Transcript No Slide Title

Weekly Mass
Friday, 4:00 Service
201 Brace Hall
The Higgs Mechanism
Dan Claes
April 8 & 15, 2005
An Outline
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?
The precise dynamical behavior of a system of particles
can be inferred from the Lagrangian equations of motion


d  L  L

0


dt  q  qi
 i
derived from the Lagrange function: L  T  V
here for a classical systems of mass points

Extended to the case of continuous (wave) function(s) k (t , x )

L(t )   dx L (t , x )
 k
3
“The Lagrangian”
L  L (k , x )

an explicit function only of the dynamical variables of
the field components and their derivatives
Euler-Lagrange 
equation
x


L

  L  0
 ( / x )  
 k
L  L (k , x )

• Does not depend explicitly on spatial coordinates (absolute positions)
- the “Ruler Postulate”
translation invariance
- otherwise would violate relativistic invariance
r
 P

3 
d x   r   g 0  L  
0

t
x
 r
 t
Conservation of Linear Momentum!
• Does not depend explicitly on t (absolute time)
- time translation invariance
dH
0
dt
Conservation of Energy!
• Similarly its invariance under spacial rotations xi  R( )ij x j
- guarantees Conservation of Angular Momentum!
 k
L  L (k , x )



• The value of L( x, t ) , i.e., at ( x, t ), must depend only on value(s) of
k ( x , t ) and its derivatives at ( x, t ).
 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  x
 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
n
 n
and
( ) ( x ) than n = 5.
 k
L  L (k , x )

•
L should be real (field operators Hermitian)
 guarantees the dynamical variables (energy, momentum, currents)
are real.
•
L should be relativistically invariant
L(x)  L(x)
 so restrictive is this requirement, it guarantees the derived
equations of motion are automatically Lorentz invariant
Real Scaler Field
the simplest Lagrangian with (x) and   dependence is

£  (  )(  )   
from which
yields
2 2
 
mc 2 2

( ) 


x x 
 £ /   [ £ /  ( ) ]  0
mc 2

2( )  2(   )  0
h
mc 2 


    ( h )   0
the (hopefully) familiar Klein-Gordon equation!
From the starting point for
a relativistic QM equation:
2 2
2 4
E  p c m c
2
together with the quantum mechanical prescriptions
pk  i k
E  i / t
1920
E. Schrödinger
O. Klein
W. Gordon
Matter fields (like the QM wave function of an electron)

i
are known only up to a phase factor  e
(t , x )
a totally non-geometric attribute


If we choose, we can write this as

(  i )
1
2
* 
1
2
₤
1
2
(  i )
1
1 
2
or
2 
1
(   *)
2
(   *)
1
i
2
  







2 2 
2 2
1
1
2
2


  1 
  2


 x x
  x x




 

 * 
2
or
₤  x  x    *

 * 
2
 
   *
x x 
₤
Then treating  and * as independent fields, we find field equations
₤
₤
 ₤ /* [ ₤/ (

/   [
/  ( ) ]  0


*) ]  0
 *   *  0
2
  0
2
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
i

 e 
 *  ei *
or
1  1 cos   2 sin 
2  1 sin   2 cos 
₤
i

 e 
 *  ei *
 * 
2
 
   *
x x 
For an infinitesimally small rotation

₤
₤
₤

₤
₤




 

( / x  ) x
 



( )
₤ )
₤
 *  i *
  i
₤
₤


 *
 * 


 ( * / x  )
x
  *
₤
(
₤ )



)









 *


  
  * x  ( * / x  ) 
  x  ( / x  ) 

  


 
 *

x   ( / x  )
 ( * / x  )



   *

 i

 * = 0 satisfying the continuity equation!

x  x 
x  


and changing sign with  *
(
a conserved 4-vector
(
₤
a charged current density
This is easily extended to 3 (or more) related, but independent fields
for example:
₤
  

       2 

 1  x x 

3
allows us to consider a class of unitary transformations
wider than the single-phase U(1)
Vector Field the field now has 4 components
Though
 
(  )(  )   
2
Spin-1 particles
γ, g, W, Z

has all the 4-vectors, tensors contracted energy is not positive definite
unless we impose a restriction

   0
i.e., only 3 linearly independent components
This is equivalent to replacing the 1st term with the invariant expression
 


£  (     )(     )   
 F  F
[ 



   
2
]
2



  0
2
the (hopefully still) familiar Klein-Gordon equation!
Dirac Field
Spin- ½ particles
e, , quarks
this field includes 4 independent
components in spinor space
L

DIRAC

  (i    mc )
 p  m c  0
2
Dirac’s equation
 ( x)  * ( x)
 ( x)  ( x)
with a current vector: J ( x )  

0
These have been single free particle Lagrangians
We might expect a realistic Lagrangian that involves systems of particles
L(r,t) = L
Vector
describes
photons
need something like:
L
+
L
DIRAC
describes
e+e objects
 

but each term
describes free
non-interacting
particles

 12 (      )(     )    

  (i    m )
2
+
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:

~    
Or consider this:
Our free particle equations of motion were all
homogeneous differential equations.
 
