Section VIIISection VIIIThe Weak InteractionThe Weak Interaction

2

The Weak Interaction The WEAK interaction accounts for many decays in particle physics

Examples:

Characterized by long lifetimes and small cross-sections

eμ

τe-

μe-

νpenνμπ

ννeτννeμ

;

;

3

Two types of WEAK interaction:

CHARGED CURRENT (CC): W Bosons

NEUTRAL CURRENT (NC): Z0 Boson

The WEAK force is mediated by MASSIVE VECTOR BOSONS:

MW ~ 80 GeV

MZ ~ 91 GeV

Examples:

Weak interactions of electrons and neutrinos:

W

e

eνW e

eν

0Z

eν

eν0Z

e

e

4

Boson Self-Interactions In QCD the gluons carry “COLOUR” charge. In the WEAK interaction the W and Z0 bosons carry the WEAK

CHARGE W also carry EM charge

BOSON SELF-INTERACTIONS

W

W

0Z

W

W

W

W

W

W

W

W

0Z

0Z

W

W0Z

W

W

5

Fermi Theory

Weak interaction taken to be a “4-fermion contact interaction” No propagator Coupling strength given by the FERMI CONSTANT, GF

GF = 1.166 x 10-5 GeV-2

Decay in Fermi Theory

Use Fermi’s Golden Rule to get the transition rate

where Mif is the matrix element and (Ef) is the density of final states.

n

p

e

eν

GF

f2

fi EρM2Γ π

e-νpen

f

f dE

dNEρ

6

Density of Final States: 2-body vs. 3-body (page 54)

TWO BODY FINAL STATE:

Only consider one of the particles since the other fixed by (E,p) conservation.

THREE BODY FINAL STATE (e.g. decay):

Now necessary to consider two particles – the third is given by (E,p) conservation.

dEdΩ

2π

EdN 3

2

ee3

2e

3

22 dEdΩ

2π

EdEdΩ

2π

ENd

Relativistic (E ~ p)i.e. neglect mass of finalstate particles.

7

In nuclear decay, the energy released in the nuclear transition, E0, is shared between the electron, neutrino and the recoil kinetic energy of the nucleus:

Since the nucleus is much more massive than the electron /neutrino:

and the nuclear recoil ensures momentum conservation.

For a GIVEN electron energy Ee :

recoilνe0 TEEE

νe0 EEE

0ν dEdE

ee3

2e

ν3

2ν

ν0

dEdΩ2π

EdΩ

2π

E

dE

dN

dE

dN

8

Assuming isotropic decay distributions and integrating over ded gives:

Matrix Element

In Fermi theory, (4-point interaction)

and treat e, as free particles

e4

2e

2

e0e4

2e

2ν

e3

2e

3

2ν2

0

dE4π

EEEdE

4π

EEdE

2π

E

2π

E4π

dE

dN

3

2e

2

e02

fie

e4

2e

2

e02

fi

2π

EEEM

dE

dΓ

dE4π

EEEM2πdΓ

rdψψψψGM 3νepnFfi

rpirpie eψeψ e

rdψeψGM 3n

rpppFfi

νe

9

Typically, e and have energies ~ MeV, so << size of nucleus and

Corresponds to zero angular momentum (l = 0) states for the e and . ALLOWED TRANSITIONS

The matrix element is then given by

where the nuclear matrix element accounts for the overlap of the nuclear wave-functions.

If the n and p wave-functions are very similar, the nuclear matrix element

Mfi large and decay is favoured:

SUPERALLOWED TRANSITIONS

-210rp

1ψ,ψe

222

nuclear2F

3np

2Ffi MGrdψψGM

2

nuclearM

12 nuclearM

10

Here, assume (superallowed transition):

SARGENT RULE:

e.g. - and - decay (see later)

By studying lifetimes for nuclear beta decay, we can determine the strength of the weak interaction in Fermi theory:

12 nuclearM

3

50

2F

50

50

50

3

2F

E

0 e2e

2

e03

2F

2e

2

e03

2F

e

60π

EGΓ

4

E2

5

E

3

E

2π

GdEEEE

2π

GΓ

EEE2π

G

dE

dΓ

0

5 0Eτ

25- GeV10003.0136.1 FG

2Ffi GM

2

11

Beta-Decay Spectrum

Plot of versus is linear

2e

2

e03e

EEE2πdE

dΓ

2FG

e02ee

EEE

1

dE

dΓ

2ee E

1

dE

dΓ e0 EE

KURIE PLOT

keVEe

e33 eHeH ν 2

ee E

1

dE

dΓ

0E

12

Mass

m= 0 (neglect mass of final state particles)

End point of electron spectrum = E0

m 0 (allow for mass of final state particles)

