The Weak Interaction
Sun sun sun
Rising sun the creator
Mid day blazing sun the destroyer
Rudra
Setting sun the maintainer and
continuance
Greatest of all
Sun sun sun
(Gajanan Mishra)
1. Costituents of Matter
2. Fundamental Forces
3. Particle Detectors
4. Symmetries and Conservation Laws
5. Relativistic Kinematics
6. The Static Quark Model
7. The Weak Interaction
8. Introduction to the Standard Model
9. CP Violation in the Standard Model (N. Neri)
1
Simple facts
The Weak Nuclear Interactions concerns all Quarks and all Leptons
The Weak Interaction takes place whenever some conservation law (isospin,
strangeness, charm, beauty, top) forbids Strong or EM to take place
In the Weak Interaction leptons appear in doublets:
Q
L(e) = +1
L(μ) = +1
L(τ) = +1
0
e


-1
e


Doublets are characterized by electron, muon, tau numbers (each conserved,
except in neutrino oscillations)  whose sum is conserved.
…and the relevant anti-leptons. For instance:
     
(see the section on Fundamental Interactions)
2
Simple facts
Weak Nuclear Interaction violates Parity
The Parity violation is maximal
The fundamental weak couplings are to fully
left-handed fundamental fermions (and fully
right-handed fundamental antifermions).
Discovered first in the Wu
experiment.
Confirmed in all other
experiments on Weak
Interactions.
Weak Nuclear Interaction violates CP
This fact will need to be incorporated in the theory:  a phase in the CKM matrix.
CPT is conserved by Weak Interactions
Weak Interactions violate P, C, CP, T but not the combination CPT.
3
Weak Interactions allow
for processes otherwise
impossible
At low energy: Fermi Theory
At high (and low) energies:
Electroweak Theory
Neutron decay
103 s
Long lifetime
due to the small
mass difference
Inverse n decay
10-43
cm2
 Has only weak
interactions
Lamda decay
p-
10-10 s
S=1:
strong/e.m.
interactions
forbidden
Pion decay
+
10-8 s
Leptons are the
lightest
particles
The first theory of Weak Interactions was developed by
Enrico Fermi in close analogy with Quantum
Electrodynamics.
The process to be explained was the nuclear beta decay.
Nature rejected his paper “because it contained speculations too remote
to be of interest to the reader.”
‘Tentativo di una teoria…’ ,
Ric. Scientifica 4, 491, 1933.
4
Fermi Theory of the Beta Decay
A(Z , N )  A(Z  1, N 1)  e   e
n  p  e   e
d  u  e   e
At the fundamental (constituents) level
u
d
g
J weak
1
M W2 Z
GF
'
J weak
g
g 2  '
LFermi  GF J J   2 J J 
MW

'
The rate of decay (transizions per unit time) will be:
2 2
2 dN
W
GF M

dE0
M
E0
2
Integration over spins and angles
Energy of the final state
5
F: Fermi transitions.
No nuclear spin change
∆J (Nuclear Spin) = 0
Leptonic state: spin singlet ↑↓
│M│2 ≈ 1
GT: transizioni alla Gamow-Teller.
Nuclear Spin change
∆J (Nuclear Spin) = +1,-1
Letponic state: spin triplet ↑↑
│M│2 ≈ 3
Several transitions are mixed transitions (F e GT).
In the assumption of no interference, one typically has :
2

2
G M  G c M F  c M GT
2
F
2
F
2
V
2
A
2

With weights:
cV 1
c A / cV 1.25
6
Beta Decay Kinematics
dE0 arises from the finite lifetime of the initial state
dN
dE0
Q-value
E0

electron, p, E

proton, P, T

neutrino, q, E
Final state
In the rest frame of the neutron :
  
P  q  p 0
T  E  E  E0
The recoil kinetic energy of the nucleon
Is negligible :
2
P / 2 M  10 3 MeV
Energy carried away by the neutrino :
cq  E0  E
7
Number of neutrino and electron available states with electron and neutrino
momenta in the ranges p,p+dp e q,q+dq
Vd 2
p dp
h3
Vd 2
q dq
h3
Choosing a normalized volume and integrating over the angles :
4 2
p dp
3
h
4 2
q dq
3
h
Neglecting dynamical correlations between p,q…
Moreover, there is no free phase space for the proton, since given p,q its

momentum is fixed: P    p  q 
The phase space is :
16  2 2 2
d N  6 p q dp dq
h
2
Now, expressing q as a function of the total available energy and E :
cq  E0  E
dq  dE0 / c
8
16  2 2
d N  6 3 p ( E0  E ) 2 dp  p 2 ( E0  E ) 2 dp dE0
hc
N
 E0  E
Kurie plot
2
p
2
N ( p)dp F ( Z , p)  p 2 ( E0  E ) 2 dp
N ( p)dp F ( Z , p)  p ( E0  E )
2
2
General form
Coulombian Correction F(Z,p)
m2
Coulombian Correction and non1
dp
zero neutrino mass
( E0  E ) 2
Kurie plot
9
Beta Decay Spectrum in short
The coupling constant enters here
10
Total decay rate
The total decay rate depends on the coupling constant and the phase space.
For a fixed coupling constant, the rate is the integral of :
d 2 N 16  2 2
 6 3 p ( E0  E ) 2 dp  p 2 ( E0  E ) 2 dp
dE0 h c
over the electron spectrum.
This quantity features a sharp dependence on the Q-value E0
This can be quickly appreciated in the (somewhat crude) relativistic electron (E
= pc) approximation :
N 
E0
0
E05
dE E ( E0  E )  E  dE E   dE E  2 E0  dE E 
30
2
2
2
0
2
4
3
Sargent’s rule
11
Coupling constants : Eelectromagnetic and Weak
A reminder :
e2
1
 
c 137
In rationalized and natural units
e is adimensional :
The Weak Fermi constant
   dyne cm cm 

erg cm

e2
1
 
4 137
 e  0.09
GF
5
2

1.2

10
GeV
( c )3
GF
2 g2

3
(c ) 8M W2 c 4
GF  9.1105 MeV  fm3
The Weak Coupling
constant is actually bigger
than the fine structure
constant.
But at low energies it is
damped by the W mass
into the small GF constant
g  GF
2
w

8
MW c2
2

2
 g w  0.65
g w2
1
w  
4 29.5
12
Weak Decays and Phase Space in the low energy regime
According to the Sargent’s rule one has – roughly :
The neutron lifetime :
n 

n
 G  m  m  GF2 (m) 5
4
2
F
And this has a general validity. In fact :
The muon lifetime :
 ( )   (n)
(mn  m p )5
(m  me )5
103 2.71010 s 107 s
For a charmed particle :
(mn  m p )5
5
3  1.3 
11
 ( D)   (n)

