Particle Physics II -The higgs mechanism Lecture 5 review fermion by nml23533

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									Particle Physics II – The higgs mechanism

Lecture 5: review fermion masses & quark masses in some detail




                            Slides based on (stolen from) the course from
                            Marcel Merk, Wouter Verkerke and Niels Tuning
The Standard Model Lagrangian




      Introduce the massless fermion fields
      Require local gauge invariance     existence of gauge bosons




      Introduce Higgs potential with <φ> ≠ 0   GSM = SU (3)C × SU (2)L ×U (1)Y → SU (3)C ×U (1)Q

      Spontaneous symmetry breaking            W+, W-,Z0 bosons acquire a mass




      Ad hoc interactions between Higgs field & fermions


                                                                         Ivo van Vulpen (2)
                         (2)




Notation of the fields
          Fields: Notation
                             ⎛ 1− γ 5 ⎞           ⎛ 1+ γ 5 ⎞
 Fermions:              ψL = ⎜        ⎟ψ   ; ψR = ⎜        ⎟ψ         with ψ = QL, uR, dR, LL, lR, νR
                             ⎝ 2 ⎠                ⎝ 2 ⎠
                                                                        Interaction rep.
                                                                        I t    ti
 Quarks:                                                                                               bord

Under SU2:
Left handed doublets
                        •
Right hander singlets
                                                Li
                                                                         SU(3)C SU(2)L Hypercharge Y
                                                        Left-handed
                                                                      generation index


                        •                                       •

 Leptons:
                        •
                                                Li


                        •                                       •
Scalar field:                                                         Interaction representation:
                        •                                             standard model Ivo van Vulpen (4)
Fields: explicitly

Explicitly:
• The left handed quark doublet                  :
                    ⎛ urI , u g , ubI ⎞ ⎛ crI , cg , cbI ⎞ ⎛ trI , t g , tbI ⎞
                              I                  I                   I
                                                                                                   T3 = + 1   2
   QLi (3, 2,1 6) = ⎜ I I I ⎟ , ⎜ I I I ⎟ , ⎜ I I I ⎟
    I
                                                                                                                   (Y = 1
                    ⎜ d , d , d ⎟ ⎜ s , s , s ⎟ ⎜b ,b ,b ⎟                                         T3 = − 1   2
                    ⎝ r g b ⎠L ⎝ r g b ⎠L ⎝ r g b ⎠L

• Similarly for the quark singlets:

   u Ri (3,1, 2 3) =
     I
                            (u , u , u ) , (c , c , c ) , (t , t , t )
                               I
                               r
                                   I
                                   r
                                        I
                                        r    R
                                                     I
                                                     r
                                                         I
                                                         r
                                                             I
                                                             r   R
                                                                       I
                                                                       r
                                                                           I
                                                                           r
                                                                                   I
                                                                                   r       R
                                                                                                          ( Y = 2 3)
   d Ri (3,1, −1 3) =
     I
        (         )         ( d , d , d ) , ( s , s , s ) , (b , b , b )
                               I
                               r
                                    I
                                    r
                                         I
                                         r   R
                                                     I
                                                     r
                                                         I
                                                         r
                                                             I
                                                             r   R
                                                                       I
                                                                       r
                                                                               I
                                                                               r
                                                                                       I
                                                                                       r       R
                                                                                                              (Y   = − 1 3)

                                               ⎛ν eI ⎞ ⎛ν μ ⎞ ⎛ν τI ⎞
                                                          I
                                                                                                        T3 = + 1 2
• The left handed leptons: LLi (1, 2, − 1 2) = ⎜ I ⎟ , ⎜ I ⎟ , ⎜ I ⎟
                            I
                                               ⎜ ⎟ ⎜ ⎟ ⎜ ⎟                                              T3 = − 1 2
                                               ⎝ e ⎠ L ⎝ μ ⎠ L ⎝τ ⎠ L


• And similarly the (charged) singlets:                          lRi (1 1 −1) = eR , μ R ,τ R
                                                                  I
                                                                     (1,1,       I     I    I
                                                                                                                       (Y
                                                                                                        Ivo van Vulpen (5)
                             (2)




Lagrangian:   kinetic part
                                                                                        L_kin 1/2


                                                 :The Kinetic Part

                  : Fermions + gauge bosons + interactions

Procedure: Introduce the fermion fields and demand that the theory is local gauge invariant
           under SU(3)CxSU(2)LxU(1)Y transformations.


