Particle Physics

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

2. 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 Unification Grand Unification ? (See Feynman Lect. Vol.2 13-8) e- e- S p e- p S` e- Force from B-field Force from E field

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

4. Divergences See Perkins, Intro to HEP, 3rd edition chapter 9 • 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 !

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 Propagator term W- Coupling strength, propagator

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

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. Unification Conditions (gZ depends on particles at vertex, discuss form later) Predicts mass MW At low energy W interaction strength given by GF Fermi constant

9. Higher Orders     Z W H q t q q Z/W Z/W W W b • Measure neutrino – nucleon scattering • NUTEV expt sin2θW=0.22770.0013(stat)0.0009(syst) MW =78.10.2 GeV/c2 using unification formula BUT measurements (LEP,TeVatron) giveMW =80.390.03 GeV/c2 Why ? Higher order diagrams e.g. Hence, MW sensitive to mass top quark, mass Higgs boson

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

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: • Change of phase change in E,p • Exactly compensated by changes in emag. Field • Emag field carries changes away • Virtual photons • 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 eqn of motion for free particle

12. Local Gauge Transform - QCD • This time use colour state of quark • 3 component vector Λin 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. 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. Weak Transform Generators of SU(2) T are Pauli matrices: 3 basis states W+,W-,W0 Arrange particles in pairs in generations: Left-handed doublet Right-handed singlet Weak force acts on LH Caveat: RH neutrinos? Weak Isospin space – up and down components e.g.

15. Electroweak Transform • Combined Electroweak • Symmetry SU(2)xU(1) • Triplet (W,W0) and singlet (B0) of massless ( range) fields • Predicts W+,W-, neutral currents, photon • Explains Fermi theory, cancels divergences • Two problems remain: • W0 same form and strength as W • But not true experimentally • W+,W-,W0 all predicted massless • But heavy, W ~ 80 GeV/c2 , Z ~91 GeV/c2 And gZ depends on particles at vertex

16. Problem 1: neutral bosons • Clearly ,Z0 related to W0,B0 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 gZ variation with particle type

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. Higgs Mechanism http://hepwww.ph.qmw.ac.uk/epp/higgs.html • Cocktail party • People at party ! • Higgs field is NOT empty • 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. 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. 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 H f f Standard Model does not predict Higgs mass, W/Z mass, fermion masses

21. Electroweak Summary • 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 ?