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)


Let’s start with...


- 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



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 :


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) :


Example : Inverse Muon Decay

( lowest order diagram )

When the amplitude is :

Applying Casimir’s trick we find :



trace theoremstrace theorems

using :



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 :


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




DECAY OF THE MUON

As before :

The amplitude :


DECAY OF THE MUON

In the muon rest frame :

Let :

Plug in :


DECAY OF THE MUON

The decay rate given by Golden Rule* :

where :

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


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 :


DECAY OF THE MUON

Also :

The integral is trivial.For the integration, let :

and :


DECAY OF THE MUON

integration :

where :

The limits of E2 and E4 integrals :


DECAY OF THE MUON

Using :


DECAY OF THE MUON


DECAY OF THE MUON

(picture from Griffiths)


DECAY OF THE MUON

The total decay rate :

Lifetime :


DECAY OF THE MUON

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

Let’s introduce “Fermi coupling constant” :

The muon lifetime :


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


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




DECAY OF THE NEUTRON

( the same as in previous case )



In the rest frame of the neutron :

We can’t ignore the mass of the electron.

As before :

where :



The integral yields :

and :

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

and :



where :

and :



The range of E2 integral :

E is the electron energy

( exact equation)



Approximations :

Expanding to lowest order :



(picture from Griffiths)



where :

Putting in the numbers :



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”



Another correction, the quark vertex carries a factor of

is the Cabibbo angle.

Lifetime :


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




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”


DECAY OF THE PION


DECAY OF THE PION

Experimental value :

The decay rate :

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



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



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 :


CHARGED WEAK INTERACTIONS OF QUARKS

(Cabibbo, 1963)

(extra cos or sin in the vertex factor)


Example : Leptonic Decays

l is an electron or muon.

The quark vertex :

Using a previous result :

The branching ratio :

Corresponding to a Cabibbo angle :


Example : Semileptonic Decays

(semileptonic decay)

(non-leptonic weak decay)



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



The decay rate :



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



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 :


Cabibbo-GIM mechanism

Now the diagrams cancel.


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 :



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



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.


Kobayashi-Maskawa (KM) matrix

The full matrix :

Using the experimental values :


END OF PART ONEEND OF PART ONE

WEAK INTERACTION

THANK YOU !THANK YOU !

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

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