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Lecture 10Lecture 10Weak interactions, Weak interactions,

parity, helicityparity, helicity

SS2011SS2011: : ‚‚Introduction to Nuclear and Particle Physics, Part 2Introduction to Nuclear and Particle Physics, Part 2‘‘

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Weak decay of particlesWeak decay of particles

The weak interaction is also responsible for the β+-decay of atomic nuclei, whichinvolves the transformation of a proton to a neutron (or vice versa).

For free protons, this is energetically impossible (cf. the particle masses), butthe crossed reaction, the β−-decay process

is allowed and is the reason for the neutron's instability (with a mean life-time of 920 sec).

Without the weak interaction, the neutron would be as stable as the proton, which has a lifetime of τp >1030 years.

Here, one of the protons in the nucleus transforms into a neutron via

(1)

(2)evepn ++→ −

eenp ν++→ +

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Weak decay of particlesWeak decay of particles

The weak decay of π-- and μ-- :

The observed lifetimes of the pion and muon are considerably longer than those of particles which decay either through color (i.e. strong) or electromagnetic interactions:

i.e. particles decay by strong interactions in about 10-23 sec and through electromagnetic interactions in about 10-l6 sec (for example, π0 γγ).

Note: The lifetimes are inversely related to the coupling strength of these interactions

Moreover, pions are the lightest hadrons cannot decay by the strong interaction, additionally π- can not decay electromagnetically as π0 due to the charge.

The pion and muon decays provide evidence for an interaction with an even weaker coupling than electromagnetism weak interaction

(3)

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Fermi‘s theory for the weak interactionFermi‘s theory for the weak interaction

Fermi's explanation of β-decay (1932) was inspired by the structure of theelectromagnetic interaction:

The invariant amplitude for electromagnetic electron-proton scattering is

β+-decay process or its crossed formeenp ν++→ +enep ν+→+ −

where G is the weak coupling constant which remains to be determined by experiment; G is called the Fermi constant [~1/GeV2].

V-V: vector-vector coupling(or A-A axialvector-axialvector coupling)

(4)

(5)

(6)

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GSW theory for the weak interactionGSW theory for the weak interaction

• Problems with Fermi‘s picture: V-A-coupling doesn't describe the weak interaction very well, especially at high energies

•1960s - GSW-theory: Sheldon Glashow, Abdus Salam and Steven Weinberg propose the theory of the electroweak interaction by the exchange of vector bosons with huge masses (~100 GeV):

W-,W+ and Z0 bosons

• W-bosons – discovered experimentally in 1973 at CERN

eenp ν++→ +evepn ++→ −

eve ++→ −−μνμ

β-decay processes μ-decay

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GSW theory for the weak interactionGSW theory for the weak interactionPhoton propagator: Vector boson propagator:

Feynman diagram for

Matrix element:

For small q:

(7)

(8)

(9)

From (6) and (9)

g is a new coupling constant – without dimention

GeV100Meg W ≈⇒≈

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The experiment studied β-transitions of polarized cobalt nuclei:

The nuclear spins in a sample of 60Co were aligned by an external magnetic field, and an asymmetry in the direction of the emitted electrons was observed. The asymmetry was found to change sign upon reversal of the magnetic field such that electrons prefer to be emitted in a direction opposite to that of the nuclear spin.

The observed correlation between the nuclear spin and the electron momentum is explained if the required JZ = 1 is formed by a right-handed antineutrino, , and a left-handed electron eL

Parity violation by the weak interactionParity violation by the weak interaction

1956 – Lee and Yang prove experimentally that the weak interaction violates parity !

right-handed:

psrr

↑↑

left-handed: psrr

↓↑

sr

pr

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Parity violation by the weak interactionParity violation by the weak interaction

The cumulative evidence of many experiments is that indeed only (and ) are involved in the weak interactions. The absence of the ‚mirror image‘ states, and , is a clear violation of parity invariance.

Also, charge conjugation, C- invariance is violated, since C transforms a state into a state.

However, the form leaves the weak interaction invariant under thecombined CP-operation.

For instance,

The operator (1-γ5)/2 automatically selects a left-handed neutrino (or a right-handed antineutrino).

V-A (vector-axial vector) structure of the weak current

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Parity Parity

Consider the parity transformation:(10)

Transformed Dirac spinors

should follow the Dirac equation:

(11)

(12)

By multiplying (12) from the left side by we obtain:(13)

Solution of (13): Parity operator

(14)

(SP is the parity operator)

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parity transformation

Particles have a positive parity !

ParityParity

Parity transformation of Dirac spionors:

is replaced by

(15)

Consider a particle at rest:(16)

or

(17)

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ParityParity

Consider an anti-particle at rest:

(18)

parity transformation

Anti-particles have a negative parity !

