The precise dynamical behavior of a system of particles can be inferred from the Lagrangian equations of motion derived from the Lagrange function: here for a classical systems of mass points Extended to the case of a continuous (wave) function(s) “The Lagrangian” m an explicit function only of the dynamical variables of the field components and their derivatives Euler-Lagrange equation

Does not depend explicitly on spatial coordinates (absolute positions) m Does not depend explicitly on spatial coordinates (absolute positions) translation invariance - otherwise would violate relativistic invariance Conservation of Linear Momentum! Does not depend explicitly on t (absolute time) time translation invariance Conservation of Energy! Similarly its invariance under spacial rotations - guarantees Conservation of Angular Momentum!

For linear wave equations L must involve terms at least quadratic in and - To generate differential equations not higher than 2nd order restrict terms to factors of the field components and their 1st derivative - note: renormalizability demands no higher powers in the fields than and than n = 5. L should be real (field operators Hermitian) - guarantees the dynamical variables (energy, momentum, currents) are real. L should be relativistically invariant - so restrictive is this requirement, it guarantees the derived equations of motion are automatically Lorentz invariant

Real Scaler Field the simplest Lagrangian with (xm) and m  dependence is £ £ £ from which  / - m [ /  (m ) ] = 0 yields the (hopefully) familiar Klein-Gordon equation! 1920 E. Schrödinger O. Klein W. Gordon From the starting point for a relativistic QM equation: together with the quantum mechanical prescriptions

₤ ₤ or or Matter fields (like the QM wave function of an electron) are known only up to a phase factor a totally non-geometric attribute If we choose, we can write this as or ₤ ₤ or

₤ Then treating  and * as independent fields, we find field equations ₤ ₤  / - m [ /  (m ) ] = 0 ₤ ₤  /*-m [ / (m *) ] = 0 two real fields describing particles of identical mass There’s a new symmetry hidden here: the Lagragian is completely invariant under any arbitrary rotation in the complex plane or

₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ = 0 For an infinitesimally small rotation satisfying the continuity equation! and changing sign with f f* a conserved 4-vector a charged current density

£ Vector Field the field now has 4 components Spin-1 particles γ, g, W, Z Though has all the 4-vectors, tensors contracted energy is not positive definite unless we impose a restriction i.e., only 3 linearly independent components This is equivalent to replacing the 1st term with the invariant expression £ the (hopefully still) familiar Klein-Gordon equation!

LDIRAC Dirac Field Spin- ½ particles this field includes 4 independent components in spinor space e, m, quarks LDIRAC Dirac’s equation with a current vector:

L(r,t) L + L INT = LVector + LDIRAC These have been single free particle Lagrangians We might expect a realistic Lagrangian that involves systems of particles L(r,t) = LVector + LDIRAC but each term describes free non-interacting particles describes photons describes e+e- objects need something like: L + L INT But what should an interaction term look like? How do we introduce the interactions they experience? Again: a local Hermitian Lorentz-invariant construction of the various fields and their derivatives reflecting any additional symmetries the interaction has been observed to respect The simplest here would be a bilinear form like:

Our free particle equations of motion were all Or consider this: Our free particle equations of motion were all homogeneous differential equations. When the field is due to a source, like the electromagnetic (photon!) field you know you need to make the eq. of motion inhomogeneous: charged 4-current density and would do the trick! exactly the proposed bilinear form! Dirac electron current Or just crack open Jackson: antiparticle (hermitian conjugate) field particle field The electromagnetic potential serves as the photon wave function

LDirac=iħc gm   mc2 Now let’s look back at the FREE PARTICLE Dirac Lagrangian LDirac=iħc gm   mc2 Dirac matrices Dirac spinors (Iso-vectors, hypercharge) Which is OBVIOUSLY invariant under the transformation ei (a simple phase change) because  e-i and in all  pairings this added phase cancels! This one parameter unitary U(1) transformation is called a “GLOBAL GAUGE TRANSFORMATION.”

LDirac=iħc gm   mc2 What if we GENERALIZE this? Introduce more flexibility to the transformation? Extend to: but still enforce UNITARITY? eia(x) LOCAL GAUGE TRANSFORMATION Is the Lagrangian still invariant? LDirac=iħc gm   mc2 (ei(x)) = i((x)) + ei(x)() So: L'Dirac = -ħc((x)) gm  + iħce-i(x)gm( )e+i(x)  mc2

L'Dirac = +q gm  () + LDirac -ħc((x)) gm  + iħcgm( )  mc2 LDirac For convenience (and to make subsequent steps obvious) define: -ħc q (x)  (x) then this is re-written as L'Dirac = +q gm  () + LDirac recognize this????