10
s

10
s


5
(mD  mK  m )
 1000 
13
Electromagnetic
e
f

Weak
f

f
 ig 
f
q
2
e
2
f
e

f
g
q 2  M 2c 2
g2
f
Low Energy Matrix Element
 ig 
q
f
High Energy Matrix Element
 i ( g   q q / M 2c 2 )
f

f
f
W
e
f
g
2
f
g
f
e2
f
g
f
i ( g   q q / M 2c 2 )
q 2  M 2c 2
g 
2
 ig 
M 2c
2
2
g

G
F
2
e
14
Inverse Beta Decay
e  p  n  e

e
e
p

n
2 2
2 dN
W
GF M

dE0

1

2
GF2 M p 2
p is the momentum of the neutron/positron system in their CM
This is a mixed (Fermi + Gamow-Teller) transition
  1043 (cm2 ) p 2 ( MeV / c) 2
M 4
2
A very small cross section
The cross section increases with E
15
Neutrino discovery:
Principle of the experiment
In a nuclear power reactor, antineutrinos come from  decay of radioactive nuclei
produced by 235U and 238U fission. And their flux is very high.
1. The antineutrino reacts
with a proton and forms n
and e+
 e p  n e
Inverse Beta Decay
2. The e+ annihilates
immediately in gammas
3. The n gets slowed down
and captured by a Cd
nucleus with the emission of
gammas (after several
microseconds delay)
Water and
cadmium
Liquid
scintillator
4. Gammas are detected by the scintillator:
the signature of the event is the delayed
gamma signal
 ( e p  ne  ) 10 43 cm 2
1956: Reines and Cowan at the Savannah nuclear power reactor
16
17
The size of the detector might be important. And this is because of the
small cross neutrino section.
Not a specific detector. But… the
typical configuration of a low energy,
low background undergound
neutrino detector :
 Neutrino beam
 Massive, instrumented detector
 Detector transparent to signal
carriers
 Background control!
« I went to the general store
but they did not sell me
anything specific»
18
Parity violation in Beta Decay
1956: Lee-Yang, studying the decay of charged K mesons hypotesized that
Weak Interactions cold not conserve Parity.
1957: esperiment by Wu et al.
60
Co  28
Ni  e   e
60
27
A sample of Co-60 nuclei at 10 mK in a magnetic field.
The Co-60 spin (J=5) get statistically aligned by the magnetic field.
The daughter nucleus (Ni*) has spin 4
The experimentally observed distribution for the emitted electron has the form :
I ( ) 1  
Hz

p
v
 1   cos 
E
c
p
e


J (Co60)
19

p
v
I ( ) 1  
 1   cos 
E
c
Hz
p
e


J (Co60)


P: B  B
P:   


P: p   p


P :  p   p
This term violates Parity,
by correlating the
momentum of the electron
to the Co-60 spin. This
alignment fades away with
increasing energy.
20
V-A structure of Weak Interactions
The helicities of neutrino and electron are :
e

v/c
e

v/c

1
Neutrinos are
 considered
 1 massless !
This property must be part of a consistent theory of Weak Interaction: the
description of Dirac-type elementary constituents
«Electroweak analogy». What is the structure of the weak current(s) ?
e p  e p
Electromagnetic
2
e
M  2 J baryonsJ leptons
q
J baryon  p   p
 e p  n e
Weak
g w2
weak
weak
M 2
J
J
baryons
leptons
q  M W2
J lepton  e   e
21
g w2
weak
weak
M 2
J
J
baryons leptons
q  M W2
M weak 
Charged weak currents
According to the original idea by Fermi :
GF
 pO n  eO  
2
At low energy
O  
In the earliest days of the parity violation discovery, it was natural to guess that the
violation itself might be a special property of neutrinos.
The two component neutrino theory: if neutrinos were massless , then they could
be polarized only parallel to the direction of motion (positive helicity) or antiparallel
to it (negative helicity).
But parity violation was seen also in reactions like
  p 
And was found to be a general property of the Weak Interactions.
A theory of the Weak Interactions had to be based on concepts like universality
and parity violation.
22
The two-component theory of the (massless) Neutrino
i 
The spin-1/2 pointlike particle wave
function obeys the Dirac Equation :
Four components : two spin states of particle
two spin states of antiparticle


  m   0
          2 
• Massive particle: both spin states must be described by the same wavefunction
because the spin direction is not Lorentz-invariant.
• Massless particle: it always travel at the speed of light, so its spin direction can
be defined in a Lorentz-covariant way (parallel or antiparallel to the direction of
the momentum, i.e. positive or negative helicity).
In the Weyl representation of the Gamma Matrices:
 0  1
0
 


1
0


 0
k
  k
 
 
k

0
23
Introducing the bispinors (upper and lower
components) :
i 

u 
   
v

u
i  i  u   mv
t
v  
i  i v   mu
t

  m   0
Dirac Equation in the Weyl
representation
For a massles fermion, the upper and lower components are decoupled :

u
i   i  u
t

v
i   i  v
t

E   p

E   p
u 
 R   
0
Right-handed
spinor
For a massles particle, E= p
for u

for v

0
 L   
v


 p p


 p p
Left-handed
spinor
24
Let us now introduce the Gamma-5
matrix (in the Weyl representation) :
1 0 
  i     

0

1


5
One can then build right-handed or left-handed
wavefunctions by using the projectors
0
1
2
3
u 
1  5
      R
2
0
0
1  5
      L
2
v
More in general, in the case of massive particles :
1  5
PR 
2
gives a v/c polarization along the direction of p (+1 when v=c)
1  5
PL 
2
gives a -v/c polarization along the direction of p (-1 when v=c)
Before the Parity violation experiments, there was no reason con consider the
right and left-handed spinors as particularly useful. However, detailed evidence
was found that only the left-handed spinor occurs in Weak Interactions
25
Only left-handed spinor particles (and right-handed spinor antiparticles) take part
in the Weak Interactions. This has several consequences :
a) The existence of a two-component massless neutrino theory
(see before)
b) Maximal Parity violation
If we carry out the P operation on the neutrino described
by ψL, we obtain a neutrino described by ψR, which is
unallowed in the theory.
c) Maximal C violation
If we carry out the C operation on the neutrino described
by ψL, we obtain an antineutrino described by ψL, which is
unallowed in the theory.
d) T conservation
This is because T reverses both spin and linear momentum.
e) CP conservation
(see the lecture on Symmetries and Conservation Laws)
There exists – however – tiny violations of CP and T invariance in the Weak
Interactions (see lecture on CP violation)
26
1  5
PR 
2
Notable properties of the projection operators
 5  i 0  1  2  3
 


