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Home , Weak interaction, Weinberg angle

Weak Outline

• Charged Leptonic Weak – Decay of the – Decay of the – Decay of the • Charged Weak Interactions of – Cabibbo-GIM Mechanism – Cabibbo-Kobayashi-Maskawa (CKM) • Neutral – Elastic - Scattering – Electron- Scattering Near the Z0 Pole • Electroweak – Chiral States – Weak and – Electro-Weak Mixing The Weak 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

E.g. neutron β decay

Weak decays often involve • 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)

Neutrinos would not be directly detected for 25 years: Reines & Cowan, using Savannah River nuclear reactor Decay of the Neutron Helicity Helicity is the component of the () angular momentum along momentum vector. For , the value is –1/2 or +1/2, depending on whether the spin S is antiparallel or parallel to direction of motion p

Solutions of the :

where

are helicity eigenstates:

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

with

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

obeys

Thus the following “” 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 and EM Interaction

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

Allowed QED vertices in the high energy limit

mirror reflection

Helicity is reversed under :

Equalities are due to parity Charged Leptonic Weak Interaction

The mediators of weak interactions are “intermediate vector ”, which are extremely heavy: 2 MW = 80.385 ± 0.015 GeV/c 2 MZ = 91.176 ± 0.002 GeV/c The 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 µ ε pµ = 0 Feynman Rules

Note: also 3 and 4 vertices exist 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 . The fundamental leptonicvertex is:

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

(the weak vertex factor)

“Weak ” (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:

also

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. Inverse Muon Decay

(lowest order diagram)

When the amplitude is:

Simplifies because MW = 80 GeV much larger than q < (100 MeV). Inverse Muon Decay Applying Casimir’s trick we find:

trace theorems trace theorems

using: Inverse Muon Decay

In the CM frame, and neglecting the 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

The amplitude:

As before: Decay of the Muon In the muon rest frame:

Let:

Plug in: Decay of the Muon

The decay rate given by Fermi’s 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 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 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 the electromagnetic fine structure constant! Weak Interactions

• The weak force is not inherently weak, we just live in a 2 2 world where typical interactions have q ≪ M W

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

• Weak Interactions are Parity violating

• Small CP-violation also observed Decay of the Neutron

(the same as in previous case) Decay of the Neutron

In the rest frame of the neutron:

We can’t ignore the mass of the electron. As before:

where: 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 Decay of the Neutron

where:

and: Decay of the Neutron

The range of E2 integral:

E is the electron energy

(exact equation) Decay of the Neutron

Approximations:

Expanding to lowest order: Decay of the Neutron Decay of the Neutron

(picture from Griffiths) Decay of the Neutron

where:

Putting in the numbers: Decay of the Neutron But the and neutron are not point-like particles.

Replacement in the vertex factor:

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

Another correction, the vertex carries a factor of is the Cabibbo angle.

cosθC = 0.97 Lifetime: 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 Decay of the Pion Decay of the Pion

The decay rate:

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

Experimental value: Decay of the Pion Decay of the Pion

Experimentally the ratio is 10-4 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:

Coupling within a generation:

There exist cross-generational coupling as: Charged Weak Interactions of Quarks

(Cabibbo, 1963) (extra cos or sin in the vertex factor) Leptonic Decays l is an electron or muon.

The quark vertex:

Using a previous result:

The branching ratio:

Corresponding to a Cabibbo angle: Leptonic Decays

()

(non-leptonic weak decay) 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 Semileptonic Decays

The decay rate: GIM Mechanism Cabibbo-GIM Mechanism The decay is allowed by Cabibbo theory. Amplitude: , far greater. GIM introduced the fourth quark c (1970). The couplings with s and d :

GIM = GLASHOW, ILIOPOULOS, MAIANI 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-Kobayash-Maskawa Matrix CKM is a generalization of Cabibbo-GIM for three generations of quarks.

The weak interaction quark generations 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. The CKM 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 or quark

Not allowed: Neutral Weak Interactions

• Like the the Z couples to f and f and NOT to f and f’. No flavour change. • Unlike the photon the Z couples differently to left and right handed fermions. • Like the W the Z coupling too violates parity but unlike the W not necessarily maximally. Neutral Weak Interactions

It doesn’t if we use physical states or Cabibbo rotated states. Neutral Weak Interactions First attempt to see weak : D. H. Perkins, Veltman. Failed.

Discovery of neutral currents: In an experiment at CERN with a Bubble Chamber. Experiments with μ beams obtained from π decays.

First process mediated by Z0

(Bubble chamber photograph at CERN, 1973) Neutral Weak Interactions 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 higher-order process (which corresponds to a far smaller cross section). Neutral Weak Interactions

The coupling to Z0 :

where :

(“Weak mixing angle” or “”) Neutral Weak Interactions Neutral vector and axial vector coupling in GWS model: Neutral Weak Interactions

( Z0 propagator )

When:

the propagator is simply:

The of the bosons are related by the formula: Elastic Neutrino- Neutral Weak Interactions

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

(E is the electron or neutrino energy)

Using: Neutral Weak Interactions

The total cross section:

Compare to:

(computed in the earlier)

(0.11 , experimental) Electron-Positron Scattering Near the Z0 Pole

Most neutral processes are “masked” by electromagnetic ones. 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: 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: Electron-Positron Scattering Near the Z0 Pole

problems at Z0 pole 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. 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: Electron-Positron Scattering Near the Z0 Pole Electron-Positron Scattering Near the Z0 Pole

Well below the Z0 pole:

Right on the Z0 pole: Electron-Positron Scattering Near 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) Chiral Fermion States

If the particle is massless:

helicity

where:

“projection operator”

Using: we can compute the following table è Chiral Fermion States 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: and Hypercharge

Negatively charged weak current

Positively charged weak current Weak Isospin and Hypercharge

We can express both by introducing the left handed doublet:

and the matrices: Weak Isospin and Hypercharge We could have a full “weak isospin” if only there is a third weak current, “neutral weak current”.

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

Weak hypercharge: Weak Isospin and Hypercharge

Everything could be extended to the other leptons and quarks:

Weak isospin currents:

Weak hypercharge current:

where Electroweak 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 .

wave functions representing the particles.