Start with the Dirac Lagrangian:


Replace:

Fields:        Gaμ : 8 gluons
               Wbμ : weak bosons: W1, W2, W3
               Bμ : hypercharge boson

Generators:       La : Gell-Mann matrices:          ½ λa      (3x3)     SU(3)C
                  σb : Pauli Matrices:              ½ τb      (2x2)     SU(2)L
                  Y : Hypercharge:
                        yp        g                                      ( )
                                                                        U(1)Y

                                                           Ivo x Vulpen Y
       For the remainder we only consider Electroweak: SU(2)L van U(1)(7)
                                                                                                             L_kin 2/2


                                                                    : The Kinetic Part

      L kinetic : iψ (∂ μ γ μ )ψ → iψ ( D μ γ μ )ψ
                           with ψ = QLi , uRi , d Ri , LILi , lRi
                                     I     I      I            I



  Example: the term with QLiI becomes:

  L kinetic (QLi ) = iQLiγ μ D μ QLi
              I        I          I


                                                     i             i         i
                               = iQLiγ μ (∂ μ +
                                   I
                                                       g s Gaμ λa + gWbμτ b + g ′B μ ) QLi
                                                                     W                  I
                                                                                                                ⎛0 1⎞
                                                     2             2         6                              τ1 = ⎜    ⎟
                                                                                                                ⎝1 0⎠
                                                                                                                 ⎛ 0 −i ⎞
        g        y          p            q
  Writing out only the weak part for the quarks:                                                            τ2 = ⎜      ⎟
                                                                                                                 ⎝i 0 ⎠
                                                                                         I
                                                  ⎛ μ i                              ⎞⎛ u ⎞
                               = i( u , d ) Lγ μ ⎜ ∂ + g (W1 τ 1 + W2 τ 2 + W3 τ 3 ) ⎟ ⎜ ⎟
                                          I                 μ        μ         μ                                 ⎛1 0 ⎞
  L    Weak
                 (u, d )   I
                                                                                                 bord       τ3 = ⎜     ⎟
       kinetic             L                                                                                     ⎝ 0 −1⎠
                                                  ⎝   2                              ⎠ ⎝ d ⎠L
                                                                         g I                     g I
                               = iuLγ μ ∂ μ uL + id L γ μ ∂ μ d L −
                                    I           I       I       I
                                                                            u L γ μW − μ d L −
                                                                                           I
                                                                                                    d L γ μW + μ u L
                                                                                                                   I
                                                                                                                       −
                                                                          2                       2
uLI

                                      W+μ
                    g
dLI                                            L=JμWμ                                                 Ivo van Vulpen (8)
                           (1)




Lagrangian:   Higgs part
                                                                                               Higgs 1/1

                                                      : The Higgs Potential




                              V(φ)                                                    V(φ)
 Symmetry                                           Broken symmetry



                                                                   0
                                                                   v
                                        φ                                                          φ



Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value

P    d
Procedure:    =       =                             Substitute:


And rewrite the Lagrangian (tedious):   1 GSM : ( SU (3)C × SU (2) L × U (1)Y ) → ( SU (3)C × U (1) EM )
                                        1.        .
                                        2. The W+,W-,Z0 bosons acquire mass
                                        3. The Higgs boson H appears                  Ivo van Vulpen (10)
                            (4)




Lagrangian:   Yukawa part
                                                                                                            Yukawa 1/4

                                                                   : The Yukawa Part

   Since we have a Higgs field we can add (ad-hoc) interactions
   between Higgs field and the fermions in a gauge invariant way
                                 doublets
    The result is:                               singlet



− L Yukawa =         Yij ψ(      Li   φ     )ψ   Rj            +      h .c .