      2
1  5
PL 
2

1  5 1  5 1
1
1
2
5
5
5
PL PL 
 1   5  2  1   (  2)  1   5  PL
2
2
4
4
2






1  5 1  5 1
1
1
2
5
5
5
PR PR 
 1   5  2  1   (  2)  1   5  PR
2
2
4
4
2






1  5 1  5 1
1
5
5
2
PR PL  PL PR 
 1       5  1   5 5  0
2
2
4
4




1  5 5 1 5 2 1 5
PR 
     5    1  PR
2
2
2
5

 

5
1


1 5 2 1 5
5
5
PL 
     5    1   PL
2
2
2

 

27
5
1




 1
PR 
 
1   5    PL
2
2

And this is because :

(an odd number of exchanges with a
different matrix)
 5   i 0 1 2 3     (1) i  0 1 2 3      5
5
1




 1
PL 
 
1   5    PR
2
2


The Universal Four-Fermion Matrix Element
Propagator and
coupling constant
i ( g   q q / M 2c 2 )
q M c
2
2 2

g 
2
 ig 
2 2
M c
g G
2

GF
M weak 
 DLO B  CLO AL
2
2
F

A
B
g
g
C
D
Now, which is the form of the current ? We know that it has to be of the form :

L
C

O AL   C PR OPL A 
28
Electromagnetism
Weak Interactions
 C O A    C  
 C PROPL A 

Scalar
  
Vector


i  
        Tensor
2
    5
Axial Vector
Now, which is the form of the current ? We
know that it has to be of the form :

(because of Lorentz invariance requirements)
  5
Pseudoscalar
In the case of the Weak Interactions :
 Scalar (originates F transitions)
PROPL  O PR PL  0
 Vector (produces F transitions)
PROPL  PR  PL    PL PL    PL


 Axial Vector PROPL  PR   5 PL    PL 5 PL    PL 5 PL     PL PL     PL
(GT transitions)
 Pseudoscalar
 Tensor
(GT transitions)
PROPL  PR 5 PL  PR PL  0
PROPL  PR (         ) PL    PL  PL    PL  PL 
    PR PL      PR PL  0
29
The Universal Four-Fermion Matrix Element :
..can be constructed with the only non-zero matrix elements (V and A). A general
form could be :

CV   5C A
2
The fact that a massless neutrino is produced in a pure helicity eigenstate requires
CA= - CV giving precisely the helicity projector in the current :
5
1 

2

In general, this holds for any massive fermion, leading to the general form :
M   D  (1   5 ) C GF  B  (1   5 ) A C A   1
CV
Low-E «propagator»
Weak Current
Weak Current
30
Corrections to the V-A current structure ?
They need to be considered when the Weak Interaction involves Hadrons !
Let us first consider the electric charge of a proton
The proton is a complicate object,
continually emitting and absorbing
quark-antiquark pairs as well as gluons
The charge of the proton – however – is equal
to the charge of the (elementary) electron !
The electric current (a vector current V) is
conserved by the Strong Interaction
What about the Weak interaction V-A current ?
  CV   5C A 
The general experimental situation indicates that the V part is conserved
(Conserved Vector Current, CVC hypotesis. Goldberger-Treiman).
The A part of the corrent gets (most or all of) the Strong Interaction corrections :
n  p e  e
   n e  e
C A / CV   1.26
  p e  e
C A / CV   0.72
C A / CV   0.34
31
Pion decay and V-A structure of Weak Interactions
Let us compare the decays :
    
   e 



H Negative

Pion has spin 0  Neutrino and muon must have antiparallel spins (J conserved)
Neutrino has -1 helicity  For a massless neutrino helicity is an exact quantum
number  Muon MUST have negative helicity (the «wrong» helicity!)
If we compare the two processes :
  
  e
l ( e,  )
u
d
W

From fundamental physics viewpoint, coupling constant, Feynman diagrams,
they are essentially the same thing! The main difference is the phase space.
32
From the point of view of the phase space, the decay in the electron is
largely favored
But…in this decay the LEPTON is forced to have an «unnatural» helicity !


 p, , m  0

H Negative

p, l  , m
Experimentally, one has the following
electron energy spectrum from stopping
pions
  e
R 
 1.310  4
  
( Anderson et al., 1960)
33
Introducing the W and the Z0
M W  80.385  0.015 GeV
M Z  91.1876  0.0021 GeV
W  2 GeV
Z  2.5 GeV

 1
1
And the relevant expression for the
propagator :
 i ( g   q q / M 2c 2 )
q 2  M 2c 2
g 
2
i g 
( Mc ) 2
g2
Low energy limit
Lifetimes ?

200 MeV  fm
100 10 15
 25
  
s

s

3

10
s
8
8
3
 2 GeV  310 m
3 10 10
34
The Weak Charged Current and
the Weak Neutral Current
States connected by a W
States connected by a Z
(no flavor change whatsoever)
In fact, there is no (flavor
changing) tc,tu,bs,bd,cu
35
Now let us consider – as a meaningful example – the
neutrino scattering process in ordinary matter :
 e   e
If the neutrino is an electron neutrino :
e
e
e
W
e
e
e

Z
e
e
If the neutrino is a muon neutrino :
There is no annihilation diagram possible,
leaving only the Z possibility (exchange of
a Z between the two leptons). Only NC


Z
e
e
36
Recalling the discovery of the third leptonic family: the Tau
SLAC, 1975, Martin Perl et al.,
studying the products of e+e- collisions
With hindsight :
e e     
        
   e     e
Detection of final states featuring an electron and a muon
This indicates intermediate states emitting invisibile leptons (neutrinos).
This is because the Lepton Numbers (elettronic, muonic) are violated.
Is this the only possible interpretation of an eμ final state?
37
The important point was that these events took place when the energy was
greater than 3.56 GeV :
3.56 GeV  2 m( )  2 1.78 GeV
This has to be disentangled from events with two charged particles produced
by the process :
e  e   Psi(3740)  D  D 
D  K 0    
D   K 0 e  e
 Energy threshold of 3740 MeV
(as opposed to 3560)
 Additional hadronic particles in
the final state (K, pions, muons)
Featuring the same leptonic final state
With the discovery of the Tau (and the Tau Neutrino in 2002) the fundamental
leptons are :
e 
 
 e 
 
 
 
 
 
 
  
38
A note on neutrino experimental characterization : flavors and currents
Key point: different interaction in materials of a neutrino beam.
Charged Currents (CC) and Neutral Currents (NC)


Muon in the final state (CC event).
Muonic neutrino arriving!

e
Electron in the final state (CC)
Electron neutrino arriving!
No final state lepton

Neutral current (NC) !



