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
      – …
    You already know about the
     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
                                                                    7
     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
                                                                                     11
    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


                         water and cadmium-chloride
      – 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
                                                                            16
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
                                                                28
    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
                                                        32
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
    Rabbi’s unanswered question:
     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
Hadronic Tau Decay
    If CC WI is universal, can we predict
     hadronic decay ratio?
         hadrons
         ( ,  , , 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?
    Answer:
     QCD radiative correction!
     (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
      – Tau/muon relative lifetime
    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 )
      – Hadronic decay width:
          ( 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
      – QCD radiative corrections
      – bound state effects

    Lifetime
      ( 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
    Cabibbo-Kobayashi-Maskawa quark
     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
        long b lifetime
    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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