Section VIII
The Weak Interaction
The Weak Interaction
 The WEAK interaction accounts for many decays in particle physics
Examples:
μ -  e  ν e ν μ ; τ -  e  ν e ντ
π   μ  ν μ ; n  pe  ν e
 Characterized by long lifetimes and small cross-sections
181
 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:
e
W
νe
νe
νe
W
e
Z0
νe
e
Z0
e
182
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
Z
W
0


W
W
W
W
Z0
W
W
0
Z
W
W

W  Z0
W 
W

W
W

183
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
p
b Decay in Fermi Theory
n  pe ν e
-
GF
n
e
νe
Use Fermi’s Golden Rule to get the transition rate
Γ  2 π M fi ρE f 
2
dN
ρE f  
dE f
where Mif is the matrix element and r(Ef) is the density of final states.
184
Density of Final States: 2-body vs. 3-body
TWO BODY FINAL STATE:
E2
dN 
dΩ dE
3
2π 
(page 54)
Relativistic (E ~ p)
i.e. neglect mass of final
state particles.
Only consider one of the particles since the other fixed by (E,p)
conservation.
 THREE BODY FINAL STATE (e.g. b decay):
2
2
E
E
e
d 2 N   3 dΩ dE
dΩe dEe
3
2π 
2π 
Now necessary to consider two particles – the third is given by (E,p)
conservation.
185
In nuclear b decay, the energy released in the nuclear transition, E0, is
shared between the electron, neutrino and the recoil kinetic energy of
the nucleus:
E0  Ee  Eν  Trecoil
Since the nucleus is much more massive than the electron /neutrino:
E0  Ee  Eν
and the nuclear recoil ensures momentum conservation.
For a GIVEN electron energy Ee :
dEν  dE0
Eν2
Ee2
dN dN


dΩν
dΩe dEe
3
3
dE0 dEν 2π 
2π 
186
Assuming isotropic decay distributions and integrating over dWedW gives:


2
E
E E
E0  Ee Ee2
dN
2 E
 4π 
dEe 
dEe 
dEe
3
3
4
dE0
4π
4π
2π  2π 
2 2
2 E  E  E
0
e
e
dΓ  2π M fi
dEe
4
4π
2
ν
2
e
2
ν

2
e
4

2
2
2 E E
E
dΓ
0
e
e
 M fi
dEe
2π 3
Matrix Element
In Fermi theory,
M fi  GF  ψ n ψ p ψ e ψ ν d 3 r (4-point interaction)
and treat e,  as free particles
ψe  e
 M fi  GF  ψ e

p
 
ipe r
  
  pe  pν r
ψ  e
 
ip   r

ψn d r
3
187
 
-2
 Typically, e and  have energies ~ MeV, so p  r  10<< size of
nucleus and
ψe , ψ  1
Corresponds to zero angular momentum (l = 0) states for the e and .
 ALLOWED TRANSITIONS
The matrix element is then given by
M fi
2
2
2
 G  ψ ψ n d r  GF2 M nuclear
2
F

p
3
2
where the nuclear matrix element M nuclear accounts for the overlap of
the nuclear wave-functions.
 If the n and p wave-functions are very similar, the nuclear matrix
element
2
M nuclear  1
 Mfi large and b decay is favoured:
 SUPERALLOWED TRANSITIONS
188
Here, assume M nuclear  1(superallowed transition):
2


2
2
2
dΓ GF2
2
M

G
 3 E0  Ee Ee
fi
F
dEe 2π
5
5
5
2


2
E
E
E
G
GF2 E0
2
0
0
0
F
2 
Γ  3  E0  Ee Ee dEe  3  
0
4 
5
2π  3
2π
GF2 E05
Γ
60π 3


SARGENT RULE:
τ  E05
e.g. m- and t- decay (see later)
By studying lifetimes for nuclear beta decay, we can determine the
strength of the weak interaction in Fermi theory:
GFb  1.136  0.00310-5 GeV 2
189
Beta-Decay Spectrum


2 2
dΓ G 2F
 3 E0  Ee Ee
dEe 2π
Plot of
dΓ 1
dEe Ee2


versus E0  Ee is linear

dΓ 1
 E0  Ee
2
dEe Ee
dΓ 1
dEe Ee2
3

KURIE PLOT
H  3He  e  ν e
E0
Ee keV 
190
 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

E2
dN 
dΩ dE
3
2π 



p2 E
dN 
dΩ dE (page 54)
3
2π  p
2
dΓ GF2
mν2
2
 3 E0  Ee Ee 1 
dEe 2π
E0  Ee


2
1
me2
Ee2
me known, m small  only significant effect is where Ee  E0
191
1
4
 m2
  m2 
1 dΓ
ν
e
 E0  Ee 1 
2  1 
2
2
E
Ee dEe
E

