# Weak Interaction

```					      Weak Interaction
Part 1
HT 2003

http://www-pnp.physics.ox.ac.uk/~weber/teaching

A Weber                                 07/04/2010
1
Introduction
   This lecture will give an introduction
to the theory of weak interaction.
   At the end you will know the basics
of
– nuclear decays
– weak particle decays
– effects of weak interactions at high
energies
– …
radioactive decays and this will be
put into the greater context.

A Weber                          07/04/2010
2
Agenda (part 1)
   Charged current weak interaction
– W exchange
– Fermi theory
(4 particle point-like interaction)
   V-A theory
   Nuclear beta decay
   Parity violation
   Test of V-A theory
– Neutrino helicity
– π and K decays
– W decays
   Unitarity violation at high energies

A Weber                           07/04/2010
3
The Standard Model
   Three generation of quarks and
leptons
   interaction via g, γ, Z, W±
   mass generation via Higgs

0, ½      -1, ½        +2/3, ½        -1/3, ½
υe         e              u             d
0 eV   0.511 MeV       0.3 GeV       0.3 GeV
0, ½     -1, ½         +2/3, ½        -1/3, ½
υμ        μ               c             s
0 eV   106 MeV         1.5 GeV       0.5 GeV
0, ½     -1, ½         +2/3, ½        -1/3, ½
υτ        τ               t             b
0 eV   1.78 GeV        175 GeV       4.7 GeV

g

γ

Z or W
H
A Weber                           07/04/2010
4
V-A Theory
   Charged Current (CC) weak inter-
action is due to W exchange
1

mW  q 2
2

   At low energies: 4 point interaction

1   G
      2
 F
mW    2

   current current interaction
H weak  GF j  j with j    (1   5 )

combination of vector (V) and
axial-vector (A) current
V             A    5

A Weber                                 07/04/2010
5
Non-relativistic limit
   Consider non-relativistic limit of
theory, e.g. nuclear beta decay:
    
   p    
         
 EM    0 
– V interaction
    0 component of nucleon current:
i   f  i 0 0  f  i f

  1,2,3 space components
 0               0
 i   f   i           i 0     0
  0  f
              f 
 Fermi transition (ΔS=0)

– A interaction
    0 component
i 5   f  0

    1, 2, 3 space component
i 5   f  i  f

    Gamow-Teller transition (ΔS=0,1)

A Weber                                            07/04/2010
6
Rate of weak nuclear decays
   Fermi’s golden rule
2
     4 2
h       M fi      dN
dE f

   Assume four point interaction (V):
M fi  G  e (r )   (r )  * (r )  n (r ) d 3r
*        *
p

   Electrons and neutrinos are free
particles leaving the nucleus:
 e,  L3 / 2 exp( i k e, r )
G
M fi 
L3     exp( i q r )  * (r )  n (r ) d 3 r
p

with       q  k e  k

   Typical beta decay
– q = 1 MeV and r = 5 fm
– exp( iqr ) = 1
– electron and neutrino take no orbital
momentum away
G
M fi 
L3     p
 * (r )  n (r ) d 3r

A Weber                                               07/04/2010
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    Selection rules (Fermi)
– we found ΔS = 0 and ΔL = 0
– therefore: ΔJ = 0 and ΔP = (-1)L
– allowed Fermi-transition

    Selection rules (Gamow-Teller)
G
M fi 
L3     p
 * (r )  n (r ) d 3r    ( A)

– λ=1.24 for nuclear beta decay
– we found ΔS = 0, 1 and ΔL = 0
– therefore: ΔJ = 0, 1 and ΔP = 0
    Mfi is a constant for allowed
transitions!
    Spectrum depends on phase space
only.