Neutrino flavor unknown.
Tau neutrino tau interactions


e
 Tau lepton decaying in different
ways (including muon, electron)
39
The Weak Charged Currents
Weak Vertex Factor
The weak charged coupling to leptons is
characterised by the fundamental vertex :
 ig 
 (1   5 )
2 2
W
The Weak Coupling Constant :
l ( e,  )

g w2
1
w  
4 29.5
e( p4 )
Charged Currents Weak Interactions at low
energy: the muon lifetime
W (q )
 ( p1 )
   e    e
The Weak Interaction (CC)
lowest order Feynman Diagram :
 e ( p2 )
  ( p3 )
40
g
1


M   (3) w   (1   5 ) (1) 
2
2 2

 ( M W c)
gw


5

(
4
)

(
1


)

(
2
)



2 2


e( p4 )
GF
2 g2

3
(c)
8M W2 c 4
W (q )
 e ( p2 )
 ( p1 )
The muon lifetime result is :
 M W  12 8 

  
 m c2
m
g

w



4
3
  ( p3 )
At low energies, MW and gw always enter in observable quantities as a ratio,
which makes it possible to write :
192  3  7
  2 5 4
GF m c
The best Weak Coupling Constant
determination at low energies
41
The Weak Charged Currents : Leptons and Quarks
The coupling of W to leptons takes
place strictly within a given
generation:
W
e 
 
 e 
Purely leptonic Charged Current Weak Processes only
involve leptons. Their general structure is :
   e  e  
       
W
 
 
 
 
W
 
 
  
Jl
W
Jl
Weak decays of leptons into other leptons
   e  e  
e e e e
 e  e
Scattering between leptons (observed only if
electrons are present to act as suitable targets)
42
The coupling of W to Quarks :
W
Similar to the Quark case, there is
coupling within a generation :
But cross-generational couplings
are also there (6 couplings, since
bu and td are not shown) :
u 
 
d 
W
u 
 
d 
c
 
s
c
 
s
W
t 
 
b
t 
 
b
Charged Current involving Quarks can originate :
Jh
Jh
W
Jl
Semileptonic processes
W
Jh
Hadronic processes
43
Charged Current semileptonic processes :
They all feature a leptonic and a hadronic
charged current
e
W-
n  p  e  e

d
d
u
e
u
d
u
The neutron decays (and beta decays)
l n l p
  pl l
D  K 0    
The «inversa beta decay» kind of reaction
The decay kind of a heavy baryon
B   D0 l   l
Beauty and Charm decays
44
Jh
Charged Current purely hadronic processes :
W
Jh
u d u d
There are weak processed conserving flavor ut they are
dwarfed by the much stronger Strong Interaction
They are possible (and the only possibility) when the flavor is changed. Other
forms of interactions are not allowed. They can connect quarks in the same
generation, like in a cs decay :
c u d s
D  K 0  
s
c
W
d
They can connect quarks in different
generations, like in a bu decay :
They of course involve Mesons and
Baryons as well :
d
u
u
b
W
d
K     0
d
u
  p 
45
Weak Charged Currents : the Cabibbo theory of Mixing (1963)
Weak Charged Interactions have been characterized with a unique coupling
constant (and the phase space). However, the intergenerational processes
seemed to take place less often than the decays within the same generation :
The charm quark was
not known at that time
Experiments say that :
W
occurs more frequently than
u 
 
d 
W
 
 
s
Cabibbo proposed that the quarks entered the Weak Charged Interactions as
“rotated” states :
u




 d cosC  ssin C 




 s cosC d sin C 
46
The Weak Interaction Eigenstates at the
time of the Cabibbo Theory (no Neutral
Currents, yet, no taus, no c, b and t) :
Weak Interaction Eigenstates
related to Mass Eigenstates by :
Mixing determined by C 130
the Cabibbo angle
sin C 0.22
e 
 
 e 
 
 
 
 
 sC  cosC
   
 dC   sin C
u 
 
 dC 
sin C 
cosC 
 
 
 sc 
s 
 
d 
cosC 0.97
The new interaction vertices for Weak Charged currents are :
u
ig 
 1   5  (cos C )
2 2
u
ig 
 1   5  ( sin C )
2 2
W
W
d
s
accounting for both the V-A structure and the quark mixing
47
The experimental evidence :
p  ne  e (14O )
u  d e  e
GF2 cos 2  C
Cabibbo-allowed
    0 e  e
d  u e  e
GF2 cos 2  C
Cabibbo-allowed
K    0 e  e
s  u e  e
GF2 sin 2  C
Cabibbo-suppressed
GF2
Leptonic
   e  e 
e( p4 )
Actually the rate of these processes is the
motivation for introducing the mixing.
All leptonic processes are unaffected. All
hadronic processes are affected.
In a semileptonic process like a beta-decay :
W (q )
u( p1 )
 e ( p2 )
d ( p3 )
g
gw
1 



5
M   (3) w   (1 5 )cos C (1) 

(
4
)

(
1


)

(
2
)
 GF cos C

2 

2 2
2 2

 ( M W c) 

48
In the Standard Model, all flavors
are mixed, as represented by the
CKM (Cabibbo-KobayashiMaskawa) Matrix :
The CKM 3-quark mixing is a
generalization of the 2-flavor
Cabibbo style mixing
Mass eigenstates
Weak Interaction eigenstates
Experimental values of the
magnitude of CKM elements
are close to a unit matrix :
Same-generation transitions are favoured :
DK
D 
favored
suppressed
u 
 
d 
suppressed
c
 
s
favored
t 
 
b
49
The flavor structure of Weak (and Electromagnetic) Interactions
Electromagnetism (the photon) couples to
charged particles.
The Charge Current coupling will couple
according to the weak charge
  e  Qi qi qi
i
W 
g
a ( ,  ) f  f

2 
If we are considering leptons, one should write :
a ( e , e )  a (  ,  )  a (  , )
All the other components are zero
because of the lepton numbers
conservation.
For instance, in the case of two families :
 a( e , e) a( e ,  )  1 0
 a( , e) a( ,  )   