[
 0
2
]

  0
2
  p  m c  0
When the field is due to a source, like the electromagnetic (photon!)
field you know you need to make the eq. of motion inhomogeneous:
[
]

   
2

charged 4-current density
and

LINT    
would do the trick!
 ( x)  ( x)
e

Dirac electron current
exactly the
proposed
bilinear form!
Just crack open Jackson:
A charge interacts with a field through:
 
INT  ( V  J  A)
L
  J  A

J  (; J )


A  (V ; A)

current-field interactions
the fermion
(electron)
the boson
(photon) field
LINT  (e   ) A
particle
field
antiparticle
(hermitian conjugate)
field
from the Dirac
expression for J
Now let’s look back at the FREE PARTICLE Dirac Lagrangian
LDirac=iħc    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.”
What if we GENERALIZE this?
Introduce more flexibility to the transformation? Extend to:
ei(x)
but still enforce UNITARITY?
LOCAL GAUGE TRANSFORMATION
Is the Lagrangian still invariant?
LDirac=iħc    mc2
(ei(x)) = i((x)) + ei(x)()
So:
L'Dirac = ħc((x))  
 iħcei(x)( )ei(x)  mc2
L'Dirac =
ħc((x))    iħc( )  mc2
LDirac
For convenience (and to make subsequent steps obvious) define:
 c
(x)  q (x)
then this is re-written as
 e
L'Dirac = q   ()  LDirac
recognize this????
 q / c

L'Dirac = q   ()  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mc2 ](q )A
and that A  A     defines its transformation
under the same local gauge transformation
L=[iħcmc2 ](q )A
•We introduced the same interaction term moments ago
following electrodynamic arguments (Jackson)
• the form of the current density is correctly reproduced
•the transformation rule
A' = A +  
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 A ) that “couples” to .
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]
for U(1) this is a
1×1 unitary matrix
(just a number)

+iq

/ħc

=e
(
)
q


   i c    
the extra term
that gets introduced
If we replace every derivative  in the original free particle Lagrangian
with the “co-variant derivative”
g
= + i ħc A
D
then the gauge transformation of A will
cancel the term that appears through 
i.e.
(D )/ = e-iq/ħcD restores the invariance of L
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”
 
i ( g / c )  
   U  e

8 3x3 “generators”
3-dimensional matrix
formed by linear combinations
of 8 independent
fundamental matrices
  The field is assumed
 1
to exist in any of 3
   2  possible
independent
 
color states
 3
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.
THEN in an effort to explain decays:
e
d
d
u
neutron decay
_
e
??
u
d proton
u

muon decay
pion
u_
d
+
??

e
_
e
??

hadron decays
involve the
transmutation of
individual quarks
as well as the observed inverse of some of these processes:
neutrino capture by protons
d
u
u
_
??
e
e
e

??

neutrino capture by muons
e+
d
u
d
in terms of the gauge model of photon-mediated
charged particle interactions
e
e


e
p
???
e
e
n
p

p
required the existence of 3 “weakons” :
W , W , Z
e
W
e
u
e
W ?
W
d
e
d
W ?
u

 0 1

1
0


 x  
SU(2) electro-weak symmetry


i  / 2
   U  e
 
 0 1
0  i 1 0 
     




   

1
0
i
0
0

1



 

u
d
L
e
e
L

0  i
 y  

i 0 
1 0 
 z  

 0  1
1
2
“Rotational symmetry”within weakly coupled
left-handed isodoublet states introduces 3 weakons:
W+, W, Z and an associated weak isospin “charge”
This SU(2) theory then
L=[iħc
2 ]


mc


1 
 F
F

4




(g )·G
2

describes doublet Dirac particle states in interaction with
3 massless vector fields
(think of something like the -fields, A)
G


This followed just by insisting on local SU(2) invariance!
In the Quantum Mechanical view:





