Transcript Document

Introduction to the Standard Model
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
What is the Standard Model?
A quantum relativistic theory of fundamental constituents
It is based on concepts of gauge symmetry (and fields) describing three
out of four fundamental interactions. Actually… two out of three.
The Modello Standard is the best candidate we have to
A complete theory of fundamental interactions
Fundamental constituents
and interactions between
them
2
Construction of a QED Lagrangian
L  i c     mc  

The Dirac Lagrangian:
 L  L
(Eulero-Lagrange)




0
     A    A


2
(i     m)  0 (Free Dirac equation)
The Lagrangian is invariant for global gauge transformations:
  ei 
(phase transformation)
We require that this global property also holds true locally. Gauge invariance
becomes a dynamical principle.
i ( x )
 e
 e
 iq ( x )
c

Now the gauge transformation depends on the
spacetime pointOra la trasformazione di gauge
dipende dal punto dello spaziotempo
Let us see how L behaves
3
 iq
 iq
 iqc 
iq
   e    e c    e c     
c


Using
L  L'  i c e
 iq
c
 iq



c
     e    mc 2  L  q     




This lagrangian is not gauge-invariant
If we want to have a gauge-invariant lagrangian, we need to introduce a
compensating field with a suitable transformation law:
L  i c     mc2  (q  ) A
A  A   
This new lagrangian is locally gauge-invariant.
It was necessary to introduce a new field (the Electromagnetic Field).
4
L  i c    mc   (q  ) A


2
Local gauge transformation
  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 also have a free-field term.
This term will be the free Electromagnetic Field.
5
Lagrangian of the free Electromagnetic Field :
1 
L
F F
16
F     A   A
(Eulero-Lagrange)
 L  L

 
0
     A    A


 F   0
(Free field Maxwell equation)
This new term is gauge-invariant by itself :
F  F  2   A   A  2   A  A  2    A          A        2    A         A       
 2  A  A  2  A  A  F  F
It is interesting to note that the presence of a mass for the photon would spoil the
gauge invariance.
6
1 
1  mAc  
F F 

 A A
16
8 

2
L
A vector massive field (…if the photon
had a mass!). The Proca lagrangian
Recall: the e.m. field interacting with a current would be :
L
1 
1
F F  J  A
16
c
The mass term violates gauge invariance.
  F  
4 
J
c
A  A   
A A   A      A      A A  2 A        A A
Gauge invariance and masslessness of the photon are connected
7
The Dirac Lagrangian with the interaction with the e.m. field (gauge-invariant):
 1 

L  i c     mc 2   
F F   (q  ) A
16

Free massive ½
fermion field
Free e.m. field
Interaction
J   cq (  )
In order for the gauge-invariance to be preserved, the gauge field must be
massless.
This is the U(1) gauge symmetry
8
A little to-do list for the construction of the Standard Model Lagrangian:
Build an electroweak lagrangian based on the symmetry group SU(2)xU(1).
Add QCD.
Solve the mass problem for the W, Z.
Mass must be generated in a gauge-invariant way
Solve the mass problem for the fermion constituents.
Mass must be introduced in a gauge-invariant way
9
The SU(2) symmetry of the Yang-Mills Field
Let us consider two non-interacting Dirac fields :
L  i c 1     1  m1c 2 1 1   i c 2   2  m2c 2 2  2 
It can be written as :
 1 
  
 2 

  1  2

Rewriting the Lagrangian :
L  i c     c2 M 
If the two masses are equal, M = m (scalar)
 m1 0 
M 

0
m

2
(mass matrix)
We can apply the SU(2) symmetry among the two fields (Yang-Mills)
10
  U
UU  1
  U 
U a unitary matrix to be written as :
U  ei ei a
τ : the Pauli matrices
U(1)
SU(2)
Let us concentrate on SU(2):
L  i c     mc2 
The initial lagrangian has an
SU(2) global invariance :
  S
S  ei a
Let us now impose the local (x-dependent) invariance :
  S
S e

 iq 
c

iq 
1 
c
11
In order to keept this lagrangian (locally) invariant, it is necessary to introduce a
triplet of compensating fields, such that:
2q
A  A        A
c


L  i c     mc2   (q   ) A
The transformation law of
the compensating fields
The new interaction term
is introduced
One should introduce ther free-field terms of the three compensating fields
1 
L
F F
16
F     A   A 
The free fields need to preserve the gaugeinvariance. Therefore they should transform :
2q  
A A
c
F   F  
2q
  F 
c
(a consequence of the transformation law of A)
12
This is the Yang-Mills, SU(2) invariat, lagrangian describing two Dirac fields,
iteracting by means of three massless gauge fields:
1

2


L  i c    mc    
F  F  (q   ) A
16
The structure of the SU(2) fields and of their transformation law depends on the
fact that the symmetry group is not abelian.
This structure bears a strong analogy with the Standard Model lagrangian, in
particular as long as the symmetry group is concerned.
If we did not worry about the mass, we could in principle write the Electroweak
lagrangian just by using the SU(2) and U(1) symmetries
13
The mass problem
Termine esplicito di massa VIETATO nella Teoria Elettrodebole !
14
Quantum Chromodynamics as a Gauge Theory
QCD is the Theory of Strong Interactions
The quarks are structureless spin ½ elementary particles
A quark is described by a 4-component spinor
obeying (in a free theory) the Dirac Equation

(i     m)  0
i
f
The general form of the free wavefunction
for a fixed flavor and a fixed color
 if (x)
f 1,2,3 : the different flavors
i  green, red , blue
 

 k , ( x)  u (k ,  ) exp  i Et  k x

(k ,  )
Four-momentum and polarization
Spin-dependent momentumspace wavefunction
15
The free-quark Lagrangian
L 0   f  fi (i      m) if
Is invariant under a SU(Nc=3) global
transformation
 ( x)  U i j  jf ( x)
'f
i
U is an SU(3) matrix acting on the color part of the wavefunction
A generic SU(3) matrix requires 8 real parameters :


  

U i j  exp i  
2 i j
 
The group SUC(3) depends on 32 – 1 = 8 real parameters.
The matrices
 

 2


i j
are unitary, hermitian, traceless.
They are the generators of the fundamental representation of SU(3), to
which the field ψif belongs.

With the SU(3) generators
(hermitian 3x3 matrices) :

2
 1,.....8
Gell-Mann matrices
  
 ,
2 2


 
 i f
2

Antisymmetric structure constants
16
Since SUC(3) is an exact symmetry, we can «gauge» it (which means: makes
it local) by requiring invariance :
If we require local gauge invariance



  (x) U i j  exp i  ( x)
2
 





i j

L 0   f  fi (i      m) if  Lq   f fi i  D  m  if
Where




D    ig A

2

i  
A  U (A
  )U
2
g



A linear combination of 8 gluon fields
A possible realization
of the Gell-Mann
matrices
17



Lq   f i D  m 
i
f

The locally SU(3) invariant
Quark lagrangian



ij
2



8 gauge potentials (gluons) and their
transformation laws
This lagrangian contains a free
quark term and an interaction term:
L q  L0  g  f fi A

i  
D    ig A
A  U (A
  )U
2
2
g

i
f
   if
j
g
A
ij
2
i
To have the full QCD Lagrangian, one needs the gluon part:
Where the field strength tensor: F     A   A  g f abc Ab Ac
1
L g   F  F
4
Contains self-interaction terms of the field with itself
f: SU(3) structure constants
The full QCD Lagrangian:
LQCD
 i
 1
ij

i
i 
 L 0  Lint  Lg   f   f i (i     m) f  g  f A
  if   F  F
2

 4
18
The costituents in the Standard Model Lagrangian
A set of elementary constituents: quarks and leptons (pointlike, no internal
structure down to 10^(-18) m)
A set of forces generated by gauge symmetries :
• Electroweak (quarks & leptons)
• Strong (quarks)
Leptonic sector characterized by the SU(2)xU(1) scheme classification :
 e 
Le   
 e L
Re  eR
  