A Weber                                         07/04/2010
8
Beta Decay Spectrum
E f  E R  E  Ee
0 P pq
M R  me  E f  E R  0
Now calculate desity of states
dN

dE f
dN e  4p 2 dpV / h 3
dN  4q 2 dqV / h 3
d 2 N  4p 2 dpV / h 3 * 4q 2 dqV / h 3
Convert to dpdE f
q  ( E f  Ee ) / c
dq / dE  1 / c
16 2 2
dN  6 3 p ( E f  Ee ) 2 dpdE f
2

hc
Result :
64 2     2
 ( p )dp  7 3 M if p 2 ( E f  Ee ) 2 F ( Z , p )dpdE f
hc
Coulomb correction : Fermi function F ( Z , p )
Now define Kurie - function K :
 ( p)
K (Z , p) 
p 2 F ( Z , p)
Linear function of Ee

A Weber                                            07/04/2010
9
Curie-Plot

A Weber      07/04/2010
10
Inverse Beta Decay
    Fermi’s Golden Rule:
 4 2                  2   dN
W          M if
  h                        dE f
dN     dN dp f

dE f dp f dE f
4 p 2 dp f
        f

h3       dE f
E 2  p 2c 2  m2c 4
dp
2 E  2 pc 2
dE
dp    E    1
 2 
dE pc     c
dp f   1

dE f v f
beam flux    na
targets (unit area)  nb
interaction rate  σΦ  σna vi
4 2              2   4 p 2 dp f
W      M if                  f
3
  vi
h                            h         dE f
16 3                 2   p 2 dp f
  4 M if                     f

h                        vi dE f
2
GF                                                   h
                                     (vi  v f  c        1)
2
M       p2
                                                   2
A Weber                                                             07/04/2010
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   Fermi transitions
ΔJ = 0  M2=1
   G-T transitions
ΔJ = 1  M2=3 (Why? Spin!)
   Total cross section
(order of magnitude)
2
4GF
                p2

   Electron extreme relativistic
( E Q )
p         c       1 MeV
5
G F  10 GeV                           -2

   Total cross section:
  O(4 *1018 fm 2 )
(tiny! tiny! tiny! tiny!   tiny! tiny   !)

A Weber                                                07/04/2010
12
Discovery of the Neutrino
   Reines & Cowan (1956)
– Inverse beta decay
Display

e

– Positron annihilation (prompt)
e  e    
– Neutron capture (delayed)
after neutron became thermal
n  Cd  Cd *  Cd  n
– Where do you get anti-neutrinos from?

A Weber                                07/04/2010
13
What have we learned today?
        Standard Model (know before)
        V-A Theory
        Charged current interactions
        Types of nuclear beta decays
–     Fermi
–     Gamow-Teller
        Kinematics of allowed decays
        Inverse beta decay
        Discovery of the neutrino

        Next Lecture:
Experimental tests of V-A theory
1.    Parity violation
2.    W decay
3.    Pion decay
4.    Helicity of neutrino

A Weber                            07/04/2010
14
Experimental Tests of V-A Theory
        We constructed a V-A theory for
charged current weak interaction
with build in Parity violation

        Now test the V-A theory

1. Parity violation in nuclear beta decay
(Maximum violation! Why?)

2. W decay angular distribution
W   l  l

3. Pion decays to electron and muons
        and   e

4. Helicity of neutrino

A Weber                         07/04/2010
15
Parity Violation in W.I.
        What is parity
     
ˆ r  r i.e. (x,y,z)  ( x, y, z )
P
ˆ            
P (r )   (r )

        Eigenvalues
ˆ ˆ
P2  1                    Eigenvalue s p  1
        Parity conservation:
–     QM tells us:
ˆ
H,P  0 ,
d P
If       ˆ           then             0
dt
–     Therefore
1.        Observed states will have definite parity.
Why?
2.        Parity is conserved in interactions
        Examples of operators
ˆ
Ps  s             (scalar : a number)
ˆ
Pp   p           (psuedoscalar : spin * momentum)
ˆ
PV  V            (vector : momentum)
ˆ
PA  A            (axial vector : angular momentum  postion * velocity)

A Weber                                             07/04/2010
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Example
   Parity conservation and helicity

s
ˆ  p
H      
s p
This is the helicity operator!