            = Y       d
                     ij   (   Q φ d +Y
                                I
                                Li     )    I
                                            Rj
                                                     u
                                                    ij     (    I ~
                                                               Q φ uRj + Yijl LILi φ lRj + h.c.
                                                                Li
                                                                    I
                                                                       )              I
                                                                                            (           )
                              i, j : indices for the 3 generations!
                                                                                        % = iσ φ * = ⎛ 0 1 ⎞ φ * = ⎛ φ ⎞
                                                                                                                       0

                                                                               With:   φ             ⎜      ⎟      ⎜ −⎟
                                                                                                     ⎝ −1 0 ⎠      ⎝ −φ ⎠
                                                                                              2

                                                                                (The CP conjugate of φ)



                                                 y     p                   p
                                     are arbitrary complex matrices which operate
    Yijd Yiju Yijl                   in family space (3x3)   flavour physics
                                                                                                     Ivo van Vulpen (12)
                                                                                      Yukawa 2/4

                                                            : The Yukawa Part


Writing the first term explicitly:


              ⎛ϕ + ⎞ I
     d       I   I
  Y (u , d )i ⎜ 0 ⎟ d Rj
    ij       L⎜ϕ ⎟
                 L                   =
              ⎝ ⎠
 ⎛ d I I                 ⎛ϕ + ⎞                    ⎛ϕ + ⎞              ⎛ϕ + ⎞ ⎞
         (
 ⎜ Y11 uL , d L
 ⎜
                     )   ⎜ϕ ⎟
                         ⎝ ⎠
                                 d
                                    (I
                         ⎜ 0 ⎟ Y12 uL , d L
                                           I
                                               )   ⎜ 0 ⎟ Y13 uL , d L ⎜ 0 ⎟ ⎟
                                                   ⎜ϕ ⎟
                                                   ⎝ ⎠
                                                           d
                                                              (I     I
                                                                      )⎜ϕ ⎟
                                                                       ⎝ ⎠⎟
                                                                              ⎟ ⎛ dR ⎞
                                                                                    I
 ⎜
                         ⎛ϕ + ⎞                    ⎛ϕ + ⎞              ⎛ϕ + ⎞ ⎟ ⎜ I ⎟
 ⎜ 21 L L(
 ⎜ Y d cI , sI
                     )   ⎜ϕ ⎟
                         ⎝ ⎠
                                 d
                                     (
                                     I
                         ⎜ 0 ⎟ Y22 cL , sL
                                          I
                                               )   ⎜ 0 ⎟ Y13 cL , sL ⎜ 0 ⎟ ⎟ • ⎜ sR ⎟
                                                   ⎜ϕ ⎟
                                                   ⎝ ⎠
                                                           d
                                                               (
                                                               I    I
                                                                      )⎜ϕ ⎟
                                                                       ⎝ ⎠⎟ ⎜ I ⎟
 ⎜
 ⎜                                                                              ⎝ bR ⎠
                         ⎛ϕ + ⎞                    ⎛ϕ ⎞                ⎛ϕ ⎞ ⎟
         (           )               (         )               (     )
                                                     +                   +
 ⎜ Y31 t L , bL
      d  I    I                   d  I    I
                         ⎜ 0 ⎟ Y32 t L , bL        ⎜ 0 ⎟ Y33 t L , bL ⎜ 0 ⎟ ⎟
                                                            d  I    I
 ⎜                       ⎜ϕ ⎟                      ⎜ϕ ⎟                ⎜ϕ ⎟ ⎟
 ⎝                       ⎝ ⎠                       ⎝ ⎠                 ⎝ ⎠⎠

                                                                                 Ivo van Vulpen (13)
                 (3)




Fermion masses
                                                                      M_fermion 1/3
               SSB
                                    : The Fermion Masses

Start with the Yukawa Lagrangian
                                  ⎛ϕ + ⎞ I
   − LYuk    = Yijd (u L , d L )i ⎜ 0 ⎟ d Rj
                       I     I
                                  ⎜ϕ ⎟         + Yiju (...) + Yijl (...)
                                  ⎝ ⎠
Spontaneous symmetry breaking



After which the following mass term emerges:




              , with
                                                                              bord