Density of states (page 54)

me known, m small only significant effect is where Ee E0

2e

2e

2

e0

2ν2

e

2

e03

2F

eE

m1EE

m1EEE2π

G

dE

dΓ

dEdΩ

2π

EdN 3

2

dEdΩ

p

E

2π

pdN 3

2

13

eE

2ee E

1

dE

dΓ

KURIE PLOT

Most recent results (1999)Tritium decay:

me < 3 eV

If neutrinos have mass, me << me

Why so small ?

eE

2ee E

1

dE

dΓ

4

14

1

2

e

2e

2

e0

2ν

e0e

2e

Em1

EEm1EE

dE

dΓ

E

1

Experimentalresolution

14

Neutrino Scattering in Fermi Theory (Inverse Decay).

where Ee is the energy of the e in the centre-of-mass system and is the energy in the centre-of-mass system.

In the laboratory frame: (see page 28)

’s only interact WEAKLY have very small interaction cross-sections Here WEAK implies that you need approximately 50 light-years of water to stop a 1 MeV neutrino !

However, as E the cross-section can become very large. Violates maximum allowed value by conservation of probability at (UNITARITY LIMIT).

Fermi theory breaks down at high energies.

epnνe

n p

eeν

GF

π

sGσ

dΩ2π

EG2π

dE

dNM2πdσ

2F

3

2e2

F

2

fi

s

nνm2Es

243- cm10MeVin~ νEσ

GeV740s

Appendix F

15

Weak Charged Current: W Boson Fermi theory breaks down at high energy True interaction described by exchange of CHARGED W BOSONS Fermi theory is the low energy EFFECTIVE theory of the

WEAK interaction.

Decay:

e Scattering:

2W

2 mq

n

p

e

eν

GF

udd

udu

e

eν

n p

W

eν

GF

e

νμ

e

2W

2 Mq

1

W

Wg

Wg

eν

νμ

16

Compare WEAK and QED interactions:

CHARGED CURRENT WEAK INTERACTION At low energies, , propagator

i.e. appears as POINT-LIKE interaction of Fermi theory. Massive propagator short range

Exchanged boson carries electromagnetic charge.

FLAVOUR CHANGING – ONLY WEAK interaction changes flavour PARITY VIOLATING – ONLY WEAK interaction can violate parity

conservation.

e

2q

1γ

emg

emg μ

e

μ

egem

4

e2

e

2W

2 Mq

1

W

Wg

Wg

eν

νμ

WEAK QED

2W

2 Mq 2W

2W

2 M

1

Mq

1

fm002.0~RangeGeV4.80W

W M

1M

4π

gα

2W

W

17

Compare Fermi theory c.f. massive propagator

For compare matrix elements:

GF is small because mW is large.

The precise relationship is:

The numerical factors are partly of historical origin (see Perkins 4th ed., page 210).

The intrinsic strength of the WEAK interaction is GREATER than that of the electromagnetic interaction. At low energies (low q2), it appears weak due to the massive propagator.

-μ

μν

e

eν

GF

e

eν

W

-μ μνWg

Wg

2W

2 Mq F2W

2W G

M

g

2

G

8M

g F2W

2W

25 GeV10166.1andGeV4.80 FW GM

30

1

4π

gα0.65g

2W

WW and137

1

4π

eα

2

(see later to why different to )

FG

18

Neutrino Scattering with a Massive W Boson

Replace contact interaction by massive boson exchange diagram:

with where is the scattering angle.

(similar to pages 57 & 97)

Integrate to give:

Total cross-section now well behaved at high energies.

e

2W

2 Mq

1

W

Wg

Wg

eν

νμ

22W

2

4W

2Mq

g

32π

1

dq

dσ

22Esinq

2W

2W

2F

2W

2F

Msπ

MGσ

Msπ

sGσ

Appendix G

19

Parity Violation in Beta Decay

Parity violation was first observed in the decay of 60Co nuclei (C.S.Wu et. al. Phys. Rev. 105 (1957) 1413)

Align 60Co nuclei with field and look at direction of emission of electrons

Under parity:

If PARITY is CONSERVED, expect equal numbers of electrons parallel and antiparallel to

B e-

e-

B

P̂

e6060 νeNiCo

B

μμ;LprLpp;rr

B

J=5 J=4

20

Most electrons emitted opposite to direction of field

PARITY VIOLATION in DECAY

As 60Co heats up, thermal excitation randomises the spins

Polarized Unpolarized

T = 0.01 K

21

Origin of Parity Violation

SPIN and HELICITY

Consider a free particle of constant momentum, .

Total angular momentum, , is ALWAYS conserved. The orbital angular momentum, , is perpendicular to The spin angular momentum, , can be in any direction relative to The value of spin along is always CONSTANT.

p

SLJ

1hp

s

1hp

sp

psh

prL p

p

S

pS

“RIGHT-HANDED” “LEFT-HANDED”

Define the sign of the component of spin along the direction ofmotion as the

HELICITY

22

The WEAK interaction distinguishes between LEFT and RIGHT-HANDED states.