   If parity is conserved, expectation
value of pseudo-scalar = 0
p 0
   Proof:
p   p
ˆ
  pP 2 

ˆ ˆ
   PpP  with          ˆ     ˆ
pP   Pp

  p
 p
A Weber                       07/04/2010
17
Structure of Weak Interaction
   Weak interaction is due to vector
current V and axial-vector current A:
V           A    5

   The interaction is V-A
V-A    (1   5 )

   It is equivalent to say:

Interaction is with left-handed
particles only!

   Because:
1
 LH     (1   5 )
2
This is a chirality-projector!

A Weber                                07/04/2010
18
Parity and V-A Theory
   W couples to left handed particles!
Weyl representation for gamma
matrices:
 1 0
1
1   5   
    

2             0 0
Projects left handed states!
   Massless limit (or high energies):
0              K
                
1              0               m
  N           N     with K 
0                1               E p
                
K              0
                

Helicity and chirality are the same!
1
1   5       and
1
1   5   0
2                              2

   Weak interaction generates net
helicity!  Parity violation!

A Weber                                    07/04/2010
19
Parity Violation
   A V-A current current interaction is
violating parity:

P V = -V

PA= A

(V-A)(V-A) = VV+AA -2AV

P (V-A)(V-A) = VV+AA+2AV

   Was originally build into theory but
not understood!

   Now is understood as a consequence
of W interaction to left handed
particles! (Not understood?)

A Weber                     07/04/2010
20
Is parity conserved?
   Yes:
– Strong interaction
– Electromagnetic interaction
– Gravity?
   Everybody expected it to be
conserved in weak interaction!
   First hint was the θ-τ puzzle!
  2   l  0 P  P2  1
  3   l  0 P  P3  1

   But both particle have same mass
and lifetime, i.e. must be the same
particle

   Parity is violated !!!!!
(direct test by Wu!)

A Weber                         07/04/2010
21
Experimental test of P-violation
   Measure decay spectrum of Cobalt
beta decay
60
Co Ni  e  e
60    

– 60Co at T=0.01 K
– all spins are parallel in external field
   Measure electron angular distribution
e-
v             θ
I ( )  1  cos( )
c                            J
   Now calculate:

p  cos  0

But, this is a pseudo-scalar and has to
be 0, if parity is conserved!

A Weber                              07/04/2010
22
Wu’s experiment

A Weber           07/04/2010
23
Parity and Nuclear states
   If parity is violated in CC weak
interaction, how can we have parity
selection rules in nuclear beta decay?
   Initial an final nuclear states are
eigenstates of the strong interaction!
Eigenstates of parity:
        
(r )  (r )
   Consider allowed decays:
       
I    * r i r d 3 r
f
                
   *  r i  r   d 3 r
f

*         3
  f i   f r i r d r
  f i I
   I=0, unless i   f  1

   No change in parity of nuclear wave
function!

A Weber                             07/04/2010
24
W Decay
   Charged current weak interaction
– couples to LH particles
– couples to RH anti-particles
   Extreme relativistic approach
(valid for W decay)
– LH = helicity minus (-)
– RH = helicity plus (+)
   W production and decay
– valence quarks dominate
u  d  W   l  l
   Spin structure                          l


d                                  u

l
A( )  d1,1 cos        1  cos  
1                1
2
 A   K 1  cos  
dN           2                2

d cos 

A Weber                                       07/04/2010
25
Pion and Kaon Decay
   Angular momentum conservation

                 

   Implications:
– muon is RH, but CC WI couples to left
handed particles
   In relativistic limit
– left handed = helicity –
– decay suppressed
   We therefore expect:
– pion decays mostly to muons and
– rarely to electrons
   Now:
Let’s calculate the decay rate

A Weber                            07/04/2010
26
Decay Kinematics
   Momentum conservation in CMS
      
p  p  p       and p   p  p

           
1
m  p 2  m     2 2
p
m  m
2    2
m  m
2    2

p                    and similar E 
2m                                     2m
   Relativistic calculation of Lorenz
invariant phase space (Lips)
          
Vd 3 p1    Vd 3 p2
dLips                        (2 ) 4  4 ( p0  p1  p2 )
(2 ) 2 E1 (2 ) 2 E2
3          3


V     p
 2
8 m0

A Weber                                    07/04/2010
27
Decay Dynamics
   W couples to left handed particles,
but we have a helicity (+) lepton.
