Transcript Document

Symmetries and Conservation Laws
1. Costituents of Matter
2. Fundamental Forces
3. Particle Detectors (N. Neri)
4. Experimental highlights (N. Neri)
5. Symmetries and Conservation Laws
6. Relativistic Kinematics
7. The Static Quark Model
8. The Weak Interaction
9. Introduction to the Standard Model
10. CP Violation in the Standard Model (N. Neri)
“Five Deities Mandala
Tibet, XVIIth Century
The word is used also to indicate a circular diagram,
basically made by the association of different geometric
figures (the most used being the dot, the triangle, the
circle and the square). The drawing has spiritual and
ritual meaning in both Buddhism and Hinduism.
“Mandala” from
it.wikipedia.org
1
A classification of
symmetries in particle
physics
Class
Invariance
Conserved quantity
Proper orthochronous
Lorentz symmetry
translation in time
(homogeneity)
energy
translation in space
(homogeneity)
linear momentum
rotation in space
(isotropy)
angular momentum
P, coordinate inversion
spatial parity
C, charge conjugation
charge parity
T, time reversal
CPT
time parity
product of parities
U(1) gauge transformation
electric charge
U(1) gauge transformation
lepton generation number
U(1) gauge transformation
hypercharge
U(1)Y gauge transformation
weak hypercharge
U(2) [U(1) × SU(2)]
electroweak force
Discrete symmetry
Wikipedia:
Internal symmetry
(independent of
spacetime coordinates)
SU(2) gauge transformation Isospin
SU(2)L gauge transformation weak isospin
P × SU(2)
G-parity
SU(3) "winding number"
baryon number
SU(3) gauge transformation quark color
SU(3) (approximate)
quark flavor
S(U(2) × U(3))
[ U(1) × SU(2) × SU(3)]
Standard Model
2
Symmetires of a physical system:
Classical
system
Lagrangian
Formalism
Hamiltonian
Formalism
Invariance of Equations of Motion
Quantum
system
Lagrangian
Formalism
Hamiltonian
Formalism
•Invarianza Eq. Dinamica
•Invarianza relazioni di commutazione
(Invarianza della probabilità)
E. Noether’s Theorem (valid for any lagrangian theory, classical or quantum)
relates symmetries to conserved quantities of a physical system
3
A “classical” example :
T 

1 2 1
m1r1  m2 r22
2
2
 
r1  r2
 
V  V (r1  r2 )

 


m1r1    V (r1  r2 )
r1

 


m2 r2    V (r1  r2 )
r2

'  
Let us do a translation :
ri  ri  ri  a
 
   
 
V (r1  r2 )  V (r1  a  r2  a) V (r1  r2 )
'
 


mi ri    ' V (r1  r2 )
ri
The equations of
motion are translation
Invariant !
4
If one calculates the forces acting on 1 and 2:

 
 
 
 
 




FTOT  F1  F2    V (r1  r2 )   V (r1  r2 )   V (r1  r2 )   V (r1  r2 )  0
r1
r2
r2
r2


dPTOT
 FTOT  0
dt
In the classical Lagrangian formalism :
L  L(qi , qi )
d L L

0
dt qi qi
L
pi 
qi
L invariant with respect to q
dpi
L

dt
qi
p conserved
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In the Hamiltonian formalism
qi   qi , H 
p i   pi , H 
d (qi , pi )
  , H 
dt
Possible conservation of a
dinamical quantity
Possible symmetry
This formalism can easily be used in Quantum Mechanics
In Quantum Mechanics, starting from the Schroedinger Equation :