L'Dirac = +q gm  () + LDirac If we are going to demand the complete Lagrangian be invariant under even such a LOCAL gauge transformation, it forces us to ADD to the “free” Dirac Lagrangian something that can ABSORB (account for) that extra term, i.e., we must assume the full Lagrangian HAS TO include a current-field interaction: L=[iħcgmmc2 ]-(qg m)Am and that defines its transformation under the same local gauge transformation

L=[iħcgmmc2 ]-(qg m)Am We introduced the same interaction term moments ago following electrodynamic arguments (Jackson) the form of the current density is correctly reproduced the transformation rule Am' = Am +  l is exactly (check your Jackson notes!) the rule for GAUGE TRANSFORMATIONS already introduced in e&m! The exploration of this “new” symmetry shows that for an SU(1)- invariant Lagrangian, the free Dirac Lagrangain is “INCOMPLETE.” If we chose to allow gauge invariance, it forces to introduce a vector field (here that means Am ) that “couples” to .

SU(3) “rotations” occur in an 8-dim “space” SU(3) color symmetry of strong interactions This same procedure, generalized to symmetries in new spaces SU(3) “rotations” occur in an 8-dim “space” 8 3x3 “generators” 3-dimensional matrix formed by linear combinations of 8 independent fundamental matrices 8-dim vector Demanding invariance of the Lagrangian under SU(3) rotations introduces the massless gluon fields we believe are responsible for the strong force. The field is assumed to exist in any of 3 possible independent color states

e e  + THEN in an effort to explain decays:  u _ e d u d ?? u d proton neutron decay u e _  e ?? muon decay  pion + hadron decays involve the transmutation of individual quarks u d _ ?? 

e+ e  as well as the observed inverse of some of these processes: neutrino capture by protons u d d u e+ _ ?? e e e ??   neutrino capture by muons

required the existence of 3 “weakons” : in terms of the gauge model of photon-mediated charged particle interactions e- e- p p  ??? e- p e n required the existence of 3 “weakons” : W +, W -, Z m e e+ W + m+ u e e- W - d d e+ W +? W -? u e

1 2 SU(2) electro-weak symmetry u d e e- “Rotational symmetry”within weakly coupled left-handed isodoublet states introduces 3 weakons: W+, W-, Z and an associated weak isospin “charge” L L

L=[iħcgmmc2 ]- F F-(gg m )·Gm This SU(2) theory then     L=[iħcgmmc2 ]- F F-(gg m )·Gm 1 4  2 describes doublets of Dirac particle states in interaction with 3 massless vector fields Gm (think of something like the -fields, A)   This followed just by insisting on local SU(2) invariance! In the Quantum Mechanical view:   These Dirac fermions generate 3 currents, J = (g   )  2 These particles carry a “charge” g, which is the source for the 3 “gauge” fields

 qg  qr     so named because unlike the PROMPT processes e+ e- The Weak Force  qg e+ e- _ e+ e- rg which seem instantaneous  qr     or the electromagnetic decay: which involves a 10-17 sec lifetime path length (gap) in photographic emulsions mere nm! weak decays are “SLOW” processes…the particles involved: , ±,  are nearly “stable.” 10-6 sec 700 m pathlengths 887 sec 10-8 sec +++ 7 m pathlengths and their inverse processes: scattering or neutrino capture are rare small probability of occurrence (small rates…small cross sections!).

Such “small cross section” seemed to suggest a SHORT RANGE force…weaker with distance compared to the infinite range of the Coulomb force or powerful confinement of the color force This seems at odds with the predictions of ordinary gauge theory in which the VECTOR PARTICLES introduced to mediate the forces like photons and gluons are massless. This means the symmetry cannot be exact. The symmetry is BROKEN.

The gravitational field around a point source Some Classical Fields The gravitational field around a point source (e.g. the earth) is a scalar field An electric field is classical example of a vector field: effectively 3 independent fields

ψ↑(x,y,z) ψ↓(x,y,z) α β Ψ = e-iEt/ħψ(x,y,z)g(ms) Once spin has been introduced, we’ve grown accustomed to writing the total wave function as a two-component “vector” ψ↑(x,y,z) ψ↓(x,y,z) Ψ(x,y,z,t;ms) = Ψ = e-iEt/ħψ(x,y,z)g(ms) time- dependent part spatial part spin space α β Ψ = e-iEt/ħR(r)S(θ)T(φ)g(ms) Yℓm(θ,φ) for a spherically symmetric potential But spin is 2-dimensional only for spin-½ systems. Recognizing the most general solutions involve ψ/ψ* (particle/antiparticle) fields, the Dirac formalism modifies this to 4-component fields! The 2-dim form is better recognized as just one fundamental representation of angular momentum.

