Let’s recap: We’ve worked through 2 MATHEMATICAL MECHANISMS for manipulating Lagrangains Introducing SELF-INTERACTION terms (generalized “mass” terms) showed that a specific GROUND STATE of a system need NOT display the full available symmetry of the Lagrangian Effectively changing variables by expanding the field about the GROUND STATE (from which we get the physically meaningful ENERGY values, anyway) showed The scalar field ends up with a mass term; a 2 nd (extraneous) apparently massless field (ghost particle) can be gauged away. Any GAUGE FIELD coupling to this scalar (introduced by local inavariance) acquires a mass as well!

We then applied these techniques by introducing the scalar Higgs fields through a weak iso-doublet (with a charged and uncharged state) + 0+ 0  Higgs = 0 v+H(x) = which, because of the explicit SO(4) symmetry, the proper gauge selection can rotate us within the  1,  2,  3,  4 space, reducing this to a single observable real field which we we expand about the vacuum expectation value v.

With the choice of gauge settled: + 0+ 0  Higgs = 0 v+H(x) = Let’s try to couple these scalar “Higgs” fields to W , B  which means replace: which makes the 1 st term in our Lagrangian: † The “mass-generating” interaction is identified by simple constants providing the coefficient for a term simply quadratic in the gauge fields so let’s just look at: † where Y =1 for the coupling to B 

† recall that τ ·Wτ ·W → = W 1  + W 2  + W 3  i i = W 3 W 1  iW 2 W 1  iW 2  W = ( ) 2 0H + v0H + v 0 H +v 1818 = ( ) 2 0H + v0H + v 0 H +v † † 1818 = ( ) H +v † ( ) H +v W 1  iW 2 W 1  iW 2 ( 2g 2 2 W +  W +  + (g 1 2 +g 2 2 ) Z  Z  ) †

1818 = ( ) H +v † ( ) H +v ( 2g 2 2 W +  W +  + (g 1 2 +g 2 2 ) Z  Z  ) † No A  A  term has been introduced! The photon is massless! But we do get the terms 1818 v22g22W+ W+v22g22W+ W+ † M W = vg (g 1 2 +g 2 2 ) Z  Z  M Z = v√g g MWMZMWMZ 2g22g2 √g g 2 2 At this stage we may not know precisely the values of g 1 and g 2, but note: = 1212

 e e   W    u e e  W   d    e  + e +  N  p + e  + e ~g W = e sinθ W ( ) 2 2 g1g2 g12+g12g1g2 g12+g12 = e and we do know THIS much about g 1 and g 2 to extraordinary precision! from other weak processes: lifetimes (decay rate cross sections) give us sin 2 θ W

Notice = cos  W according to this theory. MWMZMWMZ where sin 2  W =  We don’t know v, but information on the coupling constants g 1 and g 2 follow from lifetime measurements of  -decay: neutron lifetime=886.7±1.9 sec and a high precision measurement of muon lifetime= ±  sec and measurements (sometimes just crude approximations perhaps) of the cross-sections for the inverse reactions: e - + p  n + e electron capture e + p  e + + n anti-neutrino absorption as well as e + e -  e - + e neutrino scattering

Until 1973 all observed weak interactions were consistent with only a charged boson. All of which can be compared in ratios to similar reactions involving well-known/ well-measured simple QED scattering (where the coupling is simply e 2 =1/137). Fine work for theorists, but drew very little attention from the rest of the high energy physics community 1973 (CERN): first neutral current interaction observed ν  + nucleus → ν  + p + π  + π o Suddenly it became very urgent to observe W ±, Z o bosons directly to test electroweak theory. __

The Gargamelle heavy-liquid bubble chamber, installed into the magnet coils at CERN(1970) The first example of the neutral-current process ν μ + e  →ν μ + e . The electron is projected forward with an energy of 400 MeV at an angle of 1.5 ± 1.5° to the beam, entering from the right. ν  + nucleus → ν  + p + π  + π o __ __ and interaction with neutrons produced hadronic showers with no net electic charge.

By early 1980s had the following theoretically predicted masses: M Z = 92  0.7 GeV M W = cos  W M Z = 80.2  1.1 GeV Late spring, 1983 Mark II detector, SLAC August 1983 LEP accelerator at CERN discovered opposite-sign lepton pairs with an invariant mass of M Z =92 GeV and lepton-missing energy (neutrino) invariant masses of M W =80 GeV Current precision measurements give: M W =  GeV M Z =  GeV

Z→e+eZ→e+e

Z→e+eZ→e+e

Z→e+eZ→e+e

Z→e+eZ→e+e

Z→+Z→+

Z → jet + jet

Among the observed resonances in e + e  collisions we now add the clear, well- defined Z peak! Also notice the threshold for W + W  pair production!