i   s (t )  H  S (t )
t
 s (t )  exp i(t  t0 ) H /  S (t0 )
T (t , t0 ) Time evolution (unitary)
6
Schroedinger and Heisenberg Pictures:
Q   s (t0 )* Q(t ) S (t0 ) dV   S (t )* Q0  S (t ) dV
 S (t0 )* Q(t ) S (t0 )   S (t )* Q0  S (t )
Heisenberg
Schroedinger
 S (t0 )* Q(t ) S (t0 )   S (t0 )* T 1Q0 T S (t0 )
Q(t )  T 1Q0 T
Operators in the Heisenberg picture
Taking the derivatives:
d
dT 1
dT
i Q(t )  i
Q0 T  iT 1Q0
  HT 1Q0T  T 1Q0TH
dt
dt
dt
i
d
Q(t )   HQ  QH  Q, H 
dt
d
Q
i Q(t )  i
 Q, H 
dt
t
Conserved quantities:
commute with H
In the case when there is an explicit time
dependence (non-isolated systems)
7
Translational invariance: a continuous spacetime symmetry
 
 
 (r  r )  (r )  r
 1  r
  D
r 
r 
 i

D ( r )  1  p  r 
 

The translation operator is naturally
associated to the linear momentum
For a finite translation :
n
 i

i

D (r )  lim 1  p  r   exp p r 
n
 



unitary
( r  n  r )
Self-adjoing: the generator of space translations
If H does not depend on coordinates
D, H  0
 p, H  0
The momentum is conserved
8
Rotational invariance: a continuous spacetime symmetry

   i

R ( )  1      1  J z  
   


The rotation operator is naturally
associated with the angular momentum
 


J z   i   x  y    i
x 

 y
Angular momentum
(z-component)
operator (angle phi)
Self-adjoing: rotation generator
A finite rotation
unitary
n
 i

i

R ( )  lim 1  J z    exp J z  
n
 



  n
If H does not depend on the rotation angle φ around the z-axis
R, H  0
J z , H  0
The angular momentum is conserved
9
Time invariance (a continuous symmetry)
The generator of time translation is
actually the energy!
 s (t )  exp i(t  t0 ) H /  S (t0 )
Using the equation of motion of the operators :
i
d
Q
H (t )  i
 Q, H 
dt
t
i
d
H
H (t )  i
 H , H 
dt
t
If H does not dipend from t, the energy is conserved
The continous spacetime symmetries:
Space translation
Space rotation
Time translation
Linear momenum
Angular momentum
Energy
10
Continuous symmetries and groups: the case of SU(2)
Combination of two transformations: the result depends on the commutation
rules of the group generators
 pi , p j   0
Commutative (Abelian) algebra of translations
Translation operator along x:
i

Dx ( )  exp p x  


i

i

i

i

Di ( ) D j (  )  exp pi   exp p j    exp p j   exp pi   D j (  ) Di ( )








(two translations commute). Moreover:
i

i

i

Dx ( ) Dx (  )  exp px   exp px    exp p j (   )   Dx (   )






and clearly :
i

Dx (0)  exp px 0   1


11
In the case of rotations :
Commutation rules for the generators:
i

Rz ( )  exp Lz  


L , L  i 
j
k
jkl
Ll
A non-commutative algebra
Rotations about different axes do not commute
Why SU(2) ?
In the case of a two level quantum system, the relevant internal symmetries are described by
the SU(2) group, having algebraic structure similar to SO(3).
SU(2) finds an application in the Electroweak Theory
SU(3) can be applied to QCD
12
Isospin Symmetry
Let us consider a two-state quantum system (the original idea of this came from
the neutron and the proton, considered to be degenerate states of the nuclear
force). Since they were considered degenerate, they could be redefined :
H p  E p
 p   'p   p   n
H 'p  E 'p
H n  E n
 n  n'   p   n
H n'  E n'
Degeneration
Ridefinition
Double degeneracy, similar to what happens in s=1/2 spin systems.
The degeneration can be removed by a magnetic field
One can introduce a two-components spinor :

(1 / 2 )
 p 
    p  1p/ 2   n  n1/ 2
 n 

1/ 2
p
1 
  
 0

1/ 2
n
 0
  
1 
13
 (1/ 2)   (1/ 2) '  U  (1/ 2)
The ridefinition now becomes :
A symmetry for the Strong Interactions (broken by electromagnetism)
U U  1
det U   1
U 1 i
SU(2) is a Lie group
Properties can be deduced from infinitesimal transformations
  