•These Dirac fermions generate 3 currents, J = (g  )
•These particles carry a “charge” g, which
is the source for the 3 “gauge” fields
2
The Weak Force so named because unlike the PROMPT processes

e+

e
qg
_
rg
e+
e
or the electromagnetic decay: 0 
qr

which seem
instantaneous
which involves a 1017 sec lifetime
path length (gap) in photographic emulsions mere nm!
weak decays are “SLOW” processes…the particles
involved: ,
106 sec
700 m pathlengths
±,  are nearly “stable.”
108 sec
+++
7 m pathlengths
887 sec
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.
Some Classical Fields
The gravitational field around a point source
(e.g. the earth)
is a scalar field
M earth
g ( x, y , z )  G
2
2
2
( x  x0 )  ( y  y0 )  ( z  z0 )
An electric field is classical example of a vector field:

E( x, y, z)  iˆEx ( x, y, z)  ˆjE y ( x, y, z)  kˆEz ( x, y, z)
effectively 3 independent fields
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)
timespatial
dependent part
part
Ψ=
e-iEt/ħ
spin
space
α
R(r)S(θ)T(φ)g(ms)
β
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.
That 2-dim spin-½ space is operated on by
 0 1
 x  

 1 0
0  i
 y  

i 0 
1 0 
 z  

 0  1
or more generally by
 
 0 1  0  i   1 0 
    a
  b
  c

 1 0   i 0   0  1
The SU(2) transformation group (generalized “rotations” in 2-dim space)
 
is based on operators:
i  / 2
Ue
“generated by”
traceless Hermitian
matrices
What’s the most general traceless HERMITIAN 22 matrices?
c aib
aib c
and check out:
c
aib
= a 0 1 +b 0 -i +c 1 0
a+ib c
1 0
i 0
0 -1
 
 0 1  0  i   1 0 
    a
  b
  c

 1 0   i 0   0  1
FOR
SU(3)
What’s the form of the most general traceless HERMITIAN 3×3 matrix?
Diagonal terms
have to be real!
Transposed
positions
must be
conjugates!
=
a1ia2
a4ia5
0 1 0
a1 1 0 0
0 0 0
+a3
a1ia2
a3
a6ia7
a6ia7
0 -i 0
+a2 i 0 0
0 0 0
0 0 -i
0 0 0
i 0 0
a4ia5
0 0 1
+a4 0 0 0
1 0 0
+a3
0 0 0
+a6 0 0 1
0 1 0
Must be traceless!
+a7
0 0 0
0 0 -i
0 i 0
+a8
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i· /2ħ

three 2x2 matrix
operators
3 independent
parameters
3 simultaneous gauge transformations
3 vector boson fields
SU(3) “rotations” occur in an 8-dim “space”
i·
/2ħ
e

8 independent
parameters
8 3x3
operators
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 
 ( p) 0  p
†
which has an
E m c p c
2 4
2 2
0
compared to .
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( ∂ ) – V( )
For the simple scalar field considered earlier V( )=½m2 2
is a 2nd-order parabola:
The  → 0 case corresponds to a
stable minimum of the potential.
Quantization of the field
corresponds to small oscillations
about the position of equilibrium there.
V( )

Notice in this simple model V is symmetric to reflections of   
Obviously as 0 there are no intereactions between the fields
and we will have only free particle states.
we have the empty state | 0 
And as (or in regions where)   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?
pq
=
0  pq 0
†
We will call 0| |0 = vev the
“vacuum expectation value” of an operator state.
Now let’s consider a model with a quartic (“self-interaction”) term:
£=½()()½ 2¼ 4
Such models were 1st considered
for observed interactions like
with this sign, we’ve introduced a
term that looks like an imaginary
mass (in tachyon models)
+   →  0  0
V()  ½ 2 + ¼ 4  ½ 2(  ½λ 2)
Now the extrema at  = 0
is a local maximum!
Stable minima at 0
=±



√λ
√λ
√λ
a doubly degenerate vacuum state

The depth of the potential at 0 is V 
0
4
4
V()  ½ 2 + ¼ 4


√λ
√λ
A translation (x)→ u(x) ≡ (x) – 0
selects one of the minima by moving into a new basis
redefining the functional form of  in the new basis
(in order to study deviations in energy from the minimum 0)
V()  V(u +0)  ½(u +0)2 + ¼ (u +0)4
 V0 + u2 + √ u3 + ¼u4
energy scale
we can neglect
plus new selfinteraction terms
The observable field describes
particles of ordinary mass /2.
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 )
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
1
2
2
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)


( x )  1( x )  v


 ( x )  2 ( x )
shift to the
selected ground state
expanding the field about the ground state: 1(x)=+(x)
Scalar (spin=0) particle Lagrangian
L=½(1)(1) + ½(2)(2)
½(12 + 22 )¼(12 + 22 )
with these substitutions:
v=
 /
2
becomes


