£ introducing a MASSIVE Higgs scalar field, , With  real, the field  vanishes and our Lagrangian reduces to £ introducing a MASSIVE Higgs scalar field, , and “getting” a massive vector gauge field G Notice, with the  field gone, all those extra , , and  interaction terms have vanished Can we employ this same technique to explain massive Z and W vector bosons?

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!

Higgs= +  0 to the SUL(2)×U(1)Y Lagrangian in such a way as to Now apply these techniques: introducing scalar Higgs fields with a self-interaction term and then expanding fields about the ground state of the broken symmetry to the SUL(2)×U(1)Y Lagrangian in such a way as to endow W,Zs with mass but leave  s massless. These two separate cases will follow naturally by assuming the Higgs field is a weak iso-doublet (with a charged and uncharged state) with Q = I3+Yw /2 and I3 = ±½ Higgs= +  0 for Q=0  Yw = 1 Q=1  Yw = 1 couple to EW UY(1) fields: B

Higgs= with Q=I3+Yw /2 and I3 = ±½ +  0 Higgs= with Q=I3+Yw /2 and I3 = ±½ Yw = 1 Consider just the scalar Higgs-relevant terms £ † † † with Higgs not a single complex function now, but a vector (an isodoublet) Once again with each field complex we write + = 1 + i2  0 = 3 + i4 †  12 + 22 + 32 + 42 † † † £ Higgs † † † †

L U =½m2† + ¹/4 († )2 12 + 22 + 32 + 42 = -2m2  † † † † † † Higgs † † † † just like before: U =½m2† + ¹/4 († )2 -2m2  12 + 22 + 32 + 42 = Notice how 12, 22 … 42 appear interchangeably in the Lagrangian invariance to SO(4) rotations Just like with SO(3) where successive rotations can be performed to align a vector with any chosen axis,we can rotate within this 1-2-3-4 space to a Lagrangian expressed in terms of a SINGLE PHYSICAL FIELD

Higgs= +  0 v+H(x) v+H(x) 1 or 2 3 or 4 Were we to continue without rotating the Lagrangian to its simplest terms we’d find EXTRANEOUS unphysical fields with the kind of bizarre interactions once again suggestion non-contributing “ghost particles” in our expressions. +  0 Higgs= So let’s pick ONE field to remain NON-ZERO. 1 or 2 3 or 4 because of the SO(4) symmetry…all are equivalent/identical might as well make  real! v+H(x) v+H(x) Can either choose or But we lose our freedom to choose randomly. We have no choice. Each represents a different theory with different physics!

Let’s look at the vacuum expectation values of each proposed state. v+H(x) v+H(x) or Aren’t these just orthogonal? Shouldn’t these just be ZERO? Yes, of course…for unbroken symmetric ground states. If non-zero would imply the “empty” vacuum state “OVERLPS with” or contains (quantum mechanically decomposes into) some of + or  0. But that’s what happens in spontaneous symmetry breaking: the vacuum is redefined “picking up” energy from the field which defines the minimum energy of the system.

= v v.e.v.! 1 This would be disastrous for the choice + = v + H(x) a non-zero v.e.v.! = v 1 This would be disastrous for the choice + = v + H(x) since 0|+ = v implies the vacuum is not chargeless! But 0| 0 = v is an acceptable choice. If the Higgs mechanism is at work in our world, this must be nature’s choice.

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 ( ) e sinθW 2 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

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!

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”? Theorist 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