(1  i )(1  i )  1

Tr   0
det U   1

Which can be written in a general form :

 
2
Pauli matrices
0
1  
1
1
0
0  i 
2  

i
0


1 0
3  

0

1


 i  j 
k
 ,   i  ijk
2
2 2
14
1  2 (1/ 2 ) 3 (1/ 2 )
 p  p
4
4
1  2 (1/ 2 ) 3 (1/ 2 )
 n  n
4
4
1
1 (1/ 2 )
(1 / 2 )
3  p   p
2
2
Isospin
1
1 (1/ 2 )
(1 / 2 )
 3 n   n
2
2
 
 'p  
 
   1  i    p 
 
 '  
2

 n
 n
Infinitesimal rotation of the p-n doublet :
A finite rotation in SU(2):
  n

 

U  lim 1  i
  expi / 2
n
n 2

• Generalization of a global
phase transformation
• Three phase angles
• Non-commuting operators
(Non-abelian phase invariance)
example

(1/ 2) '

 expi / 2 (1/ 2)
 
 'p  cos 2
 
 '   
 n
sin
 2

 sin  
2  p 
 
   n 
cos
2 
15
The two nucleon system
A one-nucleon state can be described
with the base of the nuclear spinors

1/ 2
p
1 
  
 0

1/ 2
n
 0
  
1 
A two nucleon state can be :
1
 pn  np 
2
1
 pn  np 

2
 1  pp
2 
 3  nn
4
In an Isospin rotation:
 4'   4
I  1 ( I 3  1, 0 ,  1)
I  0 ( I 3  0)
Isospin singlet
The other three states transform into one another in Isospin rotations, similar
to a 3-d vector for ordinary rotations
Isospin Invariance means: there are two amplitudes, I=0 e I=1
I=1 states cannot be distinguished by the Strong Interactions
16
Nucleons and Quarks
I3
particle
antiparticle
particle
+1/2
p
u
-1/2
n
n
p
d
antiparticle
d
u
An Isospin triplet : the Pion
Wave Function
I
1
1
I3
Q/e
1
ud  
1
-1
 u d  
-1
1
0
0
o
1
dd  uu    0
2
1
dd  uu   
2
o
0
17
Building up strongly interacting particles (hadrons) using Quarks as
building blocks
18
19
 (1,1)  p (1) p ( 2)
More on the two nucleon state:
S
Isospin part
The total wave function :
A
 (tot)   (space)  (spin)  (isospin)
1
 p(1)n(2)  n(1) p(2)
2
 (1,1)  n(1) n( 2)
1
 p(1)n(2)  n(1) p(2)
 (0,0) 
2
 (1,0) 
(non relativistic decomposition)
The Deuton case:
 As it is known that spin = 1  α symmetric
 φ has (-)l symm. It is known that the two nucleons are l=0 or l=2, φ is symm.
  must be antisymmetric.
 This is because ψ tot must be antisymmetric in nucleon exchange.
I 0
The Deuton is an Isospin singlet
Let us then consider the two reactions :
(i) p  p  d   
Isospin I
1
1
The reaction can proceed only via I=1
(Deuton has I=0, Pion has I=1)
(ii) p  n  d   0
0,1
1
 ( pn  d 0 ) /  ( pp  d  )  1/ 2
20
Gauge symmetries (global and local)
Gauge symmetries are continuous symmetries (a continuous symmetry
group). They can be global or local.
Global symmetries: conserved quantities (electric charge)
Local symmetries: new fields and their transformation laws (Gauge theories)
Let us consider the Schoedinger
equation
 2  2
  