0
1






50
In the case of quarks, we now have all four
couplings that are different from zero.
W 
This matrix is not diagonal and this is because the
mass states are not eigenstates. It can be diagonalized
by means of a rotation:
d '  d cos C  s sin C
g
a ( ,  ) f  f

2 
 a(u, d ) a(u, s) 
 a ( c , d ) a ( c, s ) 


s '  s cos C  d sin C
The rotation by the Cabibbo angle θC bring us to the Weak Eigenstates
In this base :
 a(u, d ' ) a(u, s ' )  1 0



'
' 
0
1
a
(
c
,
d
)
a
(
c
,
s
)




In considering d’,s’,b’ (eigenstates of the Weak Interaction) instead of d,s,b, we
can maintain the concept that Quarks and Leptons have the same coupling to the
W boson (Universality of the Weak Interactions)
51
The full expression for the
Weak Charged Current :
W 


g
e e        d ' u  s ' c  b' t
2
In case of just two families of Quarks and Leptons :
g
2
g
2
 e
e
    d ' u  s ' c

 e
e
    cos  C  d u  s c
  sin   s u
C
 dc

The study of the relative intensities of
weak decays (comparison of different
decay odes) allows to determine the
Cabibbo Angle: about 130.
When just two families are considered,
one can divide all Charge Current
Weak Decays of Quarks into Cabibbo
“allowed” and Cabibbo “suppressed”
decays
52
Introducing the Weak Neutral Currents
Theoretical and experimental problems showed up when considering a Weak
Interaction theory with Carged Currents alone
1) Theoretical inconsistencies : divergences in the Weak Interaction theory
2) Experimental problems: the discovery of weak processes that cannot be
explained by the charged currents
The problem of divergences
We require for a Quantum Field Theory to be renormalizable.
Renormalizability (e.g. the QED case) consists in the possibility of re-absorbing
divergent diagrams by redefining bare charges and masses of the theory.
A theory is renormalizable if (at all orders of the perturbative expansions, and
possibly at all energies) the amplitudes of the processes can be kept finite by
suitably tuning a finite number of parameters (charges and masses).
53
Let us consider the weak process
 e  e  e  e
with a cross section given (in the Fermi theory) by :
 tot 
GF2 s

This cross section increases arbitrarily with energy, ultimately violating the Unitary
Limit
The W propagator has the effect of mitigating the divergence by
introducing a term of this kind in the scattering amplitude :
1
q2
1 2
MW
The Fermi pointlike interaction gets “spread out” in a finite range having a size
1
proportional to
MW
This mitigates the divergence problems. However, divergences of the type
still remain, as in the process
 tot  s
  W W 
For these reasons, Glashow, Salam, Weinberg started to develop a theory that
would unify Weak and Electromagnetic Interactions. These theory is renormalizable
(as demonstrated later by t’Hooft) and predicts the existence of a massive neutral
boson and of Weak Neutral Currents
54
The observation of weak neutral current processes
All interactions observed up to 1973 were compatible with just weak processed
induced by the W
Weak neautral process are instead mediated by th Z0:


The rate of these processes
was about one-third of the
rate of the related CC
events

 N   X
Z0
X (Hadrons)
lending credibility to the
idea of a NC process taking
place
N
Processes of this kind were observed in 1973 with the Gargamelle bubble
chamber, at CERN.
55
Gargamelle was a giant particle detector at
CERN, designed mostly for the detection of
neutrinos. With a diameter of nearly 2 meter
and 4.8 meter in length, Gargamelle was a
bubble chamber that held nearly 12 cubic
meters of freon (CF3Br). It operated from 1970
to 1978 at the CERN Proton Synchrotron and
Super Proton Synchrotron. Weak neutral
currents were predicted in 1973 and confirmed
shortly thereafter, in 1974, in Gargamelle.
The name derives from the giantess
Gargamelle in the works of Rabelais; she was
Gargantua's mother. (www.wikipedia.org)
A Neutral Current ecent in
E815-NuTeV at Fermilab
A muon neutrino is coming from the left-hand side. An hadronic shower
with no muons is generated (but a neutrino is present in the final state)
56
An event of the type :
 e  e
can only proceed :


Z0
e
e
Note that at low
energies, Z induced
events are dwarfed
by e.m. interactions
(unless neutrinos
are involved)
f
f

e
e
57
The GIM (Glashow-Iliopolous-Maiani) mechanism
Particles known in 1970 :
If neutral currents are admitted in
the model, one should have them in
forms like :
A charged current is of the type:
e 
 
 e 
 
 
 
 
 sC  cosC
   
 dC   sin C
J   u dC
u 
 
 dC 
sin C 
cosC 
 
 
 sc 
s 
 
d 
A neutral current can be formed in the more general way as :
u
u 


 
J  u dC    u d cos C  s sin C 
 dC 
 d cos C  s sin C 
 uu  dd cos 2   ss sin 2   sd  d s sin  cos 
0


ΔS=0
ΔS=1
58


J 0  uu  dd cos 2   ss sin 2   sd  d s sin  cos 
It seems that the neutral current should have both ΔS=0 and ΔS=1 components .
K     ( NC )
5

10
K    0   (CC )
Experiments however say that when ΔS=1,
Neutral currents are suppressed :
The GIM proposal : a fourth quark to complete the doublet.
u
u  

   

 dC   d cos C  s sin C 
And the new neutral current built in this
way, does not have any ΔS=1 terms :
c
c  

   

 sc   s cos C  d sin C 
u 
c
J  u dC    c sC  
 dC 
 sC 
0
The charged current now has the form :
 cos C
J  u c  
 sin C

sin C   d 
 dC 
   u c  

cos C   s 
 sC 
59
Isospin e Hypercharge of fundamental fermions
We introduce the concept of Weak isospin, to classify the states of the fundamental
fermion. This in fact can be considered as “spin”. As usual, the transformation
between the two states bear a formal analogy with space rotations.
Starting with the electron and its neutrino:
 e 
 
e
e
 
 
 e
T3   1/ 2
T3  1/ 2
T3   1/ 2
T3  1/ 2
T = ½ is the Weak Isospin for this
doublet of fermion states
e  e
e  e
The anti-electron and anti-neutrino doublet can be
obtained from the electron/neutrino one by
changing charge, lepton number T3
An equivalent SU(2) structure is considered for the quark doublets
Let us now form composite states, using the rule of addition of the spin:
60
11   e e
T = 1, T3= +1