Remember:
K
 
0                   m
  N       with K 
1                   E p
 
0
 
Lorenz invariant normalisation
    2E
N 2 (1  K 2 )  2 E
2 El
N 2

1 K 2
l

N2  2 E

   Use Weyl representation
1               1 0
1   5        to project left handed states
2               0 0

A Weber                                    07/04/2010
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   LH state                     K
 
1
 LH  1   5     N 0
2                   0
 
0
 Matrix element:                 
M    lLH
RH

ml
M  Nl2 N2 K 2                 K
2
with
El  pl
K2
 4 Ev El
1 K 2
 Cpl ml2
   Decay rate
  M Lips  Cpl2 ml2
2

   Decay ratios
(  e ) me2 m  me2
2
R            2 2
(   ) m m  m 2

 1.283 104                         (theory)
= (1.230  0.004) 104               (experiment)
(similar for K decays)
   Striking evidence for V-A form of
CC weak interaction
A Weber                                      07/04/2010
29
Helicity of the Neutrino
   Can we measure the helicity of the
neutrino?
   Consider the following decay:
e
152
Eu  152 Sm*   e  152 Sm   (960 KeV)

J=0                  1   1/2        0     1
   Conservation of angular momentum
– Neutrino spin is opposite to
direction of J in 152Sm*
– Spin of γ is parallel to J
152
Sm*  152 Sm  
J= 1            0    1
   Therefore:
– γ emitted forward has same polarisation
as 152Sm*
– γ emitted forward has same helicity
as νe
   Forward γ measures neutrino helicity

A Weber                                      07/04/2010
30
Neutrino Helicity (exp.)
   Goldhaber et al.
   Tricky bit: identify forward γ
   Use resonant scattering!
  152 Sm  152 Sm*  152 Sm  
   Measure γ polarisation with different
B-field orientations
152Eu

magnetic
field
Fe

γ           γ
Pb

152Sm    NaI         152Sm

PMT

A Weber                                 07/04/2010
31
Problems at High Energy
   Fermi theory is base on 4 point
contact interaction.
   Consider:
 e  e   e  e
2
GF s
 tot 

   Unitarity limit:
scattering probability > 1
2              h
 u .l .       with   
8             pcms

   At p=300 GeV CC WI violates the
unitarity limit!
   Solution: The W-Boson
1
 2
mW  q 2
A Weber                                    07/04/2010
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Summary (Part 1)
       We constructed a V-A theory for
charged current weak interaction
with build in Parity violation

       Different applications:

1. Nuclear Beta decay

2. Parity violation in nuclear beta decay

3. W decay angular distribution

4. Pion decays to electron and muons

5. Helicity of Neutrino

6. Unitarity violation at high energies

A Weber                             07/04/2010
33
Weak Interaction
(3 Families)
Part 2
HT 2003

A Weber                     07/04/2010
34
Content
        So fare we have only considered
weak interaction involving u and d
quarks and electrons and neutrinos.

        Now we will learn about:

1. 3 generation of leptons

2. Universal coupling strength

3. LEP data & number of generations.
Why are there 3 generations???

4. The s quark and Cabibbo’s theory

5. FCNC and the need for the c quark

6. b and t quark

7. Generalised theory of quark mixing

A Weber                            07/04/2010
35
Leptons
   Muon is heavier version of electron
– me = 0.511 KeV
– mμ = 106 MeV
Who ordered the muon?