Z peak in e + e  invariant mass distribution

Z peak in  invariant mass distribution

W peak in e + e  transverse mass distribution

Z → jets cross section LEP (CERN)

Electroweak Precision Tests LEP Line shape: m Z (GeV) Γ Z (GeV)  0 h (nb) R ℓ ≡Γ h / Γ ℓ A 0,ℓ FB τ polarization: A τ A ε heavy flavor: R b ≡Γ b / Γ b R c ≡Γ c / Γ b A 0,b FB A 0,c FB qq charge asymmetry: sin 2 θ w ± ± ± ± ± ± ± ± ± ± ± ±  SLC A 0,ℓ FB A b A c pp mWmW ± ± ± ±

Can the mass terms of the regular Dirac particles in the Dirac Lagrangian also be generated from “first principles”? Theorists noted there is an additional gauge-invariant term we could try adding to the Lagrangian: A Yukawa couplingwhich, for electrons, for example, would read which with  Higgs = 0 v+H(x) becomes Gv[e L e R + e R e L ] + GH[e L e R + e R e L ] _ _ __

e ee e _ e ee e _ from which we can identify: m e =Gv or Gv[e L e R + e R e L ] + GH[e L e R + e R e L ] _ _ __

u d ee e _ u u d d WW W links members of the same weak isodoublet within a single generation The decay conserves charge, but does NOT conserve iso-spin (upness/downness)

u d ee e _ u u d d WW However, we even observe some strangeness -changing weak decays! d u u d u s d u u d s s s s s u _ _

  →   + ν  63.43% of all kaon decays →  0 +   21.13% _ → e  + ν e _ % u s K K  ν e    W W  u s KK u  W W  u d  _

u d ee e _ u u d d WW d u u d u s d u u d s s s s s u _ WW _ WW

Cabibbo (1963) Glashow, Illiopoulous, Maiani [GIM] (1970) Kobayashi & Maskawa [KM] (1973) Suggested the eigenstates of the weak interaction operators (which couple to Ws) are not exactly the same as the “mass” eigenstates participating in the STRONG interactions (free space states) The weak eigenstates are QUANTUM MECHANICAL admixtures of the mass eigenstates d weak = c 1 d + c 2 s where, of course c c 2 2 = 1 = sinθ c d + cosθ c s

To explain strangeness-changing decays, Cabibbo (1963) introduced the redefined weak iso-doublet udcudc = u Intended to couple to the J weak current in the Lagrangian u d WW cos  c u s WW sin  c “suppressed” sin  c  cos  c  dcos  c + ssin  c θ c  13.1 o

The relevant term, J weak W , then comes from: †† † †   ig 1 B  YR2YR2 †† W 3  W 1   iW 2  W 1   iW 2   W 3  †† W 3  0 0  W 3  0 W 1   iW 2  W 1   iW 2  0 +

From which follows a NEUTRAL COUPLING to = uu - d c d c __ d s K 0K 0      Z 0Z 0 a coupling to a strangeness changing neutral current! u s K+K+ u d ++ Z 0Z 0    

BUT we do NOT observe processes like: u s K+K+ u d Though we do see the very similar processes: ++ Z 0Z 0     u s K+K+ u 00 WW    d s K 0K 0 u WW WW    d s K 0K 0      Z 0Z 0 These are suppressed, but allowed (observed). Compare to  0  e + e   0   Also  e + e 

Glashow, Illiopoulous, Maiani [GIM] (1970) even before charmed particles were discovered (1974) and the new quark identified, proposed there could be a 2 nd weak doublet that followed and complemented the Cabibbo pattern: So that the meaured Cabibbo “angle” actually represented a mixing/rotation! so that: orthogonal!

then together these doublets produce interactions of: uu – d c d c + cc  s c s c _ _ _ _ = uu + cc – (d c d c + s c s c ) = uu + cc – (ddcos 2 θ c + sinθ c cosθ c (ds + sd) + sssin 2 θ c _ _ _ _ ddsin 2 θ c  sinθ c cosθ c (ds + sd) + sscos 2 θ c ) = uu + cc – dd – ss _ _ _ _ __ _ __ absolutely NO flavor-changing neutral current terms!

1965 Gellmann & Pais Noticed the Cabibbo mechanism, where was the weak eigenstate, allowed a 2 nd order (~rare) weak interaction that could potentially induce the strangeness-violating transition of K o a particle becoming its own antiparticle! uu s d s d KoKo KoKo WW u u s d s d KoKo KoKo WW WW