1
 
 e e  ee
2
0
1
T = 1, T3= 0
11  e e
0 
The composition gives
origin to the usual tripet
and singlet states
T = 1, T3= -1

1
 e e  ee
2

Rotationally invariant in
the T space
T = 0, T3= 0
We can now see that the Weak Charged Current :

g
W 
e e        d ' u  s ' c  b' t
2


can be written (for the leptons of the first family) as the SU(2) current:
W 

g
e e
2


g
11
2
W 

g
ee
2


g
11
2
61
W 

g
e e
2


g
11
2
W 
This term of course correspond to processes like :

g
ee
2

g
11
2
e W   e
e W   e
0
It is interesting to note that Isospin invariance REQUIRES the existence of 1
W0 

g
g
10 
 e e  ee
2
2

which implies the existence of processes like :

e W e
0

W 0  e  e
 e  e W 0
W 0  e  e
and similar processes for other Isospin doublets
62
In a Neutral Current vertex the very same
fundamental fermion enters and exits (unaltered)
f
In the case of a NC process it does not matter if
one uses the mass or the charged weak
interaction eigenstates. In fact, for the case of two
generations one can easily verify that :
Z
f
u 
c
u 
c 
J  u dC    c sC    u d    c s  
d 
s 
 dC 
 sC 
0
which generalizes to the case of three families (since the CKM matrix is unitary)
While the coupling of quarks and leptons to the W is the universal coupling
described before, the coupling of the Z has the form :
e
ν
 ig w 
 1  5
2 2

W

f
f
 ig Z  f
 cV  c Af  5
2


Z
63
u
d
 ig Z  f
 cV  c Af  5
2


The coupling of the Z depends on the
specific fermion being considered
Z
But what are gz and the c coefficients ?
All these couplings (and the M,Z mass relaitionship) depends on the very same
single parameter, which is part of the Glashow-Salam-Weinberg theory of the
Electroweak Interactions.
This parameter is the Weinberg angle θW
The Weinberg angle is a characteristic of nature :
W  28.750
sin 2 W  0.2314
64
Electroweak Z parameters are defined by means of the Weinberg angle
W vertices
igW
2 2
Z vertices
ig Z  f
 (cV  c Af  5 )
2
  (1   5 )
gW
gZ 
cos W
 e ,  , 
e,  ,
MW
MZ 
cos W
cV
cA
1
2
1
2
1
  2sin W
2
u, c, t
1 4 2
 sin W
2 3
d , s, b
1 2
  sin 2 W
2 3

1
2
1
2

1
2
65
The concept of Electroweak Unification
Weak Isospin states
Weak Isospin fields
  ee

1
 
 e e  ee
2
0
1

g
ee
2
W 
1
1

11  e e
W0 


g
11
2


g
g
10 
 e e  ee
2
2
W 

g
e e
2


g
11
2
We now introduce a field corresponding to the T=0 state as well :
0 

1
 e e  ee
2

<Q> is the average charge of the
Isospin multiplet (-1/2). A different
coupling constant is introduced,
called g’
B0  2 g ' Q 0
A weak Isospin scalar
U(1) group symmetry
implied here
66
Note: the B0 field averages on the particles of the multiplet
The value of <Q> :
Lepton doublets :
Q 
Quark doublets :
1
1
(1  0)  
2
2
Q 
1
1
(2 / 3  1 / 3)  
2
6
We have introduced four spin-1 fields fields dictated by the Isospin Symmetry:
W+,W-,W0,B0
These are not the physical fields.
For instance B0 does not look like any physical field, with its coupling to
electrons and neutrinos :
0 

1
 e e  ee
2

B0  2 g ' Q 0
An electromagnetic field – for instance – should have a
coupling like this :
A  e e e
67
A  e e e
The e.m. interaction should have the form:
g B0  g ' W 0
But let us consider the combination
g 2  g '2
And calculate it for the first leptonic generation

 gg ' (  e e  ee   e e  ee )
And the result is
A 

g
1
10 
gg '   0  10 
2
2
g B0  g ' W 0  g 2 g ' Q 0  g '
1
2
g B0  g ' W 0
g g
2
'2

 gg '
g g
2
'2
ee
The Electromagnetic Interaction can be introduced as a linear combination of the
T3=0 isospin states if we just assume:
'
e
gg
g 2  g '2
68
u
 '
d 
What about the first generation of Quarks?
T3= + 1/2
T3= - 1/2
d' 
 
u
 
Let us build states with T3 = 0 (one with T = 0 and the other with T = 1)
10 

1
uu  d ' d '
2

0 

1
uu  d ' d '
2

Making the calculation as before, we obtain the electromagnetic interaction of
Quarks :
A 
g B0  g ' W 0
g g
2
(recalling <Q>=1/6).
em
'2

1 ' ' 
1 ' ' 
2
2
uu  d d   e  uu  d d 
2
'2  3
3
3

3

g g 
gg '
So, this is the Electromagnetic Interaction
g 2  g '2
1
1
0
 1  2  Q   0  

A

A
'


gg
e
2
69
Let us now consider the combination orthogonal to
Z 
0
Z0 

g W 0  g ' B0
This is a neutral field which is independent from the one
of the photon!
g 2  g '2
g W 0  g ' B0
g 2  g '2
1
g 2  g '2
A

1
g 2  g '2
g
g
1
10 
g ' 2g '  Q  0 
2
g 2  g '2
1
 g 2 10  2 g '2  Q   0 

2
And one can show that :
it is the physical Weak Neutral
Current !
A Z0 0
70
g
11
2
g
W0 
10
2
W 
W 
To summarize, we started from:
g
11
2
B0  2 g ' Q 0
and we made a rotation between the neutral fields (the Weinberg angle):
A 
Z0 
g B0  g ' W 0
g g
2
'2
 g ' B0  g W 0
g g
2
'2
 cos W B  sin W W
0
cos W 
0
sin W 
  sin W B 0  cos W W 0
g
g 2  g '2
g'
g 2  g '2
The physical fields (A and Z) as a function of the Weak Isospin fields
Z
0

 em 
1
g 2  g '2
1
 g 2 10  2 g '2  Q   0 

2
g 2  g '2
1
A

g g'
2

0
1
 2  Q  0
W 
g
11
2

71
 e 
 
e
Let us elaborate the concept a bit more, using
the first generation (Quarks and Leptons)
The electromagnetic field is given by :
g g
gg '
2
 em 
'2
A 

u
 '
d 

1
10  2 Q  0 
2
e
 
 
 e
d' 
 
u
 
T3= + 1/2
T3= - 1/2
T3= + 1/2
T3= - 1/2
1  1
1 1
 1  1
' '
' ' 





e
e

u
u

d
d

2



e
e

2
(
u
u

d
d )
 
e e
e e

6 2
2 2
 2  2

2
1
  ee  uu  d ' d '   Q f f
3
3


In addition :
And also :

