1. £ 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?

2. 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!

3. 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

4. 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 † † † †

5. 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

6. 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!

7. 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.

8. = 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.

9. 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

10. † ( ) ( ) ( ) ( 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
W W1-iW2
W1-iW 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

11. ( ) ( 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

12. ( ) 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

13. Notice = cos W according to this theory.MW MZ
Notice = cos W according to this theory.
where sin2W=0.2325
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= ± 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

14. 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.

15. 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)

16. 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 =  GeV
MZ =  GeV

19. 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!

21. 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]

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