E
e


0
e



KURIE PLOT

dΓ 1
dEe Ee2


1
4
dΓ 1
dEe Ee2
Experimental
resolution
Ee
Ee
Most recent results (1999)
Tritium b decay:
me < 3 eV
If neutrinos have mass, me << me
Why so small ?
192
Neutrino Scattering in Fermi Theory (Inverse b Decay).
νe
ν  n  p  e
e
e
dσ  2π M fi
GF2 s
σ
π
2
2
dN
2 Ee
 2π GF
dΩ
3
dE
2π 
Appendix F
GF
n
where Ee is the energy of the e in the centre-of-mass system and
energy in the centre-of-mass system.
p
s is the
In the laboratory frame: s  2Eν mn (see page 28)
 σ ~ Eν in MeV 10- 43 cm 2
 ’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
s  740 GeV
(UNITARITY LIMIT).
 Fermi theory breaks down at high energies.
193
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 q 2  mW2 EFFECTIVE theory of the
WEAK interaction.

b Decay:
m
e
p
GF
n
e
Scattering:
e
νe

νe
u
n d
d
u
d p
u
W
e
gW
μ
νm
e
νe
νe
1
q 2  M W2
W
GF
νm

gW
μ
194
Compare WEAK and QED interactions:
e
WEAK
αW 
2
W
g
4π ν
m
gW
W
νe
1
q 2  M W2

gW
μ
e
QED

μ
e
g em
g em  e
1
q2
γ
g em
μ
e2

4
CHARGED CURRENT WEAK INTERACTION 1
1
2
2
 At low energies, q  M W , propagator q 2  M 2   M 2
W
W
i.e. appears as POINT-LIKE interaction of Fermi theory.
 Massive propagator  short range
M W  80.4 GeV  Range 
1
~ 0.002 fm
MW
 Exchanged boson carries electromagnetic charge.
 FLAVOUR CHANGING – ONLY WEAK interaction changes flavour
 PARITY VIOLATING – ONLY WEAK interaction can violate parity
conservation.
195
Compare Fermi theory c.f. massive propagator
νμ
μ-
GF
μ
gW
-
e
W
νe
For q 2  M W2 compare matrix elements:
GF is small because mW is large.
The precise relationship is:
2
W
2
W
g
M
νμ
gW
e
νe
 GF
gW2
GF

8M W2
2
The numerical factors are partly of historical origin (see Perkins 4th ed., page 210).
M W  80.4 GeV and GF  1.166 105 GeV 2
 gW  0.65 and
gW2
1
αW 

4π 30
(see later to why
b
different to G F )
e2
1
α

4π 137
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.
196
Neutrino Scattering with a Massive W Boson
Replace contact interaction by massive boson exchange diagram:
e
gW
1
q 2  M W2
W
νm
νe
gW
μ
gW4
dσ
1

2
dq
32π q 2  M W2


2
with q  2Esin  2 where q is
the scattering angle.
(similar to pages 57 & 97)
Integrate to give:
GF2 s
σ
π
GF2 M W2
σ
π
s  M W2
Appendix G
s  M W2
Total cross-section now well behaved at high energies.
197
Parity Violation in Beta Decay
Parity violation was first observed in the b decay of 60Co nuclei
(C.S.Wu et. al. Phys. Rev. 105 (1957) 1413)
60
Co60Ni  e   ν e
J=5
Align 60Co nuclei with
electrons
m
B

B
J=4
field and look at direction of emission of
e
P̂
B
m
eUnder parity:

 

r  r ; p 
p

 


L  r  p  L; μ  μ
If PARITY is CONSERVED,expect equal numbers of electrons
parallel and antiparallel to B
198
Polarized
Unpolarized
T = 0.01 K
As 60Co heats up,
thermal excitation
randomises the spins
Most electrons emitted opposite to direction of field
 PARITY VIOLATION in b DECAY
199
Origin of Parity Violation
SPIN and HELICITY

p
Consider a free particle of constant momentum, .
  
 Total angular momentum, J  L  S , is ALWAYS conserved.
  