                                                                   Ivo van Vulpen (15)
                                                                                                                                               M_fermion 2/3
                           SSB
                                                                         : The Fermion Masses
  Writing in an explicit form:
                                                             ⎛dI ⎞                                      ⎛uI ⎞                                 ⎛ eI ⎞
                                                     ⎛   ⎞                                                                           ⎛
            LMass = ( d                     ) Md     ⎜   ⎟
                                                             ⎜ I⎟
                                                                      (u , c , t ) M
                                                                                               ⎛   u⎞   ⎜ I⎟
                                                                                                                 ( e , μ ,τ ) M      ⎜   l⎞
                                                                                                                                          ⎟
                                                                                                                                              ⎜ I⎟
        −                                                    ⎜s ⎟ +                                     ⎜c ⎟ +                                ⎜μ ⎟      + h.c.
                           I
                               , s I , bI                                I   I   I
                                                                                               ⎜    ⎟
                                                                                                                  I      I   I

                                                     ⎜   ⎟                                     ⎜    ⎟                                ⎜    ⎟
                                                 L
                                                     ⎝   ⎠   ⎜ bI ⎟
                                                                                      L
                                                                                               ⎝    ⎠   ⎜ tI ⎟
                                                                                                                                 L
                                                                                                                                     ⎝    ⎠   ⎜τ I ⎟
                                                             ⎝ ⎠R                                       ⎝ ⎠ R                                 ⎝ ⎠R


The matrices M can always be diagonalised by unitary matrices
                       y        g          y       y                                                             VLf and VRf                   such that:
                                                                                                    ⎡                                            ⎛dI ⎞         ⎤
                V M V =M
                     L
                      f           f
                                             R
                                              f†               f
                                                              diagonal                                  (
                                                                                                    ⎢ I I I
                                                                                                    ⎢ d , s ,b     )                             ⎜ ⎟
                                                                                                                         VLf † VLf M f VRf † VRf ⎜ s I ⎟
                                                                                                                                                               ⎥
                                                                                                                                                               ⎥
                                                                                                                       L
                                                                                                    ⎢                                            ⎜ bI ⎟        ⎥
                                                                                                    ⎣                                            ⎝ ⎠          R⎦

Then the real fermion mass eigenstates are given by:                                                                                                   bord

                 dLi = (VLd ) ⋅ dLj
                                 I
                                                                 dRi = (VRd ) ⋅ dRj
                                                                                 I
                                            ij                                            ij

                 uLi = (VLu ) ⋅uLj
                                I
                                                                uRi = (VR ) ⋅uRj
                                                                        u     I
                                            ij                                       ij

                  lLi = (VLl ) ⋅lLj
                                 I
                                                                 lRi = (VR ) ⋅lRj
                                                                         l     I
                                        ij                                           ij

   dLI , uLI , lLI        are the weak interaction eigenstates
   dL , uL , lL           are the mass eigenstates (“physical particles”)
                                                                                                                                          Ivo van Vulpen (16)
                                                                                                              M_fermion 3/3
                     SSB
                                                        : The Fermion Masses
 In terms of the mass eigenstates:
                                                ⎛ md         ⎞ ⎛d ⎞                       ⎛ mu           ⎞   ⎛u ⎞
               − L Mass =    ( d , s, b )   L
                                                ⎜
                                                ⎜    ms
                                                             ⎟
                                                             ⎟
                                                               ⎜ ⎟
                                                                          (
                                                               ⎜ s ⎟ + u , c, t   )   L
                                                                                          ⎜
                                                                                          ⎜      mc
                                                                                                         ⎟
                                                                                                         ⎟
                                                                                                             ⎜ ⎟
                                                                                                             ⎜c⎟
                                                ⎜            ⎟
                                                          mb ⎠ ⎜b⎟                        ⎜              ⎟
                                                                                                      mt ⎠   ⎜t ⎟
                                                ⎝              ⎝ ⎠R                       ⎝                  ⎝ ⎠R
                                                ⎛ me         ⎞ ⎛e⎞
                        +    ( e, μ ,τ )    L
                                                ⎜
                                                ⎜    mμ
                                                             ⎟⎜ ⎟
                                                             ⎟ ⎜ μ ⎟ + h.c.
                                                ⎜                  ⎟
                                                          mτ ⎟ ⎜ τ ⎠ R
                                                ⎝            ⎠⎝
              − L Mass =         mu uu              +     m c cc      +       mt tt
                         +       m d dd             +     m s ss      +       mb bb
                         +       m e ee            +      m μ μμ       +      mτ ττ
In flavour space one can choose:

Weak basis: The gauge currents are diagonal in flavour space, but the flavour
            mass matrices are non-diagonal