(19)

or

Particles and anti-particles have an opposite parity!

Example:

In the quark model : quarks u,d,s,c,b have a Parity P = +1

antiquarks have a Parity P = -1b,c,s,d,u

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Helicity Helicity

Introduce the helicity λ:

Note: property of the γ5 matrix :

(20)

(21)

Helicity = projection of spin on the direction of motion

prsr

q

prsr

qZ

Right-handed: λ >0 Left-handed: λ <0psrr

↑↑ psrr

↓↑

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Helicity Helicity

Dirac spinors for positive and negative helicity:

has

has

(22)

For E >> m :

(23)

(24)

positive helicity

negative helicitypsrr

↓↑

psrr

↑↑

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Helicity Helicity

(26)

(27)

General properties of projection operators:

With the help of the projection operators PL and PR the spinor can be decomposed into R and L parts:

(25)

Operator is the projection operator for negative helicity

Operator is the projection operator for positive helicity

(28)

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VV--A form of the weak currentA form of the weak current

Consider the process

From experiment νe must be left-handed, i.e.

)(u)1(21)(u 5L νγν −= (29)

(30)Current

Consider the current (30) under Lorentz transformations:

(31)

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VV--A form of weak currentA form of weak currentFrom (30) current in the neutrino-muon vertex has two terms:

1) vector currentwhich under Lorentz transformation (31) transforms as

2) axialvector current

which under Lorentz transformation (31) transforms as

(32)

(33)

Thus, the parity transformation (or space reflection ) leads to

(34)

The neutrino-muon vertex has two terms: (V-A) coupling

(35)

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Projection operator sorts antineutrinos with positive helicity

Helicity of antineutrinosHelicity of antineutrinos

Helicity of antineutrinos

We know from experiment that antineutrinos are right-handed positive helicity

Dirac spinors for antiparticles with positive and negative helicity read:

(36)

(37)

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Helicity of electronsHelicity of electrons

Helicity of the electron

The currect for the neutrino-electron vertex:

(38)

Electron with positive helicity reads as:

for

(39)

0uu γ+≡

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LeftLeft--rightright--handed assimetryhanded assimetry

Electron with negative helicity:

for

Asymmetry in left-right-handed electron production:

(40)

(41)

electron has preferentially a negative helicity (i.e. left-handed)

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The reason: in the decay an electron must have a positive helicity, i.e. to be right-handed (due to angular momentum conservation), whereas the V-A theory (41) shows that it is suppressed by a factor of (1-υ/c),

i.e. from V-A theory: particles have a negative helicity – left-handed (l.h.)antiparticles have a positive helicity – right-handed (r.h)

Pion weak decayPion weak decay

Pion weak decay in the electron channel

is suppressed by a factor of 1.3.10-4 relative to its decay in the muon channel

whereas from phase-space arguments it should be the opposite since the muon is muchheavier than the electron !

μνμπ +→ −−

ee νπ +→ −−

ee νπ +→ −−

before decay after decay

direction of spin

direction of momentum

ee νπ +→ −−

Pion weak decay is an experimental check of the V-A theory!

ps rr↑↑

ps rr↓↑

(r.h)(r.h) !

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Pion weak decayPion weak decay

Feynman diagram for

(42)

ee νπ +→ −− Matrix element:

π

μπ m

p~j

Pion currect – from Klein-Gordan equation for spinless particles:

(43)−π

GF is the coupling constant for the 4-point like fermion vertex )eud( eν−

fπ is the pion decay constant: fπ =93 MeV

In the rest frame of pion: )0,m(pr

πμ =

(44)

For the decay in the rest frame of the pion

(45)Lips= Lorentz invariant phase spaceΓ - decay width

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Pion weak decayPion weak decay

Antineutrino has a positive helicity, i.e. right-handed

(46)

z

(r.h)

(r.h) ksrr

↑↑

kdirectionzr

↓↑−

Electron for this process must have a positive helicity, too,i.e. to be right-handed u1 Dirac spinor:ps

rr↑↑

(47)

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Pion weak decayPion weak decay

Indeed, by considering

one finds that

(48)

the process is only possible if the electron has a positive helisity

ee νπ +→ −−

(49)

(50)

Matrix element:

Lorentz invariant phase space:

(51)

In the rest frame of pion or CMS ee ν+−

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Pion weak decayPion weak decay

By substituting (50) and (51) into (45), we find

(52)

where p is the momentum of the electron in the rest frame of the pion:

(53)

(54)

In a similar way (by replacing me by mμ ), one can calculate the decay width for

the ratio

decay width for

ee νπ +→ −−

μνμπ +→ −−

(55)

ee νπ +→ −−Pion decay strongly suppressed relative to μνμπ +→ −−