   Experimental facts:
– Not seen:       e
– Normal decay:   e  e
– electron neutrino  muon neutrino
– neutrino  anti-neutrino
   One more lepton neutrino pair was
discovered (SLAC)
e  e      
  e e or  
Signature: electron and muon in one
event
   Tau neutrino discovered in 2001!
A Weber                        07/04/2010
36
Lepton Universality
   Leptons are all the same, just heavier
and unstable!
   Experimental test:
– Measure W boson decay ratios!
Br (W  e e )  Br (W    )  Br (W   )
Experimental data (Jan 2002)
Br (W  e e )  10.72  0.16%
Br (W    )  10.57  0.22%
Br (W    )  10.74  0.27%
– Measure tau decay ratio!
Br (  e e )  Br (    )
Experimental data
Br (  e e )  17.84  0.06%
Br (    )  17.37  0.06%
– Compare
  e e and   e  e

A Weber                                07/04/2010
37
Leptonic lepton decay
   Decay of tau/muon into electron +
neutrinos is 3 body decay
(like nuclear beta decay)
d
 Cp 2 ( E0  E )2
dp
   Extreme relativistic approx.: p=E
d
E0
C 5
   dp        dp Cp 2 ( E0  p ) 2       E0
dp     0
30
5
 m 
(  e )          (   e )
m  
 
(  e )
Br (  e ) 
total

   Requires precise determination of
Tau mass!!! (Threshold scan at BES)
   Results:
   291.3 fs (predicted)
   290.6 fs (measured)
A Weber                                   07/04/2010
38
   If CC WI is universal, can we predict
( ,  , , a,....)

   Count number of final states:
(     hadrons)  N c  (  e e  )
NC
Br (     hadrons) 
NC  2
 0.60    (theory)
 0.64    (experiment)

   Question:
Why is there a 4% difference?
(Can be used to measure αs(mτ)
A Weber                             07/04/2010
39
Neutrinos and Lepton Number
   Questions:
– Are muon neutrinos and
electron neutrino the same?
– Are neutrino and anti-neutrino
the same?
   Facts:
– In SM
 e  n  p  e         (possible)
 e  p  n  e        (possible)
 e  n  p  e (NOT possible)
– Experimental search
 e  37 Cl  37 Ar  e
radioactive Argon isotope was not seen!

– Neutrinoless double beta decay
76
32   Ge  76 Se  2e  (2 )
34

A Weber                                   07/04/2010
40
Neutrinos and Lepton Number
   Neutrinoless double beta decay
76
32   Ge  76 Se  2e
34

– Only possible, if neutrinos have a
Majorana component
e e
– The 2 anti-neutrinos could annihilate!

   Question:
– How can we distinguish SM and exotic
reaction?

   No evidence for
Majorana neutrinos yet!

   First evidence for lepton number
violation comes from neutrino
oscillations (later in course)!
A Weber                           07/04/2010
41
Lepton Number Conservation
Q          Le=1       Lμ =1              Lτ =1
0           νe         νμ                 ντ
1           e-         μ-                 τ-

   Anti-particles have opposite lepton
numbers!
   Example:    e       e
L = 1        0   1       0
Le =    0     1   0       -1
L =     1     1   1       -1

   Universal strength for all CC WI
vertices.
   All vertex factors g for the lνW
vertex are the same!

A Weber                           07/04/2010
42
Number of Families
   Are there any more generations of
particles? Maybe just too heavy to be
produced at colliders yet?
   Neutrino is always light = massless!
   Look for neutrinos!
   Studies at LEP:
e  e  Z  f  f   f  (e , e , u, d ,   ,....)

   There are only 3 generations!
Nν = 2.9841±0.0083
A Weber                        07/04/2010
43
Summary
   There are three generations of
fermions.
   They have a universal coupling
strength to the W
– W boson decay ratio
– Tau lepton decay ratios
   Lepton number is a conserved
quantum number.
Why?
   Neutrinos and anti-neutrinos are
different.