 
  V (r )  (r )  E (r )
 2m

Let us consider a global phase transformation: the
change in phase is the same everywhere


i
 (r )  e  (r )
The Schroedinger equation is invariant for this transformation. This invariance ia
associated (E. Noether’s Theorem) to electric charge conservation.
But then what happens if we consider a
local gauge tranformation ?

  k     (r )
21


i
 (r )  e  (r )



i ( r )
 (r )  e  (r )
How does one realize a local gauge invariance ?



i ( r )
 (r )  e  (r )

 2  2
  i ( r ) 

i ( r )
 
  V (r )  e
 (r )  E e  (r )
 2m

Non invariant! And this is because :


 

 i ( r ) 


 

i
i ( r )
 e  (r )  e i      e
 (r )
extra term !
22
To solve the problem (and stick to the invariance) we can introduce a new
field. The field would compensate for the extra term.
The new field would need to have an appropriate transformation law.
Since the free Schroedinger Equation is not invariant under:
 
 2

 i
 
 ( x, t )  i   ( x, t )
2m
 t 
Let us modify the Equation :

Compensating fields

Transformation laws:

i q  ( x ,t )
   e
'

  2

 i  G

 
 ( x , t )  i   R  ( x , t )
2m
 t

 ' 

G G  G  q
R  R '  R  iq


t
23
This allows us to restore the invariance


Gq A
To give the fields physical meaning :
R  iqV

It is invariant, indeed
iq
   e 
'

 2


 i  qA

 
 ( x , t )  i   iqV  ( x , t )
2m
 t


'  
A  A  A  
V  V ' V   t 

'

 i  qA
2m
  ( x, t )  i    iqV  ( x, t )
2
'
'
 t
'

The local gauge U(1) invariance of the free Schroedinger field
• Requires the presence of the Electromagnetic Field
• Dictates the field transformation law
24
This gauge symmetry is the U(1) gauge symmetry related to phase
invariance:

   '  ei q  ( x ,t ) 
There are, of course, other possibilities.
In physics, a gauge principle specifies a procedure for obtaining an
interaction term from a free Lagrangian which is symmetric with respect to a
continuous symmetry -- the results of localizing (or gauging) the global
symmetry group must be accompanied by the inclusion of additional fields
(such as the electromagnetic field), with appropriate kinetic and interaction
terms in the action, in such a way that the extended Lagrangian is covariant
with respect to a new extended group of local transformations.
FREE
INTERACTING
 
 2

 i
 
 ( x, t )  i   ( x, t )
2m
 t 
 2


 i  qA

 
 ( x , t )  i   iqV  ( x , t )
2m
 t



25
The U(1) Gauge Invariance and the Dirac field
Dirac Equation (1928): a quantum-mechanical description of spin ½ elementary
particles, compatible with Special Relativity


i    i    m 
t
 
0  
i    i 
 i 0 
i 
 




  m   0
4 component spinor
 ( x)   ( x)  0
Conjugate spinor




1
0
  

0  1 
 0 
Matrices 4x4 (gamma)
Dirac probability current
j     
 j  0
26
L  i c     mc  

The Dirac Lagrangian:
2
Features a global gauge invariance:
  ei 
(trasformazioni di fase)
Let us require this global invariance to hold locally.
The invariance is now a dynamical principle
i ( x )
 e
 e
 iq ( x )
c

Now the gauge transformation depends on the
spacetime points
Let us see how L behaves :
Since
 iq
 iq
 iqc 
iq c
c
   e    e    e     
c


27
L  L'  i c e
 iq
c
 iq



c
     e    mc 2  L  q     




This lagrangian is NOT gauge-invariant.
Se noi vogliamo una L gauge-invariante occorre introdurre un campo
compensante con una legge di trasformazione opportuna:
L  i c     mc2  (q  ) A
A  A   
This new lagrangian is locally gauge-invariant.
This was made possible by the introduction of a new field (the E.M. field).
28
L  i c     mc2  (q  ) A
  ei ( x )  e
 iq ( x )
c