( x )  1( x )  v

 ( x )  2 ( x )
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)
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
 “appears” as a scalar (spin=0)
mass term!
particle with a mass m  2v 2

 2
2
½()()
 “appears” as a massless scalar
There is NO  mass term!
Of course we want even this Lagrangian to be invariant to
LOCAL GAUGE TRANSFORMATIONS
L
D=+igG
Let’s not worry about the
higher order symmetries…yet…
1
1 2

1 
2


    igG  *   igG     *   ( *  )  F F
2
2
4
4
free field for the gauge
particle introduced
) ][(
[(
) ]
Recall: F=GG
L=
2v2
1
g
1
-1

2
2


[    v  ] + [    ] + [ F F+ GG]
2
2
2
4
 gvG
gG[
+{

1
g2
2+2v+2]G G

]
+
[



2

+ 2 [ 2 v4]  4 [4v(3)  [4 22  4]
which includes a v4
numerical constant 4
and many interactions
between  and 
The constants , v give the
coupling strengths of each


















which we can interpret as:
L=
a whole bunch
massless free Gauge
scalar field 
scalar  field with gvG + of 3-4 legged
with

m  2v 2
vertex couplings

mass=gv
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!
Remember L is U(1) invariant  rotationally invariant in , (1, 2) space –
Note:
i.e. it can be equivalently expressed
under any gauge transformation in the complex plane
/
i (x)
   e

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  ,
for example, picking:
1
2
 sin    cos
1
   tan (2 1 )
 /2  0 and  / becomes real!
ring of possible ground states
2
equivalent to
rotating the
system by
angle 

2
tan   
1
1
sin  

 2
1  2
2
1  2
2

cos 
2
2
12  2 2
 (x)
1
1  2
2
    
 i  1 22 12 2 
   
1
2


 (x) = 0
2
With  real, the field  vanishes and our Lagrangian reduces to
2 2

1
g
v
1


2 2
  

£   2      v     4 F F  2 G G 


g2 2
 4  4

2

3
   G G  g v G G  v     v
4  4
 2
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…
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!
When repeated on a U(1) and SU(2) symmetric Lagrangian
g1g2
find the terms:
shifted scalar
ψe ψeA
2
2
g1 +g1
field, (x)
† +
1 H†+v
2
+
( ) ( 2g2 W W + (g12+g22) ZZ ) ( H +v)
8
No AA term is introduced! The photon remains massless!
But we do get the terms
1 v22g 2W+† W+
2

8
MW = 2 vg2
1
2+g 2 )Z Z
(g
1
2

8
MZ = 2 v√g12 + g22
1
1
At this stage we may not know precisely the values of g1 and g2, but note:
g2
MW
=
MZ
√g12 + g22
MW = MZcosθw
and we do know THIS much about g1 and g2
g1g2
g12+g12
=e
to extraordinary precision!
from other weak processes:
 e +e +


N  p + e +e
e
e
u
e
e
W
W
d
2
e
give us sin2θW
lifetimes (decay rate cross sections) ~gW = sinθ
W
2
( )
MW
Notice MZ = 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 -decay:
neutron lifetime=886.7±1.9 sec
and
• a high precision measurement of
muon lifetime=2.19703±0.00004 sec
and
• measurements (sometimes just crude approximations perhaps)
of the cross-sections for the inverse reactions:
as well as
e- + p  n +  e
 e + p  e+ + n
electron capture
anti-neutrino absorption
 e + e-  e- +  e
neutrino scattering
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
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
91.1884 ± 0.0022
2.49693 ± 0.0032
41.488 ± 0.078
20.788 ± 0.032
0.0172 ± 0.012
0.1418 ± 0.0075
0.1390 ± 0.0089
0.2219 ± 0.0017
0.1540 ± 0.0074
0.0997 ± 0.0031
0.0729 ± 0.0058
0.2325 ± 0.0013

2.4985
41.462
20.760
0.0168
0.1486
0.1486
0.2157
0.1722
0.1041
0.0746
0.2325
0.1486
0.935
0.669
SLC
A0,ℓFB
Ab
Ac
0.1551 ± 0.0040
0.841 ± 0.053
0.606 ± 0.090
pp
mW
80.26 ± 0.016
80.40
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

  

