Physics 842, March 2006 Bogdan Popescu

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WEAK INTERACTION (2)

Presentation based on Introduction to Elementary
Particles by David Griffiths

Physics 842, March 2006

Bogdan Popescu
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WEAK INTERACTION (2)

- NEUTRAL WEAK INTERACTION - Elastic
Neutrino-Electron Scattering -
Electron-Positron Scattering Near the Z0 Pole -
ELECTROWEAK UNIFICATION - Chiral Fermion
States - Weak Isospin and Hypercharge -
Electro-Weak Mixing

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NEUTRAL WEAK INTERACTIONS
Neutral weak interaction mediated by Z0
f stands for any lepton or quark
Not allowed
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NEUTRAL WEAK INTERACTIONS
It doesnt matter if we use physical states or
Cabibbo rotated states.
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NEUTRAL WEAK INTERACTIONS
First observation of a process mediated by Z0
(Bubble chamber photograph at CERN, 1973)
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NEUTRAL WEAK INTERACTIONS
In the same series of experiments

• neutrino-quark process in the form of inclusive
  neutrino-nucleon scattering

2nd order nm e ? nm e

• The cross sections were three times smaller than
  the correspondent charged events

• Indication of a new kind of interaction, and not
  simply a high order process. (which correspond to
  a far smaller cross section)

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NEUTRAL WEAK INTERACTIONS
Glashow-Weinberg-Salaam Model (without proof)
The coupling to W
n l
W
l
The coupling to Z0
Departure from V-A
f
Z
f
where
Weak mixing angle or Weinberg angle
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NEUTRAL WEAK INTERACTIONS
Neutral vector and axial vector coupling in GWS
model

• NOTE
• If qW0
• we have
• CV CA
• back to
• V - A !!

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NEUTRAL WEAK INTERACTIONS
The Z0 propagator is
When
the propagator is simply
The masses of the bosons are related by the
formula
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WEAK INTERACTION (2)

- NEUTRAL WEAK INTERACTION - Elastic
Neutrino-Electron Scattering -
Electron-Positron Scattering Near the Z0 Pole -
ELECTROWEAK UNIFICATION - Chiral Fermion
States - Weak Isospin and Hypercharge -
Electro-Weak Mixing

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Example Elastic Neutrino-Electron Scattering
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Example Elastic Neutrino-Electron Scattering
So
Now compute in CM frame and let
(mass of the electron)
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Example Elastic Neutrino-Electron Scattering
( E is the electron or neutrino energy )
Using
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Example Elastic Neutrino-Electron Scattering
The total cross section
Compare to
(computed in the previous lecture)
Neutral current
(0.08 , experimental)
Charged current
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Example Elastic Neutrino-Electron Scattering
Most neutral processes are often masked by
electromagnetic ones. EXCEPT when q2 ¼
Mz2 (q2 is ee- invariant mass2)
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WEAK INTERACTION (2)

- NEUTRAL WEAK INTERACTION - Elastic
Neutrino-Electron Scattering -
Electron-Positron Scattering Near the Z0 Pole -
ELECTROWEAK UNIFICATION - Chiral Fermion
States - Weak Isospin and Hypercharge -
Electro-Weak Mixing

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Example Electron-Positron Scattering Near the
Z0 Pole
f is any quark or lepton (except electron we
must include one more diagram)
We are interested in the regime
The amplitude
where
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Example Electron-Positron Scattering Near the
Z0 Pole
( since we are working in the vicinity of 90 GeV )
ignore the mass of quark or lepton
Finally
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Example Electron-Positron Scattering Near the
Z0 Pole
problems at Z0 pole
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Example Electron-Positron Scattering Near the
Z0 Pole
Z0 is not a stable particle. Its lifetime has the
effect of smearing out the mass.
Replacement in the propagator
decay rate
The cross section
Because
the above correction is negligible outside Z0
pole.
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Example Electron-Positron Scattering Near the
Z0 Pole
Cross section for the same process, mediated by a
photon
( Qf is the charge of f in units of e )
The ratio
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Example Electron-Positron Scattering Near the
Z0 Pole
(problem 10.22)
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Example Electron-Positron Scattering Near the
Z0 Pole
Well below the Z0 pole
Right on the Z0 pole
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Example Electron-Positron Scattering Near the
Z0 Pole
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WEAK INTERACTION (2)

- NEUTRAL WEAK INTERACTION - Elastic
Neutrino-Electron Scattering -
Electron-Positron Scattering Near the Z0 Pole -
ELECTROWEAK UNIFICATION - Chiral Fermion
States - Weak Isospin and Hypercharge -
Electro-Weak Mixing

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CHIRAL FERMION STATES
To unify the weak and electromagnetic
interaction, lets move the matrix into the
particle spinor.
L stands for left-handed But this is just a
definition for uL
uL is not, in general, a helicity eigenstate
(UNLESS m 0)
Pauli matrices
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CHIRAL FERMION STATES
If the particle is massless
helicity
where
and
projection operator
(! Advanced Quantum Mechanics Homework)
Using
we can compute the following table
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CHIRAL FERMION STATES
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CHIRAL FERMION STATES
(inverse beta decay)
The contribution to the amplitude from this
vertex
Negatively charged weak current
and e stand for the particle spinors.
Rewrite as
(coupling between left-handed particles only)
Note that
Electromagnetic current
Physics 842, March 2006

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WEAK INTERACTION (2)

- NEUTRAL WEAK INTERACTION - Elastic
Neutrino-Electron Scattering -
Electron-Positron Scattering Near the Z0 Pole -
ELECTROWEAK UNIFICATION - Chiral Fermion
States - Weak Isospin and Hypercharge -
Electro-Weak Mixing

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WEAK ISOSPIN AND HYPERCHARGE
Negatively charged weak current
Positively charged weak current
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WEAK ISOSPIN AND HYPERCHARGE
We can express both charged currents
and
by introducing the left handed doublet
and the matrices
Check
and

?
These are the two charged currents
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WEAK ISOSPIN AND HYPERCHARGE
Introduce a neutral weak current as a third
component of j.
Introduce
where
Check

This shows that j belongs to SUL(2) group
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WEAK ISOSPIN AND HYPERCHARGE
We also look for an analog of hypercharge (Y) in
the Gell-MannNishijima formula
We can define weak hypercharge
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WEAK ISOSPIN AND HYPERCHARGE
Everything could be extended to the other leptons
and quarks
Weak isospin currents
This theory belongs in group SU(2)L U(1)
Weak hypercharge current
Weak isospin of left-handed leptons
Weak hypercharge
where
Physics 842, March 2006

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WEAK INTERACTION (2)

- NEUTRAL WEAK INTERACTION - Elastic
Neutrino-Electron Scattering -
Electron-Positron Scattering Near the Z0 Pole -
ELECTROWEAK UNIFICATION - Chiral Fermion
States - Weak Isospin and Hypercharge -
Electro-Weak Mixing

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ELECTRO-WEAK MIXING
GWS model asserts that the three weak iso-spin
currents couple to a weak isotriplet of
intermediate vector bosons, whereas the weak
hypercharge current couples to an iso-singlet
intermediate vector boson.
wave functions representing the
particles.
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ELECTRO-WEAK MIXING
Broken symmetry in GWS theory The two neutral
states W3 and B mix, producing one massless
linear combination (the photon), and an
orthogonal massive combination (the Z0).
So, the weak and electromagnetic coupling
constant are not independent.
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WEAK INTERACTION

END OF PART TWO
THANK YOU !

Physics 842, March 2006

Bogdan Popescu