 The orbital angular momentum, L  r  p , is perpendicular to p


S
p
 The spin angular momentum, , can be in any direction relative to


 The value of spin S along p is always CONSTANT.
Define the sign of the component of spin along the direction of
motion as the
 
sp
h 
p
HELICITY

s

p
h  1
“RIGHT-HANDED”

s

p
h  1
“LEFT-HANDED”
200
The WEAK interaction distinguishes between LEFT and RIGHT-HANDED
states.
The weak interaction couples preferentially to

s
LEFT-HANDED PARTICLES
and
RIGHT-HANDED ANTIPARTICLES

s

p

p
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

p
experiment:
νe
B
60Co
J=5
60Ni
J=4


pe
e
201
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 P̂ ).
PARITY is ALWAYS conserved in the STRONG/EM interactions
Example
u 0
u
u
η
u
0

u
u
u
ηπ π π
0  0 0  0 
L
L
 1 P π 0 P π 0 P π 0  1 1  1 2
0
J
P
P
0

u
0
0
   
P  1; L!  L2  0
PARITY CONSERVED
Branching fraction = 32%
η
u
u
JP
P
u 
d
d 
u
η  π π 
0  0 0 
L




1 P π P π 1
  
P  1; L  0
PARITY VIOLATED
Branching fraction < 0.1%
202
PARITY is USUALLY violated in the WEAK interaction
but NOT ALWAYS !
Example
K

u
u
s
u
W
J
P
P
0
K
s
W
u 

d
K   π 0π 
0
0 0 
L
 1 P π 0 P π   1
  
P  1; L  0
PARITY VIOLATED
Branching fraction ~ 21%
u
JP
P
u 
d
d 
u
u 

d
K   π  π π 
0
0  0 0 
L
L
 1 P π  P π - P π   1 1  1 2
   
P  1; L!  L2  0
PARITY CONSERVED
Branching fraction ~ 6%
203
The Weak CC Lepton Vertex
All weak charged current lepton interactions can be described by the
W boson propagator and the weak vertex:
νe , ν μ , ντ


e , μ ,τ

gW
W
gW2
αW 
4π
STANDARD MODEL
WEAK CC
LEPTON VERTEX
+antiparticles
 W Bosons only “couple” to the lepton and neutrino within the
SAME generation




e   μ   τ 
     
 νe   ν μ   ντ 
e.g. no We νμ coupling
 Universal coupling constant gW
204
Examples:
μ

τ   μ  ν μ ντ

μ  e ν e νμ
νμ
-
W

W
e
νe
W   e- ν e , μ - ν μ , τ - ν τ
W
νe , νμ , νt
Bc  J  μ - ν m
u
d p
u
W
μ
νμ
e- , μ  , τ 
n  pe - ν e
u
n d
d
νt
τ
e
νe
Bc c
b
c J ψ
c
W
μ
νm
205
m Decay
 Muons are fundamental leptons (mm~ 206 me).


 Electromagnetic decay μ  e γ is NOT observed; the EM
interaction does not change flavour.
 Only the WEAK CC interaction changes flavour.
 Muons decay weakly: μ   e ν eνμ
μ
gW
-
W
νμ

gW
e
νe
νμ
μ-
GF
e
νe
As mm  MW  can use FERMI theory to calculate decay width
(analogous to b decay).
2
2
206
5
FERMI theory gives decay width proportional to mm (Sargent rule).
However, more complicated phase space integration (previously
neglected kinetic energy of recoiling nucleus) gives
1
GF2
5
Γμ 

m
μ
τ μ 192π 3
 Muon mass and lifetime known with high precision.
tm  2.19703  0.00004106 s
 Use muon decay to fix strength of WEAK interaction GF
GF  1.16632  0.00002105 GeV 2
 GF is one of the best determined fundamental quantities in particle
physics.
207
Universality of Weak Coupling
Can compare GF measured from m decay with that from b decay.
gW
-
νμ
μ
W
gW
e
νe
u
n d
d
u
d p
u
W
From muon decay measure:
GFm  1.16632  0.00002105 GeV 2
e
νe
From b decay measure:
GFb  1.136  0.00310-5 GeV 2
Ratio
GFb
 0.974  0.003
m
GF
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 cos C 
208
t Decay
The t mass is relatively large
m τ  1.777  0.0003 GeV


m τ  mμ , mπ , m ρ ,
and as
there are a number of possible decay modes.
Examples
νt
τ
W

e
νe
νt
νt τ 
τ
W

μ
νμ
W
d
u
Tau branching fractions:
τ -  e  ν e ντ
τ -  m  ν m ντ
τ -  hadrons
(17.8  0.1%)
(17.3  0.1%)
(64.7  0.2%)
209
Lepton Universality
Test whether all leptons have the same WEAK coupling from measurements of
the decay rates and branching fractions.
gW
gW

νt
νμ
τ
Compare μ


W
W
e

g
gW
e
W
νe
νe
1
GF2
5
 Γ μ e 
m
μ
τμ
192π 3
1
1
1
GF2
5

Γ te 
m
t
τ t B(τ  e)
0.178 192π 3
If universal strength of WEAK interaction, expect
mμ5
tt
 0.178 5
τμ
mt
mμ , mt , τ μ are all measured precisely
Predict
Measure
τ t  2.91  0.0110 13 s
τ t  2.91  0.0110 13 s
B(τ  e)  17.8  0.1%
mμ  105.658 MeV
mt  1777.0  0.3 MeV
τ μ  2.19703  0.00004 10 6 s
SAME WEAK CC COUPLING FOR m AND t
210
Also compare
τ

gW
νt
W
gW
τ


e
νe
IF same couplings expect:


gW
νt
W
μ
νμ
gW


B τ -  μ  ν μ ντ
 0.9726

B τ  e ν e ντ
(the small difference is due to the slight reduction in phase space due
to the non-negligible muon mass).
B τ -  μ  ν μ ντ
 0.974  0.005 is consistent with the
The observed ratio

B τ  e ν e ντ
prediction.