10 
 em 
1
1
1
1

2   e e  ee  uu  d ' d '  
2
2
2
2


2  T3 f f


1
10  2 Q  0  2 Q  0  2 em  10
2
We can now write the Z current as a function of the electromagnetic and the Φ10 :
72

1
Z0 
g g
2
1
'2
g 
2
2( g 2  g ' 2 )


1 2 0
1
g 1  2 g ' 2 Q  0 
g 2 10  g '2
2
2( g 2  g '2 )
0
1

g  
'2
0
1
g '2
g 2  g '2
 em 
g'
 tan W
g
using :

2 em  10
 
g  1 0
'


g
sin W  em
1

cos W  2

Z0 
g  1 0

2


sin


1
W
em 
cos W  2

which can also be written as :
Z
0
g

cos W
2
T

Q
sin
  3  W  f f

The coupling of th Z to the
members of the isospin doublets
The free constants of the theory :
e, g , g ' ,sin W
e
gg '
g g
2
'2
Four quantities, subjected to two conditions
sin W 
g'
g g
2
'2
Two independent quantities
73
La scoperta del W+ e della Z0 (1983)
1979: decisione del CERN di convertire l’SPS in un collisore protoni-antiprotoni.
(e disponibilita’ di un significativo numero di antiprotoni grazie allo “stochastic cooling”)
I possibili processi di produzione:
l
u
ud  W

u u  Z
u
l
0
W
Z
ud  W
d d  Z0
d
l
I possibili modi di decadimento:
Protoni a 270 GeV
u
W  l  l
l
Z l l ,  l  l
Antiprotoni a 270 GeV
74
e
u
u
p
u
W
d
 ( p p W  e )  1 nb
p
u
d
e
e
u
u
p
Z
d
 ( p p  Z  e e)  0.1 nb
d
p
u
u
e
75
Ma la sezione d’urto totale e’ dell’ordine dei 40 mb, determinata dalla sezione
d’urto di interazione forte !
Gli eventi interessanti vanno estratti dal fondo adronico sfruttandone le loro
caratteristiche peculiari.
Momento trasverso elevato, bilancio energetico globale
Il calorimetro di UA1
Il rivelatore UA2
76
Un evento in cui e’
prodotto un W che
decade:
W   e  e
• Un elettrone ad alto momento trasverso
pT 
MW
 40 GeV
2
• Uno sbilanciamento in momento trasverso di tutto
l’evento consistente con il momento trasverso
dell’elettrone e corrispondente al neutrino che non
viene osservato.
77
Scoperta della Z0: decadimento
Z 0  e e
Caratteristiche dell’evento:
 Un elettrone ad alto pT
 Un positrone ad alto pT
 Nessuna energia trasversa mancante
“LEGO plot” nello spazio
, 
E naturalmente anche il decadimento
Z 0   
Caratteristiche dell’evento:


• Due muoni di segno opposto ad alto pT
• Nessuna energia trasversa mancante
Using all data from 1982-3, and combining results from UA1 and UA2:
mW = 82.1  1.7 GeV
mZ0 = 93.0  1.7 GeV
Current values (Particle Data Group 2006):
M(W ±) = 80.403 ± 0.029 GeV
M(Z0) = 91.1876 ± 0.0021 GeV
78
Summarizing the fundamental ideas of the Electroweak Unification
Idea by GSW (Glashow, Salam Weinberg): let us treat Electromagnetic and Weak
Interactions as a part of a unified theory.
Fundamental idea: SU(2) and U(1)
symmetries to predict 4 bosons :
W

,W 0 ,W   , B 0
W  ,W  , B0 , 
Neutral bosons do mix up to generate physical bosons :
W  
 0 0
W  , B
W  


W  
 0
 Z ,
W  


Z 0  W 0 cos W  B 0 sin W
  W 0 sin W  B 0 cos W
e e W W 
Neutral currents are a cure to divergent processes like :
The full set of three graphs is now convergent (which is NOT without the Z):
e

W+
e

W-
e

e
e
W+
e
W+
Z0
W-
e
W-
79
short
summary
on the Weak
EAin
particolare
le correnti
deboliNeutral
neutre:Currents :
• They are mediated by the neutral
vector boson Z0
• They do not change flavor (no flavor
changing neutral currents)
• The Z0 couplings to fermions are a
mixture of electromagnetic and weak
couplings, i.e. they are both vector
(V) and V-A
• The relative intensity of Z0
couplings depends on a single
parameter:
W  28.750
sin 2 W  0.2314
80
Electroweak effects in e+eAt low energy, neutral current effects are
not easily visible because of the
presence of electromagnetic effects
However, when we are near to the Z mass,
the propagator increases dramatically :
f
f
f

Z0
e
f
e
e
e
i  g   q q / M Z c 2 
q 2  M Z2 c 2
As an example, if one considers the 2-muons final state :
 (e  e   Z      )
E4

 (e e        ) (2 E )2  ( M Z c 2 )2  2    Z M Z c 2 2


If 2E<<MZc2
Z

 E 

2 
 MZc 
4
(negligible)
resonating at the Z0 mass
If 2E~MZc2
 Z 1  M Z c2 
 

  10   Z 
(dominating, >200)
81
The Electroweak Theory
Proposed in 1961 by Glashow-Salam-Weinberg (GSW)
Treat the Electromagnetic and Weak Interaction as only one interaction
At high-energy: Electroweak Interaction
At low energy: electroweak symmetry is broken into Weak and Electromagnetic
Some problems that needed to be solved :
• Disparity in strength between Weak and Electromagnetic forces
• The photon is massless, while W,Z are massive
• Electromagnetic interactions are V, while W couplings are V-A
The use of chiral spinors makes it easy to overcome the last difficulty :
1  5
u ( p)  u L ( p)
2
1  5
v( p )  vR ( p )
2
A particle that has helicity -1 in the ultra-relativistic limit
An antiparticle that has helicity +1 in the ultrarelativistic limit
Left-handed means helicity -1 only in the massles (ultra-relativistic) limit
82
Antiparticles
Particles
uL 
1 
u
2
5
vL 
1 
v
2
5
1  5
uR 
u
2
1  5
vR 
v
2
1  5
uL  u
2
1  5
vL  v
2
1  5
uR  u
2
1  5
vR  v
2
A Dirac free particle wavefunction :
  u( p)eipx  v( p)eipx
particle
antiparticle
By using this notation, Weak and
Electromagnetic interactions are written in a
form that makes it easy to see how they can
be unified.
Let us consider the W vertex coupling to a lepton (say an electron) :
We write the leptonic current as (for the
electron and electron neutrino case :
W
e
e
5
1


