weak interactions - purdue university

Weak Interactions

•  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-Kobayashi-Maskawa (CKM) Matrix •  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

OUTLINE

THE WEAK FORCE

Characteristics of the WEAK FORCE

The time scale of the decay is long. Radioactive decays must proceed by the weak force since the timescale ranges from 10-8 s to years

Weak decays often involve neutrinos •  do not interact by the EM force or the strong force •  cannot detect in conventional detectors •  can infer existence from conservation of E, p (Pauli, 1930)

E.g. neutron β decay

Neutrinos would not be directly detected for 25 years: Reines & Cowan, using Savannah River nuclear reactor

DECAY OF THE NEUTRON

Helicity is the component of the (spin) angular momentum along momentum vector. For fermions, the value is –1/2 or +1/2, depending on whether spin S is antiparallel or parallel to direction of motion p

HELICITY

Solutions of the Dirac equation:

where

are helicity eigenstates:

For antiparticles the relation is reversed (because v(p) ~ u(-p)):

HELICITY

In the limit m=0 (E>>m):

obeys

with

Thus the following “chirality” projection operators are also helicity projection operators for m=0:

For m=0, PL projects onto helicity –1/2 fermions but helicity +1/2 anti-fermions.

HELICITY and EM INTERACTION

We can use the projection operators to split the electromagnetic current into 2 pieces:

where

Since

we have

and

Helicity is conserved in the electromagnetic interaction in high energy (m=0) limit

HELICITY and EM INTERACTION

Allowed QED vertices in high energy limit

Equalities are due to parity

mirror reflection

Helicity is reversed under parity:

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:

WEAK INTERACTION

Lorentz condition εµ pµ = 0

also 3 and 4 boson vertices exist

FEYNMAN RULES

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) :

CHARGED WEAK INTERACTION

The charged weak interaction violates parity maximally

By analogy to EM we associate the charged weak interactions with a current, which is purely left-handed:

The charged weak interaction only couples to left-handed leptons (e,µ,τ,νi). (Also, only couples to left-handed quarks.) It couples only to right-handed anti-fermions.

also

Example: Inverse Muon Decay

( lowest order diagram )

When the amplitude is :

Simplifies because MW = 80 GeV much larger than q< (100 MeV).


trace theorems trace theorems

using :

Applying Casimir’s trick we find :


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 :

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 is fixed, 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

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 : = 2.2×10-6 s

DECAY OF THE MUON

In Fermi’s original theory of the 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!

WEAK INTERACTIONS

Weak force is weak because boson propagator is massive, not because coupling strength is weak.


( 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 neutron are 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 : cosθC = 0.97

CHOICE OF WEAK EIGENSTATES

4-Fermion INTERACTION

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 :

DECAY OF THE PION

CHARGED WEAK INTERACTIONS OF QUARKS

For leptons, the coupling to W+ and W- takes place strictly within a particular 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)

Example: Semileptonic Decays

Neutron decay : Quark process : There are two d quarks in n, and either one could couple to the W. The net amplitude 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

Example : Semileptonic Decays

The decay rate :

GIM MECHANISM

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

In matrix form :

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

Cabibbo-Kobayashi-Maskawa (CKM) Matrix

The weak interaction quark generations

CKM 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 :

THE CKM MATRIX

NEUTRAL WEAK INTERACTIONS

Neutral weak interaction mediated by the Z0 boson

f stands for any lepton or quark

Not allowed :


It doesn’t matter if we use physical states or Cabibbo rotated states.


First process mediated by Z0

(Bubble chamber photograph at CERN, 1973)


In the same series of experiments : Neutrino-quark process in the form of inclusive scattering

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)


The coupling to Z0 :

where :

(“Weak mixing angle” or “Weinberg angle”)


Neutral vector and axial vector coupling in GWS model :


( Z0 propagator )

When :

the propagator is simply :

The masses of the bosons are related by the formula :

Example : Elastic Neutrino-Electron Scattering


Now compute in CM frame and let : (mass of the electron)


( E is the electron or neutrino energy )

Using :


The total cross section :

Compare to :

(computed in the earlier)

(0.08 , experimental)


Most neutral processes are “masked” by electromagnetic ones.

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 :


( since we are working in the vicinity of 90 GeV )

Finally :

Ignore the mass of quark or lepton


problems at 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.


Cross section for the same process, mediated by a photon :

( Qf is the charge of f in units of e )

The ratio :


Well below the Z0 pole :

Right on the Z0 pole :

CHIRAL FERMION STATES

To unify the weak and electromagnetic interaction, let’s move the matrix into the particle spinor.

( L stands for “left-handed” )

(But uL is not, in general, a helicity eigenstate)


If the particle is massless :

where :

helicity

“projection operator”

Using : we can compute the following table è


(inverse beta decay)

The contribution to the amplitude from this vertex :

and e stand for the particle spinors.

Negatively charged weak current

Rewrite as :

Note that :

(coupling between left-handed particles only)

Electromagnetic current :

WEAK ISOSPIN AND HYPERCHARGE

Negatively charged weak current

Positively charged weak current


We can express both

by introducing the left handed doublet :

and the matrices :

We could have a full “weak isospin” symmetry if only there is a third weak current, “neutral weak current”.

Weak analog of hypercharge (Y) in the Gell-Mann – Nishijima formula :

Weak hypercharge :



Everything could be extended to the other leptons and quarks :

Weak isospin currents :

Weak hypercharge current :

where :

ELECTRO-WEAK MIXING

GWS model asserts that the three weak isospin currents couple to a weak Isotriplet of intermediate vector bosons, whereas the weak hypercharge current couples to an isosinglet intermediate vector boson.

wave functions representing the particles.

weak interactions - purdue university

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