A  A   
iq
c
 iqc 
L  i c e     e    mc 2   (q  ) ( A     ) 


 i ce
iq
c


 e
 iq
c




  q       mc 2   (q  ) A  q       L
The gauge field A must however include a free-field term.
This will be the E.M. free field term..
29
Discrete symmetries: P,C,T
Discrete symmetries describe non-continuous changes in a system. They
cannot be obtained by integrating infinitesimal transformations.
These transformations are associated to discrete symmetry groups
Parity P
Inversion of all space coordinates :
 x   x 
  

P:  y     y 
 z   z 
  



P (r )  (r )
The determinant of this transform is -1. In the case of rotations, that would be +1



PP (r )  P (r )  (r )
PP 1 P P
PP 1
A unitary operator.
Eigenvalues: +1, -1 (if definite-parity states)
Eigenstates: definite parity states
30
Parity is conserved in a system when
The case of the central potential:
H , P 0



PH (r )  H (r )  H (r )
Bound states of a system with radial symmetry have definite parity
Example: the hydrogen atom
Hydrogen atom: wavefunction (no spin)
 (r, ,)   (r) Yl m ( ,)
Radial part
      

P :    
      
Angular part
P: Yl m ( , ) (1)l Yl m ( ,)
Electric Dipole Transition ∆l = ± 1
P ( )  1
31
The general parity of a quantum state


P ( x,t ) Pa ( x,t )
Let us consider a single particle a.
The intrinsic parity can be represented by a phase


P ( x,t )  expi / 2 ( x,t )
Intrinsic
Spatial


P P ( x,t ) ( x,t )
Which is the physical meaning of the intrinsic parity ?
For instance, in a plane wave (momentum eigenstates) representation




P p ( x,t )Pexpi( px  Et)Pa p (x,t )Pa  p ( x,t )
The intrinsic parity has the meaning of a parity in the p=0 system
32
The parity of the photon from a classical analogy
A classical E field obeys :
Let us take the P:
To keep the Poisson Equation
invariant, we need to have the
following law for E :
 

E( x,t )  ( x,t )


 

() PE( x,t )  (x,t )


And the parity operation would give :
 
 
PE( x,t )   E( x,t )


 
A
A
E ( x ,t )   

t
t

 
 
A( x,t )  N  (k )exp i(kx  Et)
 
 
A( x,t )  P A( x,t )
In order to make it consistent with the
electric field transformationt:
P  1
On the other hand, in vacuum:


33
The action of parity on relevant physical quantities
Position:


P: r  r
Momentum:
Angular Momentum:
P: t  t




dr
dr
P: pm
   p
dt
dt
  

 
P : L  r  p  (r )  ( p)  L
Time:
Charge:
E field:
P:

P: E
B field:

P: B  k
Current:
Spin:
P: q  q




J  v  v  J



r
r
 kq 3  kq 3  E
r
r
 



s r
( s )  (r )
I  k
I B
2
2
r
r
P:   
34
Parity of a complex: product of the parities of the parts of the system
Spatial and intrinsic parity of particles : the pion
s  0, sd  1  s1  1/ 2, s2  1/ 2 (already known)
L( d )  0 (known)
P(d )  1 (known)
  d n  n

J  L  S 1
Angular momentum conservation : J=1= L+S
Exchange between
the two neutrons
(1) LS 1   1
Global n+n symmetry
P(n, n)  (1) L  1
L  1, S  1
L  S even with J  1
P(  )  1
35
Some intrinsic parities cannot be observed (p, n). They are conventionally chosen
to be +1. Because of the Baryon Number conservation the actual P value is not
important as it cancels out in any reaction.
Neutral pion parity. From the pair polarization in :
 0  , eeee
P( 0 )  1
J P  0  : pseudoscalar ( )
J P  0  : scalar