U That 2-dim spin-½ space is operated on by or more generally by The SU(2) transformation group (generalized “rotations” in 2-dim space) is based on operators: U “generated by” traceless Hermitian matrices

c a-ib a+ib -c c a-ib a+ib -c 0 1 1 0 0 -i i 0 1 0 0 -1 = a +b +c What’s the most general traceless HERMITIAN 22 matrices? c a-ib a+ib -c and check out: c a-ib a+ib -c 0 1 1 0 0 -i i 0 1 0 0 -1 = a +b +c

a3 a1-ia2 a4-ia5 a1+ia2 a6-ia7 a4+ia5 a6+ia7 SU(3) = a1 +a2 +a3 +a4 FOR What’s the form of the most general traceless HERMITIAN 3×3 matrix? Diagonal terms have to be real! a3 a1-ia2 a4-ia5 Transposed positions must be conjugates! a1+ia2 a6-ia7 a4+ia5 a6+ia7 Must be traceless! 0 1 0 1 0 0 0 0 0 0 -i 0 i 0 0 0 0 0 0 0 1 0 0 0 1 0 0 = a1 +a2 +a3 +a4 +a3 +a6 +a7 +a8 0 0 -i 0 0 0 i 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 -i 0 i 0

U(1) local gauge transformation (of simple phase) electrical charge-coupled photon field mediates EM interactions SU(2) “rotations” occur in an 3-dim “space” e-ia· /2ħ 3 independent parameters three 2x2 matrix operators 3 simultaneous gauge transformations 3 vector boson fields SU(3) “rotations” occur in an 8-dim “space” e-ia· /2ħ 8 3x3 operators 8 independent parameters 8 simultaneous gauge transformations 8 vector boson fields

Spontaneous Symmetry Breaking Englert & Brout, 1964 Higgs 1964, 1966 Guralnick, Hagen & Kibble 1964 Kibble 1967 The Lagrangian & derived equations of motion for a system possess symmetries which simply do NOT hold for a specific ground state of the system. (The full symmetry MAY be re-stored at higher energies.) (1) A flexible rod under longitudinal compression. (2) A ball dropped down a flask with a convex bottom. Lagrangian symmetric with respect to rotations about the central axis once force exceeds some critical value it must buckle sideways forming an arc in SOME arbitrary direction Although one direction is chosen, the complete set of all possible final shapes DOES show the full symmetry.

What is the GROUND STATE? lowest energy state What does GROUND STATE mean in Quantum Field Theory? Shouldn’t that just be the vacuum state? | 0  which has an compared to 0. Fields are fluctuations about the GROUND STATE. Virtual particles are created from the VACUUM. The field configuration of MINIMUM ENERGY is usually just the obvious   0 (e.g. out of  away from a particle’s location)

Following the definition of the discrete classical L = T - V we separate out the clearly identifiable “kinetic” part L = T( ∂f ) – V(f ) For the simple scalar field considered earlier V(f )=½m2f 2 is a 2nd-order parabola: V(f ) The f → 0 case corresponds to a stable minimum of the potential. Quantization of the field corresponds to small oscillations about the position of equilibrium there. f Notice in this simple model V is symmetric to reflections of f  -f

As (or in regions where)   0 we have the empty state | 0  representing the lowest possible energy state and serving as the vacuum. The exact numerical value of the energy content/density of | 0  is totally arbitrary…relative. We measure a state’s or system’s energy with respect to it and usually assume it is or set it to 0. What if the EMPTY STATE did NOT carry the lowest achievable energy?

