We’ve worked through 2 MATHEMATICAL MECHANISMS 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 2nd (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 v+H(x) Higgs= = 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.

The “mass-generating” interaction is identified by simple constants With the choice of gauge settled: +  0 Higgs= v+H(x) = Let’s try to couple these scalar “Higgs” fields to W, B which means replace: which makes the 1st 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

† ( ) ( ) ( ) ( 2g22W+W+ + (g12+g22) ZZ ) ( ) 0 1 1 0 0 -i i 0 1 0 recall that → → 0 1 1 0 0 -i i 0 1 0 0 -1 W3 W1-iW2 W1+iW2 -W3 τ ·W = W1 + W2 + W3 = 2 W1-iW2 H + v 1 2 † ( ) = 0 H +v W1+iW2 2 H + v 1 8 † ( ) = 0 H +v 1 8 † † ( ) ( 2g22W+W+ + (g12+g22) ZZ ) ( ) = H +v H +v

( ) ( 2g22W+W+ + (g12+g22) ZZ ) ( ) v22g22W+ W+ (g12+g22 )Z Z 8 † † ( ) ( 2g22W+W+ + (g12+g22) ZZ ) ( ) = H +v H +v No AA term has been introduced! The photon is massless! But we do get the terms 1 8 † 1 2 v22g22W+ W+ MW = vg2 1 8 1 2 (g12+g22 )Z Z MZ = v√g12 + g22 At this stage we may not know precisely the values of g1 and g2, but note: MW MZ 2g2 = √g12 + g22

( ) g12+g12 and we do know THIS much about g1 and g2 -g1g2 = e to extraordinary precision! from other weak processes: m- e- +e +m N  p + e- +e u e e- W - d m e e- W - m- ( ) 2 2 e sinθW lifetimes (decay rate cross sections) ~gW = give us sin2θW

Notice = cos W according to this theory. MW MZ Notice = cos W according to this theory. where sin2W=0.2325 +0.0015 -9.0019 We don’t know v, but information on the coupling constants g1 and g2 follow from lifetime measurements of b-decay: neutron lifetime=886.7±1.9 sec and a high precision measurement of muon lifetime=2.19703±0.00004 msec 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

Fine work for theorists, but drew very little attention All of which can be compared in ratios to similar reactions involving well-known/ well-measured simple QED scattering (where the coupling is simply e2=1/137). Fine work for theorists, but drew very little attention from the rest of the high energy physics community Until 1973 all observed weak interactions were consistent with only a charged boson. 1973 (CERN): first neutral current interaction observed ν + nucleus → ν + p + π- + πo _ _ Suddenly it became very urgent to observe W±, Zo bosons directly to test electroweak theory.

The first example of the neutral-current process νμ + e- →νμ + 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- + πo The Gargamelle heavy-liquid bubble chamber, installed into the magnet coils at CERN(1970)

Current precision measurements give: By early 1980s had the following theoretically predicted masses: MZ = 92  0.7 GeV MW = cosWMZ = 80.2  1.1 GeV Late spring, 1989 Mark II detector, SLAC August 1989 LEP accelerator at CERN discovered opposite-sign lepton pairs with an invariant mass of MZ=92 GeV and lepton-missing energy (neutrino) invariant masses of MW=80 GeV Current precision measurements give: MW = 80.482  0.091 GeV MZ = 91.1885  0.0022 GeV

Z→e+e-

Z→e+e-

Z→e+e-

Z→e+e-

Z→+-

Z → jet + jet

Also notice the threshold for W+W- pair production! 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)

Higgs= Gv[eLeR + eReL] + GH[eLeR + eReL] 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 coupling which, for electrons, for example, would read Higgs= v+H(x) which with becomes _ _ _ _ Gv[eLeR + eReL] + GH[eLeR + eReL]

Gv[eLeR + eReL] + GH[eLeR + eReL] _ _ _ _ Gv[eLeR + eReL] + GH[eLeR + eReL] _ _ e e e e from which we can identify: me = Gv or