 SAME WEAK CC COUPLING FOR e, m AND t
LEPTON UNIVERSALITY
211
Weak Interactions of Quarks
In the Standard Model, the leptonic weak couplings take place within a
particular generation:τ
μ
e
W
W
νe
νμ
Natural to expect same pattern for QUARKS, i.e.
d
s
W
W
b
W

νt
W
u
c
Unfortunately, not that simple !!
Example:

The decay K   μ  νμ suggests a W u s coupling
t
νμ
K
u
s
W
μ
212
Cabibbo Mixing Angle
Four-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 C  13o (from experiment)
Weak Eigenstates
Couplings become:
d cos C  s sin C
W
s cos C  d sin C
W
 d   cos C
   
 s    sin C
d

u
s

c
sin C  d 
 
cos C  s 
gW cos C
gW sin C
d
+
W

gW cos C
u
s
W
c
W
 gW sin C
+

Flavour Eigenstates
W
s
u
d
c
213
u
n d
d
b
-5
2
W
Recall GF  1.136  0.00310 GeV
GFm  1.16632  0.00002105 GeV 2
u
d p
u
Example: Nuclear b decay




strength of ud coupling
G 
b 2
F
2
gW cos C
Cabibbo Favoured
M  cos C
2
W
Cabibbo Suppressed
M  sin C
2
C  13o
d
W
gW cos C
W

gW sin C
2
νe
g W cos C
 M  cos 2 C
b
m
Hence, expect GF  GF cos C
o
1
.
136

1
.
16632

cos
13
It works,
2
e
u
s
u
 gW sin C
W
s
c
d
c
214
Example: K   μ  νμ
u s coupling  Cabibbo suppressed
M  sin 2 C
2
0
 
0
 
Example: D  K π , D  K π
u 
D0 u
K
s
c
gW cos C
W
Expect
Measure

K
u
gW sin C
W
s
D0 u
c
- gW sin C
u 
gW cos C

d
D 0  K  π   sin 4 C

0
 
D  K π  cos 4 C
 0.0028
0.0038  0.0008
gW
W
gW sin C
D 0  K  π is DOUBLY Cabibbo suppressed
νμ
μ
u π
d
u 
K
s
215
CKM Matrix
Cabibbo-Kobayashi-Maskawa Matrix
Extend to 3 generations
d
W
W
Weak Eigenstates
VCKM
c
u
b
s
W
t
 d 
d 
 
 
 s   VCKM  s  Flavour Eigenstates
 b 
b
 
 
Vud Vus Vub   cos C

 
  Vcd Vcs Vcb     sin C
V
 
 td Vts Vtb   0.01
sin C 0.01 1  λ2
 
cos C 0.05     λ

 0.05
1   λ 3
2
Giving couplings
gW Vud
W
d
u
gW Vus
W
s
gW Vub
W

u
λ3 

2
λ 

1
λ
λ2
1 2
 λ2
  sin C
b
u
216
The Weak CC Quark Vertex
All weak charged current quark interactions can be described by the W
boson propagator and the weak vertex:
q  d, s,b
q  u,c,t
gW Vqq
gW2
αW 
4π
W
STANDARD MODEL
WEAK CC
QUARK VERTEX
+antiparticles
 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
W-Quark coupling constant
gW
gW VCKM
217
gw Vud, Vcs, Vtb
gw Vcd, Vus
gw Vcb, Vts
(Log scale)
t d, b u
very small
Example: Bs0  Ds μ  ν μ
  π K K 
s   s
s
Ds Ds
Bs0
c
b
c
gW Vcb

νμ
μ
W gW
M  g sin C
2
4
W
4
gW Vcs
W

d
gW Vud
u
M  g cos C
2
s

s
s

s
4
W
4
π
O(1)
O()
O(2)
O(3)
  sin C
s K
u
gW Vus
W

gW Vus
u

K
s
M  gW4 sin 4 C
2
218
Summary
WEAK INTERACTION (CHARGED CURRENT)
 Weak force mediated by massive W bosons
M W  80.423  0.038 GeV
 Weak force intrinsically stronger than EM interaction
1
1
αW 
cf αem 
30
137
 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
GFm  1.16632  0.00002105 GeV 2
from muon decay.
219