J 1 : vector
P
Transformation properties for rotations
and space reflections. Spin-parity
J P 1 : axial vector
P(particle) = - P (antiparticle)
FERMIONS
P(particle) = P (antiparticle)
BOSONS
Parity is violated in Weak Interactions
36
Time Inversion T
It changes the time arrow
T
Classical dynamical
equations are invariant
because of second
order in time
t t
 
r r




dr
dr
pm  m  p
dt
dt
   


L  r  p  r  ( p)   L
Classical microscopic systems : T invariance
Classical macroscopic systems: time arrow selected statistically (non
decrease of entropy)
In the quantum case :

i    H
t
Is not invariant for
 T

 (r , t )  (r ,t )
37
Let us now start from the Conjugate Schroedinger Equation :
i



 (r , t )  H  (r , t )
t
i

 * 
 (r , t )  H  * (r , t )
t
 T * 
 (r , t )  (r ,t )
Let us define the T-inversion operator :



 i   * (r , t )  H  * (r , t )
t
T
So, with this definition of T operator, we have:
 * 
* 
i   (r ,t )  H  (r ,t )
t



i  T (r ,t ) H T (r ,t )
t
The operator representing T is an antilinear operator.
The square modulus of transition amplitudes is conserved
38
Wigner Theorem on Quantum Systems
Any symmetry of a quantum
system is given by:
•
 '  U
U U  1
U a   b 
  aU 
 bU 
either a unitary
• or an antiunitary operator
 '  W
W W  1
W a   b 
 aW
*
 b*W 
The Problem of Measurement in Quantum Mechanics
39
An important consequence of T-invariance at the microscopic level concerns the
transition amplitudes :
M i  f  M f i
(detailed balance)
Note: the detailed balance DOES NOT imply the equality of the reaction rates:
M i  f  M f i
Wi  f
2
2
2
2

M i f  f 
M f i  i  W f i


A “classical” test, the study of the reaction
p  27Al   24Mg
T is violated at the microscopic level il the Weak Nuclear Interactions
Physical Review Letters 109 (2012) 211801. BaBar experiment at SLAC
Comparing the reactions:
B  B0
40
Charge Conjugation C
C
An internal discrete symmetry
q q
It changes the sign of the charges (and magnetic moments)




r
r
C : E  k q 3  k (q) 3   E
r
r
 
 


s r
s r
C : B  k 2 I  k 2 ( I )   B
r
r
In the case of a quantum state
C


 (q, r , t )   (q, r , t )
C  C 
The C eigenstates are the neutral states
C ( )   1
For the photon case
41
The C-parity of a state can be calculated for a neutral state if we know the
wave function of the state
Since charge conjugation of two particles of opposite charge, swaps the
identify of the particles, one has to account for the proper quantum statistics
C    , L  (1) L   
True also in general for spin zero particles
For a couple of femions, instead :
C f f , L, S  (1) L S f f
In the pi-zero decay
This decay in 3 photons
 0  
C ( 0 )  C( )C( )  (1) (1) 1
 0 
Is forbidden if C is conserved in
electromagnetic interactions. In fact :
 0 
7

10
 0 
42
Action of C,P,T

r
t

 md r
p
dt
  
L r  p
q
C

r
t

p

L
q



J   v   v   J

qr (  q ) r   E
E 3
r3
r
 
 s  r
s r

B 2 I
(I )  B
2
r
r


P
T

r
t

r
t

md (  r )

p
dt

md r

p
d ( t )



( r )  (  p)  L



r  ( p)   L
q
r

2

I  B

L

J

E


 ( v )   J


q(r )