The weak interaction couples preferentially to

LEFT-HANDED PARTICLES

and

RIGHT-HANDED ANTIPARTICLES

In the ultra-relativistic (massless) limit, the coupling to RIGHT-HANDED particles vanishes.

i.e. even if RIGHT-HANDED ’s exist – they are unobservable !

60Co experiment:

ps

ps

B60Co 60Ni

J=5 J=4

eν

e

p

ep

23

Parity Violation

The WEAK interaction treats LH and RH states differently and therefore can violate PARITY (i.e. the interaction Hamiltonian does not commute with ).

PARITY is ALWAYS conserved in the STRONG/EM interactions

Example

PARITY CONSERVED PARITY VIOLATED

Branching fraction = 32% Branching fraction < 0.1%

P̂

u

u

u

u

uu

η

00u

u 0

21 LL000

P

000

11πPπPπP1P

0000Jπππη

0LL1;P 2!

u

u

u

u

ddη

LP

1πPπP1P

000Jππη

0L1;P

24

PARITY is USUALLY violated in the WEAK interaction

but NOT ALWAYS !

Example

PARITY VIOLATED PARITY CONSERVED

Branching fraction ~ 21% Branching fraction ~ 6%

s

u

u

u

ddK

d

W u

s

u

u

uK

0

d

W u

LP

1πPπP1P000JππK

0

0

0L1;P

21 LL-

P

11πPπPπP1P0000JπππK

0LL1;P 2!

25

The Weak CC Lepton Vertex

All weak charged current lepton interactions can be described by the W boson propagator and the weak vertex:

W Bosons only “couple” to the lepton and neutrino within the SAME generation

e.g. no coupling

Universal coupling constant gW

STANDARD MODELWEAK CCLEPTON VERTEX

W τ,μ,e

τμe ν,ν,ν

Wg

4π

gα

2W

W +antiparticles

τμe ντ

νμ

νe

μνWe

26

Examples:

e

eν

W

-μ μν

ν,ν,ν μe

W

τ,μ,e-

udd

udu

e

eν

n p

W

bc

cc

cB ψJ

ν

W μ

e-νpen

τ-

μ-

e- ντ,νμ,νeW

μeννeμ

μν

W

τ ν

μ

τμννμτ

νμJB -c

27

Decay

Muons are fundamental leptons (m~ 206 me).

Electromagnetic decay is NOT observed; the EM interaction does not change flavour. Only the WEAK CC interaction changes flavour. Muons decay weakly:

As can use FERMI theory to calculate decay width (analogous to decay).

γeμ

μeννeμ

-μ

μν

e

eν

GF

e

eν

W

-μ μνWg

Wg

2W

2 Mm

28

FERMI theory gives decay width proportional to (Sargent rule).

However, more complicated phase space integration (previously neglected kinetic energy of recoiling nucleus) gives

Muon mass and lifetime known with high precision.

Use muon decay to fix strength of WEAK interaction GF

GF is one of the best determined fundamental quantities in particle physics.

5m

5μ3

2F

μμ m

192π

G

τ

1Γ

s1000004.019703.2 6

25 GeV1000002.016632.1 FG

29

Universality of Weak CouplingCan compare GF measured from decay with that from decay.

From muon decay measure:

From decay measure:

Ratio

Conclude that the strength of the weak interaction is ALMOST the same for leptons as for quarks. We will come back to the origin of this difference

e

eν

W

-μ μνWg

Wg

udd

udu

e

eν

n p

W

25 GeV1000002.016632.1 FG

25- GeV10003.0136.1 FG

003.0974.0

F

F

G

G

Ccos

30

DecayThe mass is relatively large

and as

there are a number of possible decay modes.

Examples

Tau branching fractions:

GeV0003.0777.1 τm

,m,m,mm ρπμτ

μν

W

τ ν

μ

eν

W

τ ν

eu

W

τ ν

d

%)2.07.64(%)1.03.17(%)1.08.17(

hadronsτνντννeτ

-τ

-τe

-

31

Lepton UniversalityTest whether all leptons have the same WEAK coupling from measurements of the decay rates and branching fractions.

Compare

If universal strength of WEAK interaction, expect

are all measured precisely

Predict

Measure

SAME WEAK CC COUPLING FOR AND

e

eν

W

-μ μνWg

Wg

eν

W

τ ν

e

Wg

Wg

5μ3

2F

eμμ

m192π

GΓ

τ

1

%1.08.17 e)B(τ

53

2F

e m192π

G

0.178

1Γ

e)B(τ

1

τ

1

5

5μ

μ m

m

τ

178.0

μμ τmm ,, s1000004.019703.2

MeV3.00.1777MeV658.105

6

μ

μ

τmm

s1001.091.2 13 τ

s1001.091.2 13 τ

32

Also compare

IF same couplings expect:

(the small difference is due to the slight reduction in phase space due to the non-negligible muon mass).

