Date:

- current date: Tuesday 24.07.2012 - 14:00 – 16:00h, gr. HS

- alternative option – to be discussed: - Tuesday 24.07.2012 - 13:00 – 15:00h, gr. HS - Friday 27.07.2012 - 14:00 – 16:00h, gr. HS

Requirement: 60% of the points of the homeworks (12 exercise sheets) Exam: 90 min, 8-10 short exercises, covering the content of the lecture + homeworks

Please bring your students card!

Paper and Calculator will be provided – own calculators are not allowed.

You can bring one (double sided) sheet of paper with hand written notes

If you are ill at the date of the exam, please bring a certificate from the doctor.

If you can for a very good reason not participate in the exam, please contact me before hand.

Depending on the number of students which need a second exam, it will be a written or an oral exam.

Organisatorial Issues: Exam

Organisatorial Issues: Homeworks

- homework 11 will be put on the web Wed. 27.06. - to be handed in Thu/Fr 05.07./06.07.

- homework 12 (last one) will be put on the web Fr. 06.07. to be handed in Thu/Fr 19.07/20.07.

Lecture on Friday will be given by Prof. U. Uwer

V-A Theory

The requirement of Lorentz invariance of matrix elements severely restrictsform of interaction vertex. There are in general only 5 possible combinations of twospinors and the gamma matrices that form Lorentz invariant currents, called “bilinear covariants”

- pure vector and pure axial currents conserve parity

- QED/QCD are described by pure vector currents, thus these IA per construction conserve parity

- charge current (W± exchange) of weak IA violates parity (e.g. Wu experiment). The according vertex currents are a linear combination of vector and axial coupling. Nature has choose maximal parity violation → V-A theory

Reminder: Chirality and Helicity

Helicity is an conserved observable (measurement quantity), helicity is not lorentz invariantexperimental values are between [-1,1]

Dirac spinors for particle helicity eigenstates:

h = +1 (right handed) h = -1 (left handed)

momentum along z direction

Reminder: Chirality and Helicity

Chirality is a quantum number; every particle is in chirality +1 or chirality -1 state.

chirality operator:

Projectors of chirality eigenstates:

Chirality is not directly measurable.

Reminder: Chirality and Helicity

~ 0 for E >> m

~ √E for E >> m

In the relativistic limit helicity eigenstates are also chirality eigenstates!

momentum along z direction

E >> m

Definition: Polarization of particles with finite mass

Assume a h=1 particle, what is the probability that it has RH chirality?

Assume a h=+1 particle, what is the probability that it has LH chirality?

Connection to Fermi Theory

Strength of Weak Interaction

Structure of the CKM matrix

Most preferred transition within the same quark family doublet

In first order only Vtd and V

ub are complex numbers,

CKM matrix and their complex element important for meson mixing and CP violation

Experimental probes of V-A structure: Muon Decay

General form of matrix element:

n,m = R/L given if coupling i and handiness of electron and muon are given ( )

L/R L/R

L/RL/R

Measurement Idea:

Consider muon rest system: ;

maximum energy ( /2 ) of particle 1, if particle 2,3 fly in opposite direction

12

3

expect energy distribution from kinematical point of view identical for

Fit to energy spectra + angular distribution of final state particles to determine current current couplings.

gV

LLgV

RL

Energy spectrum of emitted electron:

Michelparameters:

V-A theory:

Experiment:

Test of V-A structure: Decay of the pion

Phase space argument: Phase phase for decay into electron significantly larger than for decay into muon

Measurement:

Qualitative explanation with V-A theory:

Angular momentum conservation forces the lepton into the “wrong” helicity state, suppressed by

S=0

pion form factor

Bounded quarks cannot be described by Dirac spinors of free particles

excellent agreement with experiment

Com

puta

tion

see

hom

ewor

k 6.

2

(Anti-)neutrino – (anti-)quark scattering

Differences in neutrino-quark and neutrino anti-quark scattering, direct consequence ofV-A structure!

Sideremark: is a hint that V-A theory cannot be complete

Weak neutral current

Gargamelle - first large bubble chamber:To detect neutrinos, needs large volume with high density gas due to small cross-section

Spin of Z0 : S =1

→ (similar to W exchange) general form of coupling is a linear combination of vector and axial coupling

Neutral weak interaction couples to left- and right-handed fermion currentsContributions are however differently.

Structure of Neutral Current

Vector and axial-vector coupling of Z0

Theory predictions for couplings to neutral current:

e.g. in case of neutrinos, pure V-A structure

Weinberg-angle:

This structure is a very strong hint that weak and electromagnetic theory should be combined!

Coupling of Z0 boson

One coupling for neutral and charged weak currents!

How to determine couplings: neutrino-electron scattering (NC)

gR

2 gL

2

Integral over angel

Theorie confirmed within uncertainties: