The number of quarks increased with discoveries of new particles and have reached 6

H

Quarks

Quarks – – “ “ the building the building blocks of the Universe blocks of the Universe ” ”

The number of quarks increased with discoveries of new particles and have reached 6

For unknown reasons Nature created 3 copies (generations) of quarks and leptons Charm came as

surprise but

completed the

picture

Discovery History Discovery History

u d

t c

s b

six quarks 1947

1974 1995

1977

e µ

ν _µ

τ ν _τ ν _e

six leptons 1956

1895

1963

1936 1975

Now we have a beautiful pattern of three pairs of quarks and three pairs of leptons. They are shown here with their year of discovery.

2000

Matter and Antimatter Matter and Antimatter

The first

generation is what we are made of

Antimatter was created together with matter

during the “Big bang”

Antiparticles are created at accelerators in ensemble with

particles but the visible Universe does not contain antimatter

Quark

Quark ’ ’ s s Colour Colour

( )

( )

( )

d d d s s s

u u u

!

! ++

" # # #

$ # # #

" # # #

Baryons are “made” of quarks

To avoid Pauli principle veto one can antisymmetrize the wave function introducing a new quantum number - “colour”, so that

( )

ijk

i j k

d d d

"

!

# = $ $ $

?

The Number of Colours The Number of Colours



The x-section of The x-section of electron-positron electron-positron

annihilation into annihilation into

hadrons is proportional hadrons is proportional to the number of quark to the number of quark

colours. The fit to colours. The fit to

experimental data at experimental data at

various colliders at various colliders at

different energies gives different energies gives

N

= 3.06 ± 0.10

The Number of Generations The Number of Generations

  Z-line shape Z-line shape

obtained at LEP obtained at LEP depends on the depends on the

number of number of

flavours and flavours and

gives the gives the

number of (light) number of (light)

neutrinos or neutrinos or

(generations) of (generations) of

the Standard the Standard

Model Model

N

= 2.982 ± 0.013

Quantum Numbers of Matter Quantum Numbers of Matter

  Quarks Quarks

  Leptons Leptons

L

L

R R

R R

Q up

down U up

D down

! "

= # $

% &

=

=

SU(3)c SU(2)_L U_Y(1)

3 3 3

2 1 1

1/ 3 4 / 3

2 / 3

!

?

L

L

R R

R R

L e

N

E e

!

!

" #

= $ %

& '

=

=

1 1 1

2 1 1

1 0 2

!

triplets doublets

singlets

3

/ 2

Q T = + Y

Electric charge

T

T

1

- 2

0 0

V-A currents in

weak interactions

The group structure of the SM The group structure of the SM

Casimir Operators For SU(N)

QCD analysis

definitely singles out

the SU(3) group as

the symmetry group of

strong interactions

Electro-weak sector of the SM Electro-weak sector of the SM

SU(2) x U(1) versus O(3)



The heavy photon gives the

neutral current without flavour

violation



Discovery of neutral currents was a crucial test of the gauge gauge model of weak interactions at CERN in 1973

After spontaneous symmetry breaking one has 3 massive gauge bosons

(W⁺ , W^- , Z⁰) and 1 massless (γ)

2 massive gauge bosons

(W⁺ , W^- ) and 1 massless (γ)

Gauge Invariance Gauge Invariance

( )

( ) exp[

( )

]

i

x U x

i x T

! # ! = " !

( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

i x x i x U x U x x

i x x x U x U x x

µ µ

µ µ

µ µ

µ µ

! " ! ! " !

! " ! ! " !

+

+

# $ #

= # + #

Gauge transformation

parameter matrix Fermion Kinetic term

Covariant derivative

! "

D

= !

I gA T #

= !