£=½+½ 2¼ 4 Now let’s consider a model with a quartic (“self-interaction”) term: £=½+½ 2¼ 4 with this sign, we’ve introduced a term that looks like an imaginary mass (in tachyon models) Such models were 1st considered for observed interactions like +  - →  0  0 V(f) = -½ 2 + ¼ 4 = -½ 2( - ½λ 2) Now the extrema at f = 0 is a local maximum!  √λ  √λ -  √λ Stable minima at f0 = ± a doubly degenerate vacuum state The depth of the potential at f0 is

V(f) = V(u +f0) = -½(u +f0)2 + ¼ (u +f0)4 √λ  √λ - A translation f(x)→ u(x) ≡ f(x) – f0 selects one of the minima by moving into a new basis redefining the functional form of f in the new basis (in order to study deviations in energy from the minimum f0) V(f) = V(u +f0) = -½(u +f0)2 + ¼ (u +f0)4 = V0 + u2 + √ u3 + ¼u4 plus new self- interaction terms energy scale we can neglect The observable field describes particles of ordinary mass m/2.

the independent states Complex Scalar field £=½+½¼ Note: OBVIOUSLY globally invariant under U(1) transformations   ei £=½11 + ½22 ½12 + 22 ¼12 + 22  1 2 Which is ROTATIONALLY invariant under SO(2)!! Our Lagrangian yields the field equation: 1 + 12 + 112 + 22  = 0 or equivalently  2 +  22 +  212 +  22  = 0 some sort of interaction between the independent states

 2  1 Lowest energy states exist in this circular valley/rut of radius v =  2  1 This clearly shows the U(1)SO(2) symmetry of the Lagrangian But only one final state can be “chosen” Because of the rotational symmetry all are equivalent We can chose the one that will simplify our expressions (and make it easier to identify the meaningful terms) shift to the selected ground state expanding the field about the ground state: 1(x)=+(x)

L=½11 + ½22 Scalar (spin=0) particle Lagrangian L=½11 + ½22 ½12 + 22 ¼12 + 22  with these substitutions: v = becomes L=½ + ½ ½2 +2v+v2+ 2  ¼2 +2v+v2 + 2  L=½ + ½ ½2 + 2 v ½v2 ¼2 +2 ¼22 + 22v+v2 ¼2v+v2

Explicitly expressed in L=½ + ½ ½(2v)2 v2 + 2¼2 +2  + ¼v4 ½  ½(2v)2 Explicitly expressed in real quantities  and v this is now an ordinary mass term!  “appears” as a scalar (spin=0) particle with a mass ½  “appears” as a massless scalar There is NO  mass term!

L = D=+igG LOCAL GAUGE TRANSFORMATIONS +{ Of course we want even this Lagrangian to be invariant to LOCAL GAUGE TRANSFORMATIONS D=+igG free field for the gauge particle introduced L = [  - v22 ] + [   ] + [ FF+ GG] - gvG Recall: F=G-G 1 2 1 2 -1 4 g2v2 2 +{ g2 2 gG[-] + [2+2v+2]GG  2 1 2  4 + [ v4] - [4v(3+2) + [4 +222 + 4] and many interactions between  and  which includes a numerical constant v4 4

                  The constants , v give the coupling strengths of each                  

L = G  plus - gvG seems to describe which we can interpret as: scalar field  with massless scalar  free Gauge field with mass=gv a whole bunch of 3-4 legged vertex couplings L = + + -gvG + But no MASSLESS scalar particle has ever been observed  is a ~massless spin-½ particle  is a massless spin-1 particle spinless , have plenty of mass! plus - gvG seems to describe G  Is this an interaction? A confused mass term?  G not independent? ( some QM oscillation between mixed states?) Higgs suggested: have not correctly identified the PHYSICALLY OBSERVABLE fundamental particles!

 /2  0 and  / becomes real!  /=(cos + i sin )(1 + i2) Remember L is U(1) invariant - rotationally invariant in , (1, 2) space – i.e. it can be equivalently expressed under any gauge transformation in the complex plane Note: or  /=(cos + i sin )(1 + i2) =(1 cos - 2 sin ) + i(1 sin + 2 cos) With no loss of generality we are free to pick the gauge a , for example, picking:  /2  0 and  / becomes real!

 2 1 ring of possible ground states equivalent to rotating the system by angle - 2  1   (x)  (x) = 0

£ 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 This is the technique employed to explain massive Z and W vector bosons…

g12+g12 ( ) ( 2g22W+W+ + (g12+g22) ZZ ) ( ) v22g22W+ W+ When repeated on a U(1) and SU(2) symmetric Lagrangian find the terms: -g1g2 g12+g12 ψe ψeA shifted scalar field, f(x) 1 8 † † ( ) ( 2g22W+W+ + (g12+g22) ZZ ) ( ) H +v H +v No AA term is introduced! The photon remains 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 g2 = MW = MZcosθw √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

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

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

Appendix

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 “OVERLAPS 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.

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

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