L     
  L
R R
  
L    
  L
R  R
I
I 0
1
2
Y ( Ll )  1
Y ( Ll )   2
(assume Massless Neutrinos)
19
The hadronic sector consists of the left-handed quarks:
u 
L(q1)  ' 
 d L
c 
L(q2 )   ' 
 s L
t 
L(q3)   ' 
 b L
Ru(1)  u R
Ru( 2) cR
Ru(3)  t R
Rd(1)  d R
Rd( 2)  sR
Rd(3)  bR
I
I0
Y ( Ru ) 
4
3
1
2
Y ( Ll ) 
1
3
Y ( Rd )  
The primes on the lower components of the quark doublets signal that the
weak eigenstates are mixtures of the mass eigenstates (CKM matrix):
 d '  Vud Vus Vub   d 
  
 
'
 s   Vcd Vcs Vcb   s 
 ' 
 b  Vtd Vts Vtb   b 

  
20
2
3
The non-interacting Standard Model Lagrangian
L0,SM  L0,leptons,SM  L0,quarks ,SM  Rl i    Rl  Ll i    Ll  Ru(n)i    Ru(n)  Rd(n)i    Rd(n)  Lq(n)i    L(qn)
Massles non
interacting Leptons
Massles non interacting Quarks
Let us require local gauge invariance of the type SU(2)xU(1)xSU(3)


  
 ( x)
 ( x)  exp(i ) exp(i  bi i ) exp i 
2 
i
 
'
g'
g'
g 

Lmassless , SM  Lmassless ,leptons, SM  Lmassless ,quarks , SM  Rl i (   i A Y ) Rl  Ll i (   i A Y  i  b ) Ll
2
2
2
gs 
g
g'
g'
(n)

(n)
(n)

 Ru i (   i A Y  i C  ) Ru  Rd i (   i A Y  i s C  ) Rd( n )
2
2
2
2
'
g
g
g 
1
1
1
(n)


 Lq i (   i A Y  i  b  i s C  ) L(qn )   Fl F l  f  f    G
G
2
2
2
4 l
4
4 

21
g'
g'
g 

Lmassless , SM  Lmassless ,leptons, SM  Lmassless ,quarks , SM  Rl i (   i A Y ) Rl  Ll i (   i A Y  i  b ) Ll
2
2
2
gs 
g
g'
g'
(n)

(n)
(n)

 Ru i (   i A Y  i C  ) Ru  Rd i (   i A Y  i s C  ) Rd( n )
2
2
2
2
'
g
g
g 
1
1
1
(n)


 Lq i (   i A Y  i  b  i s C  ) L(qn )   Fl F l  f  f    G
G
2
2
2
4 l
4
4 

The free gauge fields
The gauge potentials:
A  A 
1
 
g'
Hypercharge phase

  
1
b  b    b    
g



C  U (C

2

Weak Isospin rotation
i  
 )U
gs
Color rotation
f    A    A
Fl   bl    bl  g  ijkbj bk
G    C   C   gs fijk Aj Ak
22
Now the masses !
We introduce a doublet of scalar field:
With a Lagrangian:
  
   0 
 
Lscalar  (D ) (D ) V (  )
A two parameters potential:
 
 
V (  )   2      
2
This potential gives mass to W,Z in a gauge invariant way (more on this later)
This potential can give mass
to the constituent fermions by
adding to the Lagrangian
terms like

 

LYukawa  l (Ll )Rl  Rl (  Ll )  q (Lq )Rq  Rq (  Lq )
The Full Lagrangian of the Standard Model (prior to 1996,massless neutrinos) :
LSM  Lmassless ,SM  Lscalar  LYukawa
23
The very many
interactions of
vector bosons in
the Standard Model
24
Introduction to the Higgs Mechanism
On the mass terms in a Lagrangian
Suppose we consider the L
1

 ( )2
L       e
2
To understand where is the mass term, we
compare with the KG:
1
1  mc  2
LKG       
 
2
2


  
m2 c 2
2
2
 0
By expanding in power series :
So, this lagrangian describes a massive field with
L
1
     1   2 2 .........
2
m 2 / c
25
On the fundamental state of the Lagrangian
In general, in order to find the ground state, it is necessary to write L in the
form L = T-U and then minimize U with respecto to the fields.
For instance, in the Klein-Gordon:
2
1
1  mc  2
LKG       
   T U
2
2

 0
The minimum of U with respect to the fields
Let us now start with our prototype lagrangian :
The minima of
Are the two points :
L
1
1
1
      2 2   2 4
2
2
4
1 2 2 1 2 4
U    
2
4