   Last Lecture:
–   WI and quarks
–   Cabibbo’s theory
–   FCNC
–   CKM matrix

A Weber                        07/04/2010
44
Weak Interaction and Quarks
        Compare interaction strength of
non-strange and strange decays:
1. Beta decay

2. Strange quark decays

in quark model s becomes u quark
explains selection rules:
   ΔQ=Δs
   ΔI = 1/2

A Weber                          07/04/2010
45
Cabibbo Theory
   Measure strength of weak interaction
for different processes:
      e   e  GF
14
O  14 N  e   e  Gud  0.95GF
K    0  e    e  Gus  0.05GF
   Cabibbo theory:
– quark mass eigenstates are eigenstates
of strong interaction but NOT of weak
interaction
– CC WI couple with universal strength to
rotated quark states.
u u
d  d cos c  s sin  c

– Ratio of Gus/GF=sin2θc
– Fit to many different reactions

 c  0.25
A Weber                               07/04/2010
46
Flavour Changing Neutral Currents
   Why don’t we see FCNC?
   Naively one would expect to see
FCNC, if NC couples to uu or dcdc !

d c  d cos  c  s sin  c
J NC  uu  d c d c  sc sc
0

 uu  d d cos 2  c  ss sin 2  c
( sd  sd ) cos  c sin  c
   GIM mechanism kills unwanted
FCNC (1970), but one has to
introduce a new quark doublet:
u c               d c  d cos  c  s sin  c
 ,                sc  d sin  c  s cos  c
 d c   sc 

A Weber                                 07/04/2010
47
FCNC
J NC  uu  d c d c  sc sc  cc
0

 uu  cc  (d d  ss )(cos 2  c  sin 2  c )
( sd  sd  sd  sd ) cos  c sin  c
 uu  cc  d d  ss
   Δs=0:
No FCNC for lowest order weak
interactions, but possible as higher
order corrections!

vanishes, if mu=mc
   Measured rate of transition allowed
prediction of mc!
   Discovery of the J/ψ in 1974 was
triumph for quark model and GIM!
A Weber                                  07/04/2010
48
GIM
   Other consequences
– In charm quark decays:
cs and cd are possible, because
tan 2 c  0.05
Find Kaons in decay of charmed
particles!
– Charm production in neutrino beams
1.)    d     u
2.)    d     c followed by c     s   

Signature for 2.) is a muon pair!

 1  cos 2  c
 2  sin 2  c

 2  tan 2  c
1

(plots)

A Weber                                 07/04/2010
49
Charm Decays
   Simple spectator model assumes c
quark decays as if it was a free
quark. (Neglecting strong interaction
effects.)

   Expect:
– Lifetime of D0 and D+ are the same
Experiment:       ( D0 ) 2.5  ( D )
( D  hadrons )  N c ( D  e  hadrons )
1
 Br ( D  e  hadrons )            0.2
2 N c
Experimental values:
Br ( D 0  e X )  6.8  0.3%
Br ( D   e X )  17.2  1.9%
   Simple spectator model works for D+
but not for D0!
– charm mass to low for reliable
perturbative predictions
– D0 has extra annihilation diagrams
A Weber                               07/04/2010
50
B Decays
   One more quark was discovered
very soon
– Discovery in    p  N      X
– Studied in   e  e  J /       

   Similar story for B decays!

   Simple spectator model works better:
– mb>mc
– α(mb)<α(mc)
– perturbation theory works better
1
Br ( B  l  hadrons)            0.11
3  2N c

A Weber                           07/04/2010
51
   Naïve expectation from universality
of CC WI
– Br ( B  l X )  11%         (expected)
Br ( B  l X )  10.2  0.9% (measured)
– Expect some phase space suppression in
charm and tau decays

   Discrepancy can be understood
– bound state effects

 ( B  )  (1.653  0.028) 1012 sec
 ( B 0 )  (1.548  0.032) 1012 sec
 (B )
 1.062  0.029
 (B )
0

A Weber                        07/04/2010
52
The 6 Quark Model
   After 5th quark was discovered:
–   FCNC in theory again!
