1. Weekly Mass
Friday, 4:00 Service
201 Brace Hall
The Higgs Mechanism
Dan Claes
April 8 & 15, 2005
I. Lagrangians
Why we love symmetries, even to the point of seemingly
imagining them in all sorts of new non-geometrical spaces.
II. Introducing interactions into Lagrangians: SU(n) symmetries
III. Symmetry Breaking
Where’s the ground state? What the heck are Goldstone bosons?
An Outline

2. 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 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

3. Does not depend explicitly on spatial coordinates (absolute positions) m
Does not depend explicitly on spatial coordinates (absolute positions)
the “Ruler Postulate”
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!

4. The value of , i.e., at , must depend only on value(s) of
and its derivatives at
- no “non-local” terms
which, in general, create problems in
causality and with non-real physics quantities
- non-local terms never appear in any Standard Model field theory
though often considered in theories seeking to extend field theory
“beyond the Standard Model”
THINK: wormholes and time travel
For linear wave equations need 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.

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

6. 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

7. ₤ ₤ 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

8. ₤
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

9. ₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ ₤ = 0 For an infinitesimally small rotation
satisfying the continuity equation!
and changing sign with f f*
a conserved 4-vector
a charged current density

10. This is easily extended to 3 (or more) related, but independent fields
for example: ₤
allows us to consider a class of unitary transformations
wider than the single-phase U(1)

11. £ 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!

12. 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:

13. 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:

14. 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

15. (hermitian conjugate)
Just crack open Jackson:
A charge interacts with a field through:
current-field interactions
the fermion
(electron)
the boson
(photon) field
from the Dirac
expression for J
particle
field
antiparticle
(hermitian conjugate)
field

16. 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.”

17. 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

18. 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????

19. 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

20. 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 .

21.  (D )/ = e-iq/ħcD   / =  [e+iq/ħc] = e+iq/ħc D= + i A
We can generalize our procedures into a PRESCRIPTION to be followed,
noting the difference between LOCAL and GLOBAL transformations
are due to derivatives:
  / =  [e+iq/ħc] = e+iq/ħc 
for U(1) this is a
1×1 unitary matrix
(just a number)
the extra term
that gets introduced
If we replace every derivative  in the original free particle Lagrangian
with the “co-variant derivative”
D= + i A g ħc
then the gauge transformation of A will
cancel the term that appears through 
(D )/ = e-iq/ħcD
i.e.
restores the invariance of L

22. 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

23. 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 _ ?? 

24. 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

25. 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

26. 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

27. 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 doublet 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

28.  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 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!).

29. 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.

30. 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

31. 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.

32. 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

33. 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

34. 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 -i 0 i
= a a a a4
+a a a a8 i i i
0 i 0

35. 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

36. Spontaneous Symmetry Breaking
Englert & Brout,
Higgs , 1966
Guralnick, Hagen & Kibble 1964
Kibble
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.

37. 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)

38. 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

39. Obviously as 0 there are no intereactions between the fields
and we will have only free particle states.
And 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? =
We will call 0| |0 = vev the
“vacuum expectation value” of an operator state.

40. £=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

41. 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.

42. 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

43.  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)

44. 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

45. 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!

46. L = D=+igG LOCAL GAUGE TRANSFORMATIONS +{
Of course we want even this Lagrangian to be invariant to
LOCAL GAUGE TRANSFORMATIONS
D=+igG
Let’s not worry about the
higher order symmetries…yet…
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

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

48. 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!

49.  /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!

50.  2 1 ring of possible ground states equivalent to rotating the
system by
angle - 2  1 
 (x)
 (x) = 0

51. £ 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…

52. 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!

53. 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

54. ( ) 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

55. 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

56. 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

57. Electroweak Precision Tests
LEP
Line shape: mZ(GeV)
ΓZ(GeV)
0h(nb)
Rℓ≡Γh / Γℓ
A0,ℓFB
τ polarization: Aτ Aε
heavy flavor: Rb≡Γb / Γb
Rc≡Γc / Γb
A0,bFB
A0,cFB
qq charge asymmetry: sin2θw ± ±
± 0.078
± 0.032
± 0.012 ± ± ± ± ± ± ± -
2.4985
41.462
20.760
0.0168
0.1486
0.2157
0.1722
0.1041
0.0746
0.2325
SLC
A0,ℓFB Ab Ac ±
± 0.053
± 0.090
0.1486
0.935
0.669 pp mW
80.26 ± 0.016
80.40

58. 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]

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

60. Bibliography
Classical Mechanics, H. Goldstein Lagrangians, symmetries and conservation laws
Addison-Wesley (2nd edition) 1983
Classical Electrodynamics covariant form of Maxwell’s equations
J. D. Jackson (3rd Edition) gauge transformation on the 4-potential
John Wiley & Sons electron-photon interaction Lagrangian
Relativistic Quantum Fields Klein Gordon Equation, Dirac Equation
J. Bjorken, S. Drell
McGraw-Hill 1965
Introduction to High Energy Physics gauge transformation & conserved charges
Donald H. Perkins (4th Edition)
Cambridge University Press 2000
Advanced Quantum Mechanics neutral and complex scalar fields
J. J. Sakurai gauge transformations & conserved charges
Addison-Wesley vector potentials in quantum mechanics
Quantum Fields real scalar fields, vector fields, Dirac fields
N. Bogoliubov, D. Shirkov
Benjamin/Cummings 1983
Weak Interactions of Leptons & Quarks Electro-weak unification, U(1), SU(2), SU(3)
E. Commins, P. Bucksbaum electro-weak symmetry breaking
Cambridge University Press the Higgs field

61. Appendix

62. 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

63. 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 † † † †

64. 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

65. 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!

66. 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.

67. = 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.

68. 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.

69. 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

70. † ( ) ( ) ( ) ( 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

71. ( ) ( 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