E
r3

r
t

p
q
q


 ( v )   J


(s )  (r )
CPT


qr

E
r3


(s )  r
r

2

I  B

B

43
Positronium
Similar to the hydrogen atom.
Actually, the «true» atom.
   (space)  (spin)  (C )
The space part
 Yml ( ,  )
The spin part
C  (1)
S 1
C  (1)l
 (1,1)  1 (1 / 2) 2 (1 / 2)
Triplet
Singlet
The C conjugation balance:
1
1 (1 / 2) 2 (1 / 2)  1 (1 / 2) 2 (1 / 2)
2
 (1,1)  1 (1 / 2) 2 (1 / 2)
 (1,0) 
 (0,0) 
1
1 (1 / 2) 2 (1 / 2)  1 (1 / 2) 2 (1 / 2)
2
K  (1)l (1) S 1 C
44
K  (1)l (1) S 1 C
Singlet:
Positronium in the l=0 (fundamental) state
e e  2 
J  0, l  0, S  0
e  e   3
J  1, l  0, S  1
K  (1)l (1) S 1 C  (1)0 (1)1 (1)  1
Triplet: K  (1) (1)
l
S 1
C  (1)0 (1)2 (1)  1
C  1
C  1
Antisymmetry by
electron/positron
exchange
C-parity conservation determines the Ps decay modes :
Singlet:
 (2 )  1.2521010 s
Triplet:
 (3 )  1.374107 s
45
Photons, Spin, Helicity
  
B  A



1 A
E    
c t
Gauge - invariant
  
A  A  
1 
  
c t
Coulomb Gauge

A  0
Free propagation:

2 1  A
 A 2 2 0
c t

 
A  e A0 expi(kr   t )
2


A  0  e k  0
For instance:
ex2  ey2 1
Plane wave solution
Transversality condition

 0
k  (0,0, k )
Ax  ex A0 expi(kz   t   )
Plane polarization
Ay  e y A0 expi (kz   t )
Circular polarization
   / 2 ex  e y
46
Circular polarization
  /2
ex  ey
Which can be expressed by using the rotating vectors :
1
(ex  ie y )
2
1
eL 
(e x  ie y )
2
eR 
The polarization vectors can be associated to the photon spin states
If the wave propagates along z:
Lz  xpy  ypx  0
Let us make a rotation around the z axis :
Autostati
di Jz
ex'  ex cos  e y sin 
e 'y  ex sin   e y cos
ez'  ez
Per la trasversalità abbiamo solo:
Fotoni con Jz=0 sono i fotoni
longitudinali. Virtuali: m≠0
Jz only due to spin
R  exp(i J z )
1 '
(ex  ie'y ) 
2
1 '
eL' 
(ex  ie'y ) 
2
ez'
eR' 
e
1
(ex  ie y ) exp(i )
2
1
(ex  ie y ) exp(i )
2
1
0
e
k
J z  1 R
1
k
J z  1 L
47
Helicity
Proiezione dello spin nella direzione del momento
right handed H  1
left handed H   1
scalar
H0

p
H
 E
Un numero quantico approssimato per particelle con massa
Tanto più buono quanto la particella è relativistica
Rigorosamente buono per i fotoni
La legge di invarianza in azione.
Le Interazioni Elettromagnetiche conservano la Parità
Ma:

 

P :  p   ( p)    p
Nelle interazioni elettromagnetiche questa quantità deve essere nulla.
Nelle interazioni elettromagnetiche i fotoni right e left handed compaiono sempre
in pari ampiezze, in modo da compensarsi
48
The Neutrino
C, P sono violate nelle Interazioni Nucleari Deboli
Il neutrino partecipa solo delle Interazioni Nucleari Deboli
Peraltro, nell’approssimazione di neutrini senza massa, abbiamo : J z   1/ 2, 1 / 2
L’evidenza sperimentale indica che nelle Interazioni Deboli:
I neutrini sono sempre sinistrorsi. Gli antineutrini sono sempre destrorsi !