I g A $ #

( ) ( ) ( )

D

! x " U x D

! x ( ) ( ) ( ) ( )

( ) ( )

A x

U x A x U x

U x U x

+ +

! + "

matrix

U U

= 1

Gauge invariant kinetic term

i ! ( ) x "

D

! ( ) x

[ D D

,

] = G

= "

A

# "

A

+ g A A [

,

] ^G

^{( ) x} " ^{U x G ( )}

^{( ) x U x}

^{( )}

Gauge field kinetic term ¹

4

Tr G G

"

( )

( )

i

x

U x

! " !

Lagrangian of the SM Lagrangian of the SM

gauge Yukawa Higgs

L L = + L + L

1 1 1

4 4 4

( ) (

)

a a i i

gauge

L G G W W B B

iL D L iQ D Q iE D E

iU D U iD D D D H D H

µ! µ! µ! µ! µ! µ!

µ µ µ

" µ " " µ " " µ "

µ µ

"

" µ " µ " µ µ

# # #

# #

= $ $ $

+ + +

+ + +

L D U

Yukawa

L = y L E H y Q D H y Q U H

+

+

2 † † 2

2

( )

Higgs

L = " V m H H = "

H H ^{H = i H !}

(3) (2) (1)

c L Y

SU ! SU ! U

α,β=1,2,3 - generation index

Fermion

Fermion Masses in the SM Masses in the SM

L L R R

0

! ! = ! ! =

L R R L

! ! + ! !

c c

L L L L

! ! + ! !

Direct mass terms are forbidden due to SU(2)

invariance !

SU(2) doublet SU(2) singlet Dirac Spinors

5 5

0 0 2 *

1 1

, , , ,

2 2

c

L

!

! C i

" " = ^# " " = ⁺ " " = " !

" = ! " = ! "

left right Dirac conjugated Charge conjugated

Lorenz invariant Mass terms SU_L(2)

c c

R R R R

! ! + ! !

SU_L(2) & U_Y(1) U_Y(1)

Unless Q=0, Y=0

c

! !

Majorana mass term

( )

0 exp( )

2 v

2

H H i !# H

H S

% &

' (

) )

* = +++* =

' + (

' (

, -

r r

r r

Spontaneous Symmetry Breaking Spontaneous Symmetry Breaking

(3) (2) (1) (3) (1)

c L Y c EM

SU ! SU ! U " SU ! U

0

H H

H

!

"

= # $

% &

2 † † 2

2

( )

V = " m H H +

H H

Introduce a scalar field with quantum numbers: (1,2,1) With potential

Unstable maximum Stable minimum At the minimum

0

0 exp( )

2 v

v 2 2

H H

H S iP i S

H

!"

+

+

# $ # $

# $ % & % &

= % ' & ( = % % + + & & = % % + & &

' (

' (

r r

scalar

pseudoscalar v.e.v.

Gauge transformation Higgs boson

h

0 H ! " v

= # $

% &

3 - 3 -

1

4 + 3 + 3

gW ' 2gW gW ' 2gW 0

(0 v)

2gW -gW ' 2gW -gW ' v

g B g B

g B g B

µ µ µ µ µ µ

µ µ µ µ µ µ

! + " ! + " ! "

# $ # $

% # & + $ # ' & + $ ' # $ & '

The Higgs Mechanism The Higgs Mechanism

( )

e

e

e

e

W

W

# #

$ + %

a a

! = # "

Q: What happens with missing d.o.f. (massless goldstone bosons P,H

or ξ ) ? A: They become longitudinal

of the gauge bosons W

, i=1,2,3

Gauge transformation

Longitudinal components Higgs field kinetic term

2 ' 2

2 2

g g

D H

= !

H " W H

" B H

2 2

1

v v ( ' )

2 4

g W W

gW

g B

" + ! +

3 3

2

Z sin cos

cos sin

W W

W W

W W W

B W

B W

µ µ

µ

µ µ µ

µ µ µ

! !

" ! !

