H

QuarksQuarks –– ““the buildingthe buildingblocks of the Universeblocks of the Universe””

The number ofquarks increasedwith discoveries ofnew particles andhave reached 6

For unknownreasons Naturecreated 3 copies(generations) ofquarks and leptons

Charm came assurprise butcompleted thepicture

Discovery HistoryDiscovery History

u

d

tc

s b

six quarks

1947

1974 1995

1977e µ

νµ

τ

ντνe

six leptons

1956

1895

1963

1936 1975

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

2000

Matter and AntimatterMatter and Antimatter

The firstgeneration iswhat we aremade of

Antimatter wascreated togetherwith matterduring the “Bigbang”

Antiparticles are created at accelerators in ensemble withparticles but the visible Universe does not contain antimatter

QuarkQuark’’s s ColourColour

( )

( )

( )

d d d

s s s

u u u

!

!

++

" # # #

$ # # #

" # # #

Baryons are “made” of quarks

To avoid Pauli principle veto one canantisymmetrize the wave functionintroducing a new quantum number- “colour”, so that

( )ijk

i j kd d d!"# = $ $ $

?

The Number of ColoursThe Number of Colours

The x-section ofThe x-section ofelectron-positronelectron-positronannihilation intoannihilation intohadrons is proportionalhadrons is proportionalto the number of quarkto the number of quarkcolours. The fit tocolours. The fit toexperimental data atexperimental data atvarious colliders atvarious colliders atdifferent energies givesdifferent energies gives

Nc = 3.06 ± 0.10

The Number of GenerationsThe Number of Generations Z-line shapeZ-line shape

obtained at LEPobtained at LEPdepends on thedepends on thenumber ofnumber offlavours andflavours andgives thegives thenumber of (light)number of (light)neutrinos orneutrinos or(generations) of(generations) ofthe Standardthe StandardModelModelNg = 2.982 ± 0.013

Quantum Numbers of MatterQuantum Numbers of Matter

QuarksQuarks

LeptonsLeptons

L

L

R R

R R

upQ

down

U up

D down

! "= # $% &

=

=

SU(3)c SU(2)L UY(1)

3

3

3

2

1

1

1/ 3

4 / 3

2 / 3!

?

L

L

R R

R R

Le

N

E e

!

!

" #= $ %& '

=

=

1

1

1

2

1

1

1

0

2

!

tripletsdoublets

singlets

3/ 2Q T Y= +

Electric charge3T

3T

1

2

1

2-

0

0

V-Acurrents in

weakinteractions

The group structure of the SMThe group structure of the SM

Casimir Operators

For SU(N)

QCD analysisdefinitely singles outthe SU(3) group asthe symmetry group ofstrong interactions

Electro-weak sector of the SMElectro-weak sector of the SMSU(2) x U(1) versus O(3)

The heavy photon gives theThe heavy photon gives theneutral current without flavourneutral current without flavourviolationviolation

Discovery of neutral currents was acrucial test of the gaugegauge model ofweak interactions at CERN in 1973

3 gauge bosons 1 gauge boson 3 gauge bosonsAfter 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 InvarianceGauge Invariance( ) ( ) exp[ ( ) ]a a

iji j ij jx U x i x T! ! " !# = 1,2,...,a N=

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

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

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 a aD I gA T I g Aµµ µ µ µ µ! " = ! # = ! #$ Gauge field

( ) ( ) ( )D x U x D xµ µ! !"1( ) ( ) ( ) ( ) ( ) ( )g

A x U x