p
P
L
R
C

p
L

p
CP

R
p
In buona approssimazione le Int. Deboli conservano CP (non C e non P)
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The CPT Theorem
In a local, Lorentz-invariant quantum field theory, the interaction (Hamiltonian)
is invariant with respect to the combined action of C,P,T
(Pauli, Luders, Villars, 1957)
A few consequences :
1) Mass of the particle = Mass of the antiparticle
2) (Magnetic moment of the particle) = -- (Magnetic moment of antiparticle)
3) Lifetime of particle = Lifetime of antiparticle
Proton
Protons,
electrons
Antiproton
Electron
Positron
Q
+e
-e
-e
+e
B o L(e)
+1
-1
+1
-1
μ
 2.79(e / 2Mc)
 2.79(e / 2Mc)
 e / 2mc
 e / 2mc
σ
/2
/2
/2
/2
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Search for violations of C,P,T
A quantity is formed that would violate a conservation law
One checks if this quantity exists for a pure state.
Examle: the Electric Dipole Moment (EDM)
 
 
T :  E   E
 
 
P :  E   E
Cannot exist for a pure state is T, P invariant situations
CPT Theorem (wikipedia)
In quantum field theory the CPT theorem states that any canonical (that is, local
and Lorentz-covariant) quantum field theory is invariant under the CPT
operation, which is a combination of three discrete transformations: charge
conjugation C, parity transformation P, and time reversal T. It was first proved by
G.Lüders, W.Pauli and J.Bell in the framework of Lagrangian field theory.
At present, CPT is the sole combination of C, P, T observed as an exact symmetry
of nature at the fundamental level.
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Particle Numbers: baryonic, flavor, and leptonic
Flavor :
The flavor is the quark content of a hadron
Massa
(MeV)
Quark
U
D
S
C
B
p
938
uud
+2
+1
0
0
0
n
940
udd
+1
+2
0
0
0
Λ
1116
uds
+1
+1
-1
0
0
Λc
2285
udc
+1
+1
0
+1
0
π+
140
u-dbar
+1
-1
0
0
0
K-
494
s-ubar
-1
0
-1
0
0
D-
1869
d-cbar
0
+1
0
-1
0
Ds+
1970
c-sbar
0
0
+1
+1
0
B-
5279
b-ubar
-1
0
0
0
-1
Υ
9460
b-bbar
0
0
0
0
0
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Favor quantum numbers refer to quark content of hadrons
They are conserved in Strong and Electromagnetic Interactions
They are violated in Weak Interaction


U  N (u)  N (u ) D  N (d )  N (d )
S  N (s)  N (s )
Strangeness

C  N (c)  N (c ) B   N (b)  N (b )
Charm

T  N (t )  N (t )
Beauty
In a Stong Nuclear (or E.M.) process,
all flavors are conserved:
In Weak Interactions instead :
Top
p 
p

n 
p  
(uud)  (uud)  (udd)  (uud)  (ud )
n

p  e   e
(udd)  (uud)
Baryon Number:
1
B  U  D  S  C  B T 
3
53
Baryon Number
1
B  U  D  S  C  B  T 
3
The Baryon Number is equivalent to :
1
B  (nQ  nQ )
3
Baryons have B=1 while Antibaryons have B = -1
Mesons have B = 0
This law follows from the conservation of the Quark Number.
Quarks transform into each other. They disappear (or appear) in pairs.
Flavor quantum numbers refer to the
identity of quarks :
Violated in Weak Interactions
(Isospin: +1/2 o -1/2 in doublets)
Strangeness: -1 for the s quark
Charm: +1 for the c quark
Bottom: -1 for the b quark
Top: +1 for the t quark
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The Leptonic Numbers:
Ne  N (e )  N (e )  N ( e )  N ( e )
Electronic Lepton Number
N  N (  )  N (  )  N (  )  N (  )
Muonic Lepton Number
N  N (  )  N (  )  N ( )  N ( )
Numero leptonico tauonico
The Leptonic Numbers are conserved in any known interaction WITH THE
EXCEPTION OF Neutrino Oscillations. In Neutrino Oscillations, they are violated.
However, one can define a total lepton number :
Nl  Ne  N  N
To the best of our knowledge the Total Lepton Number (sum of the three
leptonic numbers) is conserved in every interaction.
For instance, the decay :
 e
Does not take place. .
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