0  e  
Lint  G ( e e ) L  0 eR  eR (  )  
 e L 
 

which with
_
Higgs=
_
0
v+H(x)
_
becomes
_
Gv[eLeR + eReL] + GH[eLeR + eReL]
_
_
_
_
Gv[eLeR + eReL] + GH[eLeR + eReL]
_
_
ee
from which we can identify:
or
mee e 
ee
me = Gv
me
v
e eH
Bibliography
Classical Mechanics, H. Goldstein
Addison-Wesley (2nd edition) 1983
Lagrangians, symmetries and conservation laws
Classical Electrodynamics
J. D. Jackson (3rd Edition)
John Wiley & Sons 1998
covariant form of Maxwell’s equations
gauge transformation on the 4-potential
electron-photon interaction Lagrangian
Relativistic Quantum Fields
J. Bjorken, S. Drell
McGraw-Hill 1965
Klein Gordon Equation, Dirac Equation
Introduction to High Energy Physics
Donald H. Perkins (4th Edition)
Cambridge University Press 2000
gauge transformation & conserved charges
Advanced Quantum Mechanics
J. J. Sakurai
Addison-Wesley 1967
neutral and complex scalar fields
gauge transformations & conserved charges
vector potentials in quantum mechanics
Quantum Fields
N. Bogoliubov, D. Shirkov
Benjamin/Cummings 1983
real scalar fields, vector fields, Dirac fields
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 1983
the Higgs field
Appendix
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)

Higgs=
+
0
with Q = I3+Yw /2 and I3 = ±½
for Q=0  Yw = 1
Q=1  Yw = 1
couple to EW UY(1) fields: B

Higgs=
+
0
with Q=I3+Yw /2 and I3 = ±½
Yw = 1
Consider just the scalar Higgs-relevant terms
£
Higgs
1  †
1 2 † 1
 (  )        ( † )2
2
2
4
with
2  0
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
1  †
 ( 1 1   †2 2       4† 4 )
2
1 2 †
1
†
  (11      44 )   (1†1      4†4 )2
2
4
L
Higgs
1  †
 ( 1 1   †2 2       4† 4 )
2
1 2 †
1
†
  (11      44 )   (1†1      4†4 )2
2
4
just like before:
U =½2† + ¹/4 († )2
12 + 22 + 32 + 42 =
Notice how 12,
22

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
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.
So let’s pick ONE field to remain NON-ZERO.
1 or 2

Higgs=
3 or 4
+
0
because of the SO(4) symmetry…all are equivalent/identical
might as well make  real!
Can either choose
v+H(x)
0
or
0
v+H(x)
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)
0

0 0  0 
or
0
v+H(x)

00 0  0 0
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.
0  0  0 v  H ( x) 0  0 v 0  0 H ( x)
0 0  0 v 
0H
 ( x0) H
0 ( x)
 v 0 0  0 H ( x)
0 =v
a non-zero
v.e.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)

Higgs=
+
0
0
=
v+H(x)
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.
With the choice of gauge settled:
+
0
Higgs=  0 = v+H(x)
Let’s try to couple these scalar “Higgs” fields to W, B which means
replace:
 
D

 
Y

    ig1 B  ig2  W
2
2
which makes the 1st term in our Lagrangian:
 
 
†
1 
Y

Y

  

 
    ig1 B  ig2  W       ig1 B  ig2  W  

2 
2
2
2
2
  
 


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:
†
 
 
0
0 




1 
Y

Y




 
     ig1 B  ig2  W 

    ig1 B  ig2  W 

2 
2
2
2
2
 H  v   
 H  v  
where Y =1 for the coupling to B
†
 
 
0
0 




1 
1

1 



 
     ig1 B  ig2  W 

    ig1 B  ig2  W 

2 
2
2
2
2
 H  v   
 H  v  
recall that
W3
W1iW2
τ ·W = 0 1 W1 + 0 -i W2 + 1 0 W3 =
W1iW2 W3
1 0
i 0
0 -1
→→
g1B
=
1 0
(
2
H†+v )
2
g2
2
1 0
= (
8
†
H +v )

g2
2
W3
(W1iW2)
g1B  g2W3
2 g 2W

g2
(W1iW2)
2
g1B
2

g2
2
2
W3
2 g 2W

0
H+v
2
 g12  g22 Z 
† +
1 H†+v
2
+
) ( 2g2 W W + (g12+g22) ZZ ) ( H +v)
= (
8
0
H+v
1 H†+v
2W+†W+ + (g 2+g 2) Z Z ) ( H +v)
(
)
(
2g
=
2

1
2

8
No AA term has been introduced! The photon is massless!
But we do get the terms
1 v22g 2W+† W+
2

8
MW = 2 vg2
1
2+g 2 )Z Z
(g
1
2

8
MZ = 2 v√g12 + g22
1
1
At this stage we may not know precisely the values of g1 and g2, but note:
2g2
MW
=
MZ
√g12 + g22