presentation based on “ introduction to elementary particles ” by david griffiths

Physics 842, February 2006 Bogdan Popescu

Presentation based on “Introduction to Elementary Particles”by David Griffiths

WEAK INTERACTION (1)

Physics 842, February 2006 Bogdan Popescu

Let’s start with...

Physics 842, February 2006 Bogdan Popescu

- CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion

- CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix

WEAK INTERACTION (1)

Physics 842, February 2006 Bogdan Popescu

CHARGED LEPTONIC WEAK INTERACTION

The mediators of weak interactions are “intermediate vector bosons”, which are extremely heavy :

The propagator for massive spin-1 particles is :

, where M is MW or MZ

In practice very often :

The propagator for W or Z in this case :

Physics 842, February 2006 Bogdan Popescu

CHARGED LEPTONIC WEAK INTERACTION

The theory of “charged” interactions is simpler than that for “neutral” ones.We start by considering coupling of W’s to leptons.The fundamental leptonic vertex is :

The Feynman rules are the same as for QED, except for the vertex factor :

( the weak vertex factor )

“Weak coupling constant” (analogous to ge in QED and gs in QCD) :

Physics 842, February 2006 Bogdan Popescu

Example : Inverse Muon Decay

( lowest order diagram )

When the amplitude is :

Applying Casimir’s trick we find :

Physics 842, February 2006 Bogdan Popescu

Example : Inverse Muon Decay

trace theoremstrace theorems

using :

Physics 842, February 2006 Bogdan Popescu

Example : Inverse Muon Decay

In CM frame, and neglect the mass of the electron :

where E is the incident electron (or neutrino) energy.The differential scattering cross section is :

The total cross section :

Physics 842, February 2006 Bogdan Popescu

- CHARGED LEPTONIC WEAK INTERACTION - - Decay of the MuonDecay of the Muon - Decay of the Neutron - Decay of the Pion

- CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix

WEAK INTERACTION (1)

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

As before :

The amplitude :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

In the muon rest frame :

Let :

Plug in :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

The decay rate given by Golden Rule* :

where :

* a lot of work, since this is a three body decay

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

Perform integral :

where :

Next we will do the integral. Setting the polar axis along (which isfixed, for the purposes of the integration), we have :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

Also :

The integral is trivial.For the integration, let :

and :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

integration :

where :

The limits of E2 and E4 integrals :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

Using :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

(picture from Griffiths)

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

The total decay rate :

Lifetime :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

gW and MW do not appear separately, only in the ratio.

Let’s introduce “Fermi coupling constant” :

The muon lifetime :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE MUON

In Fermi’s original theory of beta decay there was no W;the interaction was a direct four-particle coupling.

Using the observed muon lifetime and mass :

and :

“Weak fine structure constant” :

Larger than electromagnetic fine structure constant

Physics 842, February 2006 Bogdan Popescu

- CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron- Decay of the Neutron - Decay of the Pion

- CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix

WEAK INTERACTION (1)

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE NEUTRON

( the same as in previous case )

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE NEUTRON

In the rest frame of the neutron :

We can’t ignore the mass of the electron.

As before :

where :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE NEUTRON

The integral yields :

and :

Setting the z-axis along (which is fixed, for the purposes of the integral), we have :

and :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE NEUTRON

where :

and :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE NEUTRON

The range of E2 integral :

E is the electron energy

( exact equation)

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE NEUTRON

Approximations :

Expanding to lowest order :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE NEUTRON

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE NEUTRON

(picture from Griffiths)

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE NEUTRON

where :

Putting in the numbers :

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE NEUTRON

But the proton and neutronare not point particles.

Replacement in the vertex factor :

cV is the correction to the vector “weak charge”cA is the correction to the axial vector “weak charge”

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE NEUTRON

Another correction, the quark vertex carries a factor of

is the Cabibbo angle.

Lifetime :

Physics 842, February 2006 Bogdan Popescu

- CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion- Decay of the Pion

- CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix

WEAK INTERACTION (1)

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE PION

The decay of the pion is really a scattering event in which the incident quarks happen to be bound together. We do not know how the W couples to the pion. Use the “form factor”.

“form factor”

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE PION

Physics 842, February 2006 Bogdan Popescu

DECAY OF THE PION

Experimental value :

The decay rate :

The following ratio could be computed without knowing the decay constant :

Physics 842, February 2006 Bogdan Popescu

- CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion

- CHARGED WEAK INTERACTIONS OF QUARKS- CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix

WEAK INTERACTION (1)

Physics 842, February 2006 Bogdan Popescu

CHARGED WEAK INTERACTIONS OF QUARKSFor leptons, the coupling to W+ and W- takes place strictly within aparticular generation :

For example :

There is no cross-generational coupling as :

There are 3 generations of quarks :

There exist cross-generational coupling as :

Coupling within a generation :

Physics 842, February 2006 Bogdan Popescu

CHARGED WEAK INTERACTIONS OF QUARKS

(Cabibbo, 1963)

(extra cos or sin in the vertex factor)

Physics 842, February 2006 Bogdan Popescu

Example : Leptonic Decays

l is an electron or muon.

The quark vertex :

Using a previous result :

The branching ratio :

Corresponding to a Cabibbo angle :

Physics 842, February 2006 Bogdan Popescu

Example : Semileptonic Decays

(semileptonic decay)

(non-leptonic weak decay)

Physics 842, February 2006 Bogdan Popescu

Example : Semileptonic Decays

Neutron decay : Quark process :There are two d quarks in n, and either one could couple to the W. The netamplitude for the process is the sum. Using the quark wave functions :

The overall coefficient is simply cos, as claimed before.

In the decay : the quark process is the same

But :

we get an extra factor

Physics 842, February 2006 Bogdan Popescu

Example : Semileptonic Decays

The decay rate :

Physics 842, February 2006 Bogdan Popescu

- CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion

- CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism- Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix

WEAK INTERACTION (1)

Physics 842, February 2006 Bogdan Popescu

Cabibbo-GIM mechanism

GIM = GLASHOW, ILIOPOULOS, MAIANI

The decay is allowed by Cabibbo theory.

Amplitude : , far greater.

GIM introduced the fourth quark c (1970). The couplings with s and d :

Physics 842, February 2006 Bogdan Popescu

Cabibbo-GIM mechanism

Now the diagrams cancel.

Physics 842, February 2006 Bogdan Popescu

Cabibbo-GIM mechanismThe Cabibbo-GIM mechanism : Instead of the physical quarks d and s, the “correct” states touse in the weak interactions are d’ and s’ :

In matrix form :

The W’s couple to the “Cabibbo-rotated” states :

Physics 842, February 2006 Bogdan Popescu

- CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion

- CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM mechanism - Kobayashi-Maskawa (KM) matrix- Kobayashi-Maskawa (KM) matrix

WEAK INTERACTION (1)

Physics 842, February 2006 Bogdan Popescu

Kobayashi-Maskawa (KM) matrix

The weak interaction quark generations

KM is a generalization of Cabibbo-GIM for three generations of quarks.

are related to the physical quarks states by Kobayashi-Maskawa (KM) matrix

for example :

Canonical form of KM matrix depend only on three generalized Cabibbo angles and one phase factor.

Physics 842, February 2006 Bogdan Popescu

Kobayashi-Maskawa (KM) matrix

The full matrix :

Using the experimental values :

Physics 842, February 2006 Bogdan Popescu

END OF PART ONEEND OF PART ONE

WEAK INTERACTION

THANK YOU !THANK YOU !