–   expect 6th quark (bottom  top)
–   GIM like mechanism cancels FCNC
–   Top quark was discovered at FNAL
–   mt=174.35.1 GeV
   Generalise GIM mechanism to 3
generations:
– CC WI couples with universal strength
to rotated quark states!
d'     d 
 '      
 s  V  s            V is unitary CKM matrix
 b'    b
        
V + V=1
j
V jkV ji  0
*
(i  k )

V              Vkj  1  if real, orthogonal
2          2
jk
j              j

A Weber                                   07/04/2010
53
   real n x n Matrix
– ½ n(n-1) independent parameters
   n=2: 1 rotation angle
   n=3: 3 rotation angles
   But V is unitary matrix
– ½ n(n-1) mixing angles
– ½ n(n+1) complex phases
– absorb 2n-1 phases in definition of q- and q’-
fields
   n=3 case
– 3 mixing angles
– 1 complex phase (CP violation)
   No prediction. Obtain from experiment

1 0      0  c1 s1 0  1 0 0  0 c3               s3 
                                                  
V   0 c2 s2   s1 c1 0  0 1 0  1 0                0
 0  s c  0 0 1  0 0 ei  0  s                c3 
      2   2                      3                
with si  sin  i , ci  sin  i

   How can one obtain Cabibbo’s Theory?

A Weber                                07/04/2010
54
CKM Matrix
mixing matrix
   Different parameterisations
 Vud   Vus Vub 
               
V   Vcd   Vcs Vcb 
V      Vts Vtb 
 td            
    1                       A 3 (   i ) 
2

             2                                
                                              
1 
2
                                  A 2       
2
 A 3 (1    i )  A 2           1        
                                              
                                              
 0.9742  0.9757 0.219  0.226             0.002  0.005 
                                                         
  0.219  0.225 0.9734  0.9749 0.037  0.043 
 0.004  0.014        0.035  0.043 0.9990  0.9993 
                                                         
s1  0.219  0.226
s2  0.037  0.043
s3  0.002  0.005

A Weber                                 07/04/2010
55
Measurement of CKM angles
   Vud: Compare     0e e and    e e 
   Vus: Compare     0e e and K   0 e 
   Vcd: Measure di-muon production in
muon neutrino beams (see above).
   (Vcb)2+(Vub)2: from lifetime of b
quarks
    Vcb/Vub: from muon spectrum in b
decays b  u  W  u    
b  c  W  c   
– muon spectrum from b-u decay has
higher end point as mc>>mu
Vub  0.002  0.007
Vcb  0.032  0.045
   Off diagonal elements are small!
– Why????
– Prefered decay chain: bcs
– t quark: tb+Wb+l+υ (Emis)
A Weber                          07/04/2010
56
Unitarity Triangle (I)
    The CKM matrix V is a unitary
matrix! VV+ = 1
– Neatly summarize information in terms
of “the unitarity triangle”
– Unitarity of 3x3 CKM matrix applied to
the first and third columns yields
VudVub  VcdVcb  VtdVtb  0
*        *        *

– choose VcdV*cb
real = horizontal in complex plane
– Set cosines of small angels to unity
Vub  Vtd  sin 13Vcb
*                  *

– Unitarity Triangle
A                             A=(ρ,η)

α                              α
V*ub                Vtd

γ               β            γ                    β

C                            B   C                            B
s13 V*cb                           1
rescaled
A Weber                                    07/04/2010
57
Unitarity Triangle (II)
   Why all the effort?

A Weber                    07/04/2010
58
Summary
   Part I
– Weak interaction and nuclear decay
   selection rules
   decay spectra (Curie-plot)
– V-A theory
   non-relativistic limit
   ultra relativistic limit
   particle decays
– Experiments:
   discovery of neutrino
   parity violation
   Helicity of neutrino
   Part II
– 3 generations
– WI and leptons
   Lepton number conservation
– WI and quarks
   quark mixing
   CKM matrix
– Universality of WI

A Weber                                  07/04/2010
59

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