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

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

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

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

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

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

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

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

Important

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)

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.)

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

Higgs mechanism Peskin & Schoeder, sec Consider electrodynamics of a scalar field, which also has a φ 4 self-interaction: Symmetry:

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

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.

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

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)

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),

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

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

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

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?).

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

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)

Low-energy limit for charged currents: Fermi constant

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

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 1-4sin 2 θ W ~1/10.

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

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

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 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 Third generation of fermions

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”

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

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

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

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

c quark and cancellation of high-energy growth d s u d s c

d s u d s c

d s u d s c GIM mechanism (Glashow-Iliopoulos-Maiani 1970)

Electroweak loop effects New processes: meson-antimeson mixing CP violation, GIM mechanism (b  s+gamma, muon g-2) Corrections to relations among parameters: oblique corrections Higgs mass prediction

Eigenvectors and eigenvalues We will now see that the meson mixing has a complex off-diagonal matrix element, because of the KM phase.

CP versus mass eigenstates CP eigenstates: Hamiltonian eigenstates: Result: time evolution mixes CP eigenstates (does not conserve CP). CP can also be violated in a decay (direct CP violation).

Modification of relations among SM parameters due to loops Oblique corrections Higgs mass prediction

Three precisely measured quantities: determine M W and sin 2 θ W : Cornerstones: precise measurements

Three precisely measured quantities: Tests of the EW theory are sensitive to top and Higgs masses through loops

Fermi constant: progress in muon lifetime measurements muLan: New value G F = (6) × 10 −5 GeV −2 (5 ppm) Future: 2006 run: events  1 ppm. Chitwood et al., arXiv:

Deviations from the Dirac-equation predictions: experiments by Kusch: anomalous magnetic moment Spinning charged particle: magnetic dipole For elementary fermions (electron, muon): Dirac equation predicts Fine structure constant: electron g-2

g-2: first results Schwinger (1948) Kusch & Foley (1948)

Fine structure constant from g-2 (3.7 ppb) Schwinger Sommerfield Laporta+Remiddi Aoyama+Hayakawa +Kinoshita+Nio Result: the most precise value of the Fine Structure Constant Kinoshita & Nio, hep-ph/ (0.7 ppb) NEW: Aoyama et al, arXiv: Gabrielse et al, 2006 Previous: e

Fine structure constant: other methods Nature 442 (2006) 516.

M H from radiative corrections Tension with direct searches

Recent measurement of sin 2 θ W Follow-up plan: 12 GeV, Jefferson Lab Discrepancy in sin 2 θ W values from Z-pole era: LEP: A FB → (3) M H ~ 540 GeV SLD: A LR → (2) M H ~ 54 GeV

New W-mass from Tevatron E. Nurse, Moriond QCD 2007

Current state of EW theory Erler hep-ph/ NuTeV Asymmetries 90%

Another common way of displaying m H

Muon g-2: measurement vs. Standard Model QED (1)hep-ph/ & updates Hadronic LO6 934 (64)hep-ph/ & updates NLO− 98 (1)hep-ph/ LBL 120 (40)tentative, see hep-ph/ Electroweak 154 (3)hep-ph/ Total SM (75) Experiment − SM Theory = 252 (96) (2.6σ deviation) Units: SuSy?