party ! –Higgs field is NOT empty http://hepwww.ph.qmw.ac.uk/epp/higgs.html An ex-PM arrives –People cluster around her –She acquires mass from the Higgs field Rumour passes through room –Cluster of people –Excitation of Higgs field – Higgs boson

19. 19 Higgs Field Introduce doublet of scalar fields Vacuum state –Not zero Emag bowl shaped –Vacuum field 0 Higgs field, “Mexican Hat”-like –Vacuum expectation value of field, v Ground state is degenerate –Spontaneous symmetry breaking Redefine all fields wrt physical vacuum Potential Energy not symmetric about this point Symmetry between W and B fields is broken

20. 20 Higgs Mechanism Predictions 1) W and Z acquire masses –Masses from interaction of gauge fields with non- zero vac. expectation value, v, of Higgs Field 2) Neutral spin-zero Higgs bosons H –Quanta of Higgs field from gauge invariance 3) Particle masses –Particles travel through Higgs field and acquire masses –Fermions/bosons also interact with Higgs boson –Coupling proportional to particle mass Standard Model does not predict Higgs mass, W/Z mass, fermion masses f f H

21. 21 Electroweak theory provides well-behaved theory without divergences Gauge invariance leads to introduction of weak force Higgs Mechanism leads to particle masses Tests of Theory: –Find Neutral Currents  –Discover W,Z bosons  –Measure W,Z couplings and masses  –Find Higgs Boson ? Electroweak Summary