Mass basis: The fermion masses are diagonal, but some gauge currents (charged
                k interactions) are not di
            weak i        i   )                l in flavour space
                                        diagonal i fl

        What happened to the charged current interactions (in LKinetic) ?                                Ivo van Vulpen (17)
                  (3)




Charged current
                                                                                                 ChargedC 1/2


                                                   : The Charged Current
 The charged current interaction for quarks in the interaction basis is:
                          g
    − LW +       =                   I
                                   u Li       γμ     d Li Wμ+
                                                       I

                           2
  The charged current interaction for quarks in the mass basis is:
                      g
    − LW +   =                  uLi VLu       γ μ VLd † d Li Wμ+
                       2

                                  VCKM = (VLu ⋅ VLd † )                     VCKM ⋅ VCKM = 1
                                                                                     †
  The unitary matrix:                                            with:
   is the Cabibbo Kobayashi Maskawa mixing matrix:
                                                       ⎛d ⎞
                     g
    −LW +    =                 ( u , c , t ) L (VCKM ) ⎜ s ⎟
                                                       ⎜ ⎟        γ μ Wμ+          Note: alleen down-
                      2                                ⎜b⎟                         type roteert
                                                                                    yp
                                                       ⎝ ⎠L

Lepton sector: similarly                  VMNS = (VL ⋅ VLl † )
                                                   ν


However, for massless neutrino’s: VLν = arbitrary.
Choose it such that VMNS = 1  no mixing in the lepton sector
                                                                                              Ivo van Vulpen (19)
                                                                                        ChargedC 2/2


 Charged Currents

The charged current term reads:
            g I μ − I                  g I μ + I                 μ−         μ+
  LCC =        uLiγ Wμ d Li       +       d Liγ Wμ uLi       = J CC Wμ− + J CC Wμ+
             2                          2
          g ⎛ 1− γ 5 ⎞ μ − ⎛ 1− γ 5 ⎞                          g    ⎛ 1− γ 5 ⎞ μ + † ⎛ 1− γ 5 ⎞
        =   ui ⎜     ⎟ γ Wμ Vij ⎜   ⎟dj                  +       dj ⎜        ⎟ γ Wμ V ji ⎜    ⎟ ui
           2 ⎝ 2 ⎠              ⎝ 2 ⎠                           2 ⎝ 2 ⎠                  ⎝ 2 ⎠
            g                                   g
        =      uiγ μWμ−Vij (1 − γ 5 ) d j   +      d j γ μWμ+Vij* (1 − γ 5 ) ui
             2                                   2

 Under the CP operator this gives:
 (together with (x,t)   (-x,t))


                g                                               g
     LCC    ⎯⎯→CP
                   d j γ μWμ+Vij (1 − γ 5 ) ui           +         uiγ μWμiVij* (1 − γ 5 ) d j
                 2                                               2
   CP is conserved only if        Vij = Vij*

                                                                                     Ivo van Vulpen (20)
                                                       ComplexPhase 1/1


Why complex phases matter
• CP conjugation of a W boson vertex involves complex
  conjugation of coupling constant



                         W−                            W+
       b                             b
                gVub
                   b                          gV*ub
                              u                             u


           Above process violates CP if Vub ≠ Vub*

• With 2 generations Vij is always real and Vij≡Vij*
• With 3 generations Vij can be complex
    CP violation built into weak decay mechanism!
                                                       Ivo van Vulpen (21)
                        (3)




Standard Model Lagrangian
      recap
                                                                                               L_recap 1/1


    The Standard Model Lagrangian (recap)




• LKinetic : IIntroduce the massless ffermion fi ld
                t d     th      l         i fields
             Require local gauge invariance      gives rise to existence of gauge bosons




• LHiggs :
     gg      Introduce Higgs potential with <φ> ≠ 0    GSM = SU (3)C × SU (2) L × U (1)Y → SU (3)C × U (1)Q

             Spontaneous symmetry breaking               The W+, W-,Z0 bosons acquire a mass




             Ad hoc interactions between Higgs field & fermions
                 CP violating with a single phase
                              (3)




CKM matrix in more detail



          Parameters:     2
          Measurements:   2
          Reenking        2
                                                             Vckm_param 1/2


Ok…. we’ve got the CKM matrix, now what?




