Electroweak Theory Overview: Higgs Mechanism, Gauge Symmetry Breaking, Mass Generation, and Particle Interactions

1. Electroweak Theory Joint Belgian Dutch German School September 2007 Andrzej Czarnecki University of Alberta

2. Outline • Wednesday: Higgs mechanism and masses of gauge bosons. • Thursday: Fermion masses, decays, CKM matrix and CP violation. • Friday: Loop effects.

3. Fermi’s effective theory Fermi 1934 After parity violation: Feynman & Gell-Mann 58 Marshak & Sudarshan 58 Dimensional argument: unit of cross section = 1/energy squared. Typical scattering cross-section at high energy Fermi theory prediction: violates unitarity at high E. (Probability should not be > 1.) Compton scattering in QED – yesterday’s QFT lecture

4. Cure for Fermi’s theory: intermediate vector bosons. Propagator introduces inverse energy dependence, suppression at high E, But this is not the end of the problems. Processes with external longitudinally polarized vector bosons, or loops: sensitive to the part Note: if M is large enough, the coupling constant may be similar to e and a unification with electromagnetism seems natural.

5. To cancel bad high-energy behavior, more fields are needed s d u, c Charm quark Neutral vector boson Scalar particles (?)

6. The needed fields naturally occur in a theory with a gauge symmetry, spontaneously broken… • What is a gauge symmetry? • QED as an example • Spontaneous breaking • Global symmetry: m=0 Goldstone bosons • Local (gauge) symmetry: Higgs mechanism; • Goldstone bosons become longitudinal components of gauge bosons

7. Such local symmetry discovered in 1920’s in classical electromagnetism.

9. Note the fermion mass term, “put in by hand”

10. Important Note the fermion mass term, “put in by hand”

11. Spontaneous breaking of a global symmetry Classic example: ferromagnet Interaction is rotationally invariant, but the ground state (“vacuum”) breaks that invariance, and there are long-wavelength (massless, gapless) excitations: Goldstone bosons. If magnetization along z-axis, then x- and y-rotations are the broken symmetries  two Goldstone bosons. (option: more formal proof)

12. Two types of massless fields Exact local symmetry: massless gauge field (eg. photon) Spontaneously broken global symmetry: massless Goldstone boson. Now consider a spontaneously broken local symmetry. We will find that the two m=0 fields combine into one massive gauge field. (Higgs mechanism.)

13. Original papers Phys. Rev. Lett. 13, 321 (1964) Phys. Lett. 12, 132 (1964)

14. Higgs mechanism Consider electrodynamics of a scalar field, which also has a φ4 self-interaction: Peskin & Schoeder, sec. 20.1 Symmetry:

15. Expand the scalar field around the minimum Real massive field (“Higgs”): Massless Goldstone boson:

16. Now consider the kinetic term + cubic + quartic terms Mass term for the photon is the vacuum expectation value of the scalar field, is not physical: can be removed by a gauge transformation making the scalar field real.

17. Glashow-Weinberg-Salam model • Higgs mechanism in a non-abelian theory • Masses of electroweak bosons • Decays • Masses of fermions

18. Standard Electroweak Model • We need interactions which • are mediated by heavy particles (short distance) • include charged currents • (muon decay, neutrino-nucleus scattering) • violate parity (only left-handed neutrinos produced)

22. Masses of electroweak bosons Consider a doublet of complex scalar fields, What is its gauge-invariant kinetic energy term? Here c and c’ are coupling constants, and W and B are gauge potentials. Now suppose the scalar develops a vacuum expectation value (VEV),

23. Masses of electroweak bosons We get something that looks like a mass term: Let’s compute the resulting masses,

24. Photon: the orthogonal combination of the neutral fields Photon does not couple to the VEV and remains massless. We define the coefficient of the Wz field in photon as sine of the mixing angle, Note: Z predicted to be heavier than W,

25. The neutral current sector We identify the coupling of A with the electric charge, For comparison, the coupling of the charged W is

26. Summary so far Theory of weak interactions: problems at high energy Fermi theory  intermediate vector bosons  gauge theory Gauge symmetry guarantees such couplings that cancellations occur among various contributions (eg. vector + scalar). Also, a number of predictions: “closed multiplets” of quarks and leptons, neutral currents, and a symmetry-breaking sector (Higgs?).

27. Covariant derivative (summary) Acting on SU(2) doublets (Higgs scalar doublet; left-handed fermions): Acting on SU(2) singlets (right-handed fermions):

28. Plan for today • Fermi constant • Z-boson couplings  parity violation also in the neutral current • Fermion masses  mixing  Cabibbo angle, CKM matrix  CP violation • A remark about the running of the coupling (momentum dependence)

29. Low-energy limit for charged currents: Fermi constant

30. What are the neutral-boson couplings to electrons? What we have just derived are the couplings to left-handed fields, The right-handed ones couple only to the “B”field, in such way that the photon couples with strength e; but Thus the right-handed electrons couple via

31. Vector and axial couplings of Z to electrons Let’s combine the left and right-handed fields, What is the numerical value of the mixing angle? Similar to ¼ The vector coupling of Z to electrons is suppressed by 1-4sin2θW~1/10.

32. Parity violation The axial coupling connects small and large components of electron spinors. vector coupling pseudoscalar, violates parity (mirror symmetry) axial coupling

33. Møller scattering: E158 Measurement of the asymmetry, accurately determines the mixing angle.

36. Summary of some historical developments 1954 Yang & Mills: non-abelian gauge fields 1960 Glashow: electroweak unification, including the mixing angle θ, but the origin of boson masses unknown. 1964 Higgs mechanism 1967 Salam; Weinberg: Higgs mechanism incorporated to explain masses 1971 Renormalizability of a spontaneously broken gauge theory 1972 Neutral currents discovered at CERN 1973 Kobayashi & Maskawa: CP violation if the third generation exists 1974 Charm discovered: SLAC and Brookhaven 1976-95 Third generation of fermions

40. What is the value of the VEV? We have already determined the strength of the W coupling, as well as the W mass: This is sufficient to determine the VEV from the muon lifetime alone: “electroweak mass scale”

41. Yukawa couplings to fermions Electron: Why such small number? An unsolved puzzle. Top quark: Much more “natural”.

47. Summary of fermion masses We have seen how fermion masses are generated by Yukawa couplings of the scalar field  which has a VEV. When there is more than one generation, different combination of fermion fields couple to the scalar (mass eigenstates) and different ones participate in the charged interactions: * Cabibbo angle for two generations * CKM matrix for three

48. Summary of the Cabibbo angle Masses: Interactions: Remaining phases absorbed in R-fields:

49. CKM matrix In the case of three generations: generalization of the Cabibbo mixing, but now one complex phase remains.