j    
e
2
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Based on the properties of the Gamma matrices :
1  5
1  5 1  5
j   e  
e  e  
e
2
2
2
1  5 1  5
 e

e   L   eL
2
2

The weak vertex is now
purely a vector vertex, but
left-handed spinor particles
are used.
The Electromagnetic current as a function of the chiral spinors :
Since in general this expression holds :
One can easily show that :
j   l   l   lL   lL  lR   lR
em
 1  5   1  5 
u
u 
 u  uL  uR
 2   2 
The Electromagnetic Current
couples to both left and right
fermions.
And in addition :
1  5 1  5
1  5 1  5
 eR  eR   e

e  e 
e0
2
2
2
2
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Weak Isospin and Hypercharge currents
The weak charged currents can be written as :
W
W
e
e
e
e
j  eL    L
j   L  eL

And in a more compact notation, by introducing the lefthanded doublet :
0 1 
 

0
0


Introducing the two matrices :
one can write :

 e 
L   
 e L
0 0 
 

1
0



j   L    L


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j   L     L
Weak Isospin and Hypercharge currents
 
eL      L    L
 eL 
 eL 
eL      L   eL
0
j   L
For example :
  L
We note that the tau matrices are linear combinations
of the Pauli matrices
 Introducing the third matrix would give a full Weak
Isosping symmetry SU(2)
But which is the relative current ?
0 1  L 
 
eL    

0 0   e L 
1 1
  (  i 2 )
2

1 3 1 1 0 
  
2
2 0  1
Let us calculate :
j   L  
3
3
1
 L   L
2
2
 L  1
   L  L  eL  eL 
eL   
  eL  2
86
1
j   L  L  eL  eL 
2
3
It actually is a neutral current, but not the right
neutral current (for instance, it is pure V-A, so
it just couples to LH states)
The description of Weak Interactions (and Electromagnetism) with an SU(2) and
U(1) symmetry makes it possible to have a gauge theory according to a
symmetry group. This is important for the renormalization of the theory.
….and recall that the electromagnetic
current is :
In analogy with the Hypercharge, we
introduce the Weak Hypercharge :
Its relevant current being :
jem   eL  eL  eR  eR
Y
Q  I3 
2
jY  2 jem  2 j3   2e R  eR  e L  eL  L  L
In terms of the couplings to the vertices, we will introduce a coupling constant g for
the weak isospin triplet and a coupling constant g’ for the hypercharge singlet
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The underlying symmetry has a SU(2)L
Weak Isospin

U(1)Y structure
Weak Hypercharge
And the currents are:
j
 L eL
 L  L  e L eL
2 j3
e L  L
2e R eR  e L eL  L  L
jY
j
And this structure can form the electromagnetic current by means of a combination
em
j
1 Y
 j  j   e L  eL  e R  eR   e  e
2
3
as well as the weak neutral current
88
SU(2)L

U(1)Y
The Electroweak Lagrangian
The constituents of the Electroweak Standard Model
 e 
 
 e L
  
 
  L
  
 
  L
u
 '
 d L
c t 
 '  '
 s L  b L
eR,μR,τR,uR,cR,tR,d’R,s’R,b’R
We have introduced three Weak Isospin and a Hypercharge current :
1

j

  2  L    L

 j Y  2 j em  2 j 3


 
Three Weak Isospin currents
A Weak Hypercharge current
The symmetry group is related to the following fields :
1
j1   L  1  L
2
jY  2 jem  2 j3
1
j2   L  2  L
2
1
j3   L  3  L
2
89
1
j3   L  3  L
2

This Weak Isospin currents are relative to Weak Isospin charges :
T i   j0i dx
A little bith of math :
1
j1   L  1  L
2
1
j2   L  2  L
2
T , T  i 
i
And these follow an SU(2)L algebra :
j
k
T
ijk
Summary of quantum numbers (just one generation) :
Lepton
T
T3
Q
Y
νe
1/2
1/2
0
-1
e -L
1/2
-1/2
-1
-1
e -R
0
0
-1
-2
Quark
T
T3
Q
Y
uL
½
½
2/3
1/3
dL
½
-1/2
-1/3
1/3
uR
0
0
2/3
4/3
dR
0
0
-1/3
-2/3
90
The Electroweak Lagrangian is now being
written as :
The vector part :
By using :
L  g j W

g' Y 

j B
2
   1 1 2  2
jW  jW  jW  j3W  3
j  j1  i j2
The vector part becomes :

1
W 
W1  i W2
2


   1   1  
jW 
j W 
jW  j3W  3
2
2
which now contains the physical W+ vector boson of the Weak Interactions ♫
For what concern the neutral part, we have two fields here: W3 and B0
They are the symmetry group fields, not the physical fields.
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The physical fields are
generated via a Weinberg
rotation.
A  B cos W  W3 sin W
W3 W0
Z    B sin W  W3 cos W
B  A cos W  Z  sin W
inverting the rotation
W3  A sin W  Z  cos W
The neutral part becomes :
 3 3 g ' Y   

g'
g'
3
Y 
3
Y 
g
j
W

j
B

g
sin

j

cos

j
A

g
cos

j

sin

j

W 
W 
W 
W  Z
 
 

2
2
2

 



If we want that the A coupling describes
the electromagnetic interaction :
which happens if :
1 

 g e jem A   g e  j3  jY  A
2 

g sin W  g ' cos W  g e
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There is only one field, the ElectroWeak Field !
In addition, the coupling to the Z is :

ge
g'
3
Y 
3
2
em

g
cos

j

sin

j
Z

j

sin

j
Z
W 
W 

W 

2
sin

cos



W
W


The full Electroweak Interaction Lagrangian can be written as
L  g j W
L

g' Y  g
g
 jW    jW    
 j3  (sin 2 W ) jem  Z 0   ( g sin W ) jem A

j B 
2
cos W
2
g
g
 jW    jW    
 j3  (sin 2 W ) jem  Z 0   ( g sin W ) jem A
cos W
2
Weak Charged
Weak Neutral
and e.m.
Pure E.M.
Note: this Lagrangian does not include masses of W,Z and fermion masses. It is just
the Electroweak Interaction part.
93