• It’s unitary
   – “probabilities add up to 1”:
   – d’=0.97 d + 0.22 s + 0.003 b   (0.972+0.222+0.0032=1)


• How many free parameters?
   – How many real/complex?


• How do we normally visualize these parameters?



                                                             Ivo van Vulpen (25)
                                                    Vckm_meas 1/2


How do you measure those numbers?

• Magnitudes are typically determined from ratio of decay
  rates
• Example 1 – Measurement of Vud
   – Compare decay rates of neutron
     decay and muon decay
   – Ratio proportional to Vud2
     |
   – |Vud| = 0.9735 ± 0.0008
   – Vud of order 1




                                                   Ivo van Vulpen (26)
                                                                         Vckm_meas 2/2


        What do we know about the CKM matrix?

        • Magnitudes of elements have been measured over time
           – Result of a large number of measurements and calculations

                          ⎛ d ' ⎞ ⎛ Vud     Vus Vub ⎞ ⎛ d ⎞
                          ⎜ ⎟ ⎜                     ⎟⎜ ⎟
                          ⎜ s ' ⎟ = ⎜ Vcd   Vcs Vcb ⎟ ⎜ s ⎟
                          ⎜ b'⎟ ⎜V          Vts Vtb ⎟ ⎜ b ⎟
                          ⎝ ⎠ ⎝ td                  ⎠⎝ ⎠
                                                                        t
                                                               4 parameters
                                                               •3 real
                                                               •1 phase



⎛ Vud     Vus   Vub ⎞ ⎛ 0.9738 ± 0.0002 0.227 ± 0.001 0.00396 ± 0.00009 ⎞
⎜                   ⎟ ⎜                                                   ⎟
⎜ Vcd     Vcs   Vcb ⎟ = ⎜ 0.227 ± 0.001 0.9730 ± 0.0002 0.0422 ± 0.0005 ⎟
⎜V
⎝ td      Vts   Vtb ⎟ ⎜ 0.0081 ± 0.0005 0.0416 ± 0.0005 0.99910 ± 0.00004 ⎟
                    ⎠ ⎝                                                   ⎠
                                    only,
        Magnitude of elements shown only no information of phase

                                                                     Ivo van Vulpen (27)
                                                            Vckm_ranking 1/2


Exploit apparent ranking for a convenient parameterization
• Given current experimental precision on CKM element values,
  we usually drop λ4 and λ5 terms as well
   – Effect of order 0.2%...


               ⎛        λ2                               ⎞
               ⎜    1−              λ      Aλ ( ρ − iη ) ⎟
                                              3

               ⎜         2                               ⎟⎛ d ⎞
     ⎛ d′⎞
     ⎜ ′⎟      ⎜                     λ2                  ⎟⎜ ⎟
     ⎜s ⎟     =⎜      −λ          1−          Aλ  2
                                                         ⎟⎜ s ⎟
     ⎜ b′ ⎟    ⎜ 3                    2                  ⎟⎜ b ⎟
     ⎝ ⎠L      ⎜ Aλ (1 − ρ − iη ) − Aλ 2        1        ⎟ ⎝ ⎠L
               ⎜                                         ⎟
               ⎝                                         ⎠

             f    k     f t     d d              (λ
• Deviation of ranking of 1st and 2nd generation ( vs λ2)
  parameterized in A parameter
• Deviation of ranking between 1st and 3rd generation,
  parameterized through |ρ-iη|
                            i
• Complex phase parameterized in arg(ρ-iη)
                                                            Ivo van Vulpen (28)
                                                                   Vckm_ranking 2/2


Approximately diagonal form

• Values are strongly ranked:
   – Transition within generation favored
   – Transition from 1st to 2nd generation suppressed by cos(θc)
   – Transition from 2nd to 3rd generation suppressed bu cos2(θc)
   – Transition from 1st to 3rd generation suppressed by cos3(θc)


                 g
           CKM magnitudes
             d     s    b              Why the ranking?
                                       We don’t know (yet)!
      u           λ         λ3

                                       If you figure this out,
      c     λ               λ2         you will win the nobel
                                       prize
      t
            λ3    λ2


          λ=cos(θc)=0.23
                                                                    Ivo van Vulpen (29)

								
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