±

=

= # +

= +

m

2 1 2 2

2

2 1 2 2 2

2

v

( +g' )v

W Z

M g

M g

=

=

tan /

0

W

g g M

" = ^#

=

The Higgs Boson and

The Higgs Boson and Fermion Fermion Masses

Masses

2 † † 2

2

( )

V = " m H H +

H H

0

v 2

H h

! "

# $

= # # + $ $

% & ^v

_v

^v

2 2 8

V ! h ! h ! h

!

= " + + +

( )v, ( )v, ( )v

u u d d l l

i i i

M = Diag y

M = Diag y

M = Diag y

2 2 v

m

= m = !

2 2

v = m / !

E D U

Yukawa

L = y L E H y Q D H y Q U H

+

+

α, β =1,2,3 - generation index Dirac fermion mass

( )v

N N

y L N H

$ M

= Diag y

The Running Couplings The Running Couplings

( ) ( / ) ( )

Bare

Z

! " = " µ ! µ

2 2

( / ) 1 ( / ) ...

Z " µ = # b Log ! " µ +

2 2 2 2 2 2

( / ) ( / ) ( / )

Log p Log Log p

! " # ! " µ = ! µ

Radiative Corrections

~α (log Λ

/p

+fin.part)

Renormalization operation

UV cutoff Renormalization scale Renormalization constant

Subtraction of UV div

Finite

R

( )

! µ

d

( ), ( )

d

( / )

Log Z

d d

µ ! " ! " ! µ µ

µ = = # µ $

Running coupling

Renormalization Group Renormalization Group

2 2 2 2 2

( , ( )) (1 ( ) ( ))

R

Q µ ! µ = ! + b Log Q ! µ + O !

2 2 2

2

(

) 0

d d

R R

d d

µ µ µ !

µ µ µ !

" "

= + =

" "

2 2 2 2

2

2

( , ( )) (1, ( , ))

( )

RG PT

R Q R Q

Q d

dQ

µ ! µ ! µ !

! " !

=

=

Observable

RG Eq.

Solution to RG eq.

Effective coupling

! " ( )

Solution to RG eq. sums up an infinite series of the leading Logs coming from Feynman diagrams

2 2

2 2

(1, ) , (1 ( ) ...)

1 ( )

R

b Log Q

b Log Q

! ! ! ! ! ! µ

! µ

= = " + +

#

Asymptotic Freedom and Asymptotic Freedom and

Infrared Slavery Infrared Slavery

( ) b

! " = "

One-loop order 4/3 n_f QED

-11+2/3 n_f QCD

2 2

1 b Log Q ( )

! !

! µ

= "

QED QCD

α_ α_

UV Pole IR Pole

Comparison with Experiment Comparison with Experiment

Global Fit to Data Higgs Mass Constraint

Remarkable agreement of ALL the data with the SM predictions - precision tests of radiative corrections and the SM

Though the values of sin ϑw extracted from different experiments are in good agreement, two most precise

measurements from hadron and lepton asymmetries disagree by 3σ

••

Inconsistency at high energies due to Landau poles Inconsistency at high energies due to Landau poles

• Large number of free parameters • Large number of free parameters

• Still unclear mechanism of EW symmetry breaking • Still unclear mechanism of EW symmetry breaking

• • CP-violation is not understood CP-violation is not understood

• • The origin of the mass spectrum in unclear The origin of the mass spectrum in unclear

• • Flavour Flavour mixing and the number of generations is arbitrary mixing and the number of generations is arbitrary

• Formal unification of strong and electroweak interactions • Formal unification of strong and electroweak interactions

Where is the Dark matter?

The SM and Beyond The SM and Beyond

The problems of the SM:

The problems of the SM:

The way beyond the SM:

The way beyond the SM:

••

The SAME fields with NEW The SAME fields with NEW interactions and NEW fields

interactions and NEW fields GUT, SUSY, String, ED GUT, SUSY, String, ED

••

NEW fields with NEW NEW fields with NEW interactions

interactions

Compositeness,

Compositeness, Technicolour Technicolour , ,

preons preons

We like elegant solutions

We like elegant solutions