26
We can then reformulate the theory in terms of
deviations from the ground state :
  


In terms of this new variable, the lagrangian becomes
L
1
1
      2 2   3   2 4
2
4
And this brings into evidence a mass term :
m  2 /c
To correctly identify a mass term in a lagrangian, it is necessary first to expand
around the fundamental state.
27
Spontaneous Symmetry Breaking
The starting lagrangian apparently has
not mass term.
It has, however, a symmetry
  
In doing the expansion around the
fundamental state one identifies the
mass term
1
1 2 2 1 2 4

L           
2
2
4
1
1 2 4 1  2 

2 2
3
L               

2
4
4  
2
In this second form, the lagrangian has lost its symmetry.
This is because the vacuum (fundamental state) does not have the symmetry of
the lagrangian itself.
The symmetry is spontaneously broken.
In this case we have a discrete symmetry, made by two values.
We can have a continuous symmetry by using two fields.
28
1
1
1 2 2
1 2 2


2
2 2
L   1  1   2 2   1  2    1  2 
2
2
2
4
This lagrangian involves two fields
1 , 2
It is invariant by rotations in 1 , 2
2
1
The “potential energy” part is
1 2 2
1 2 2
2
2 2
U    1  2    1  2 
2
4
featuring a ring of minima along the circumference :
2

2
2
1min
 2min
 2

We make an expansion around a point of minimum that we can arbitrarily choose :

1min 

2min  0
29
L gets written as a function of fields that are fluctuations around the ground state:
  1 


  2
2 4
1  2 
1
1

2 2
  
3
2
4
2 2 
L                     (   )  (    2  )    
4
2
 2
 
 4  
Free KG with m  2


Couplings

Free KG with
c



2

m  0












There is a massless field, which is typical of the case when a continuous global
symmetry gets broken
30
The Higgs Mechanism
Spontaneous symmetry breaking applied to the case of local gauge invariance.
Equivalence between a complex field and two real
fields like this :
1  Re 

2  Im 
The prototype lagrangian that we have used :
L
1
1 2 *
1 2 * 2
*










 (  )






2
2
4
  ei 
Features the “usual” trivial U(1) global symmetry (phase)
If we now make the symmetry local :
  ei ( x )
It is necessary to introduce a compensating field (and a free field term):
L
1 
iq  *    iq    1 2 *
1 2 * 2 1 


A



A






 (  ) 
F F



 

 



2 
c   
c   2
4
16
31
Let us now expand around the minimum of U

  1 

  2
2
 1 

1  q 
1
1
 q 

2 2
 


L                   
F F  
A
A

2
i

 

     A 
2  c 
2
 2
 16
 c

2
2
2


 q 
1 q 
1 2 4
q


2
2

3
2
2 2
4    
          A      A A         A A           2       

c

c
2
c
4
2











1

2 2






 
 2 

1
 




 2 

Scalar particle with mass
2
m  2  / c
Massless Goldstone Boson
2
 1 

1  q 
 q 


F F  
A
A

2
i


 

     A
2  c 
 c
16

Compensating field term that
now has aquired a mass.
 q 
mA  2   2 
 c 
32
Origin of the mass term: the original term in the “gauged” lagrangian :

A
* A A

A
The shift of the Φ field on this term generates a mass :
2
 2 A A

This is a “Proca” term. The value of the mass depends on the Vacuum
Expectation Value (VEV) of the Higgs field.
33
Epilogue
What we have not described in these lectures (among many things)
Neutrino mass and oscillation
The discovery of the Higgs Boson
The Standard Model : a great success
A Model that accounts for all constituents and 2 out of 3 fundamental forces of
nature in a quantum-relativistic way, explaining an enormous amount of data
Drawbacks of the Standard Model
• The Model has very many free parameters (constants of Nature)
• It has few internal consistency problems (source of CP and
radiative correction of masses, understood up to the TeV scale,
strong interaction problems out of the perturbative regime)
• Gravity not included
• Dark Matter not included
• Dark Energy not included
34
Neutrino Physics: extension of the Standard Model
(M. Mezzetto)
35
Thank you for your attention
36
Backup slides
37
38
39
40
41
42