The observed ratio is consistent with the prediction.

SAME WEAK CC COUPLING FOR e, AND

LEPTON UNIVERSALITY

eν

W

τ ν

e

Wg

Wg

μν

W

τ ν

μ

Wg

Wg

9726.0

τe-

τμ-

ννeτB

ννμτB

005.0974.0

τe-

τμ-

ννeτB

ννμτB

33

Weak Interactions of QuarksIn the Standard Model, the leptonic weak couplings take place within a particular generation:

Natural to expect same pattern for QUARKS, i.e.

Unfortunately, not that simple !!

Example:

The decay suggests a coupling

eνW

-e

μνW

μ

νW

τ

uW

d

cW

s

tW

b

μνμK suW

u W

μs

μν

K

34

Cabibbo Mixing AngleFour-Flavour Quark Mixing The states which take part in the WEAK interaction are ORTHOGONAL combinations of the states of definite flavour (d, s) For 4 flavours, {d, u, s and c}, the mixing can be described by a single parameter

CABIBBO ANGLE (from experiment)

Weak Eigenstates Flavour Eigenstates

Couplings become:

oC 13

s

d

cossin

sincos

s

d

CC

CC

uW

d

uW

d

uW

sCcosWg CsinWg

CC sincos sd

cW

s

cW

d

cW

sCcosWg Csin Wg

CC sincos ds

+

+

35

Example: Nuclear decay

Recall

strength of ud coupling Hence, expect It works,

Cabibbo Favoured

Cabibbo Suppressed

udd

udu

e

eν

n p

W

25 GeV1000002.016632.1 FG

25- GeV10003.0136.1 FG

CW cosg C

222cosM

FG

Ccos FF GG

o13cos16632.1136.1 oC 13

uW

dCcosWg

uW

sCsinWg

cW

sCcosWg

cW

dCsin Wg

C2cos 2

M

C2sin 2

M

36

Example:

coupling Cabibbo suppressed

Example:

Expect

Measure

is DOUBLY Cabibbo suppressed

μνμK

su

C2sin 2

M

u

W μs

μν

K

CsinWg Wg

πKD,πKD 00

cu

su0D K

d

W u

CcosWg

CcosWg

cu

du0D π

s

W uK

CsinWg-

CsinWg

0028.0cos

sin

C4

C4

πKD

πKD0

0

0008.00038.0

πKD0

37

CKM MatrixCabibbo-Kobayashi-Maskawa Matrix

Extend to 3 generations

Weak Eigenstates Flavour Eigenstates

Giving couplings

uW

d

cW

s

tW

b

b

s

d

V

b

s

d

CKM

uW

dudWVg

uW

susWVg

uW

bubWVg

1λλ

λ1λ

λλ1

105.001.0

05.0cossin

01.0sincos

23

22

λ

32

λ

CC

CC2

2

tbtstd

cbcscd

ubusud

CKM

VVV

VVV

VVV

V

Csin

38

The Weak CC Quark VertexAll weak charged current quark interactions can be described by the W boson propagator and the weak vertex:

W bosons CHANGE quark flavour W likes to couple to quarks in the SAME generation, but quark state mixing means that CROSS-GENERATION coupling can occur.

W-Lepton coupling constant gW

W-Quark coupling constant gW VCKM

STANDARD MODELWEAK CCQUARK VERTEX

+antiparticles

W

qqWVg

4π

gα

2W

W

tc,u,q

bs,d,q

39

Example:

gw Vud, Vcs, Vtb O(1)gw Vcd, Vus O()gw Vcb, Vts O()

t d, b u O()very small

(Log scale)

KKπ

νμDB-

μs0s

μ

bs

cs0

sBsD

μνW

cbWVg

Wg

sD

u

ss

cs

dW

csWVg

udWVg π s

ss

us -K

uW

usWVg

usWVg K

C4sin 4

W

2gM

Csin

C4cos 4

W

2gM C

4sin 4W

2gM

40

Summary

WEAK INTERACTION (CHARGED CURRENT) Weak force mediated by massive W bosons

Weak force intrinsically stronger than EM interaction

Universal coupling to quarks and leptons Quarks take part in the interaction as mixtures of the flavour

eigenstates Parity can be VIOLATED due to the HELICITY structure of the

interaction Strength of the weak interaction given by

from muon decay.

25 GeV1000002.016632.1 FG

GeV038.0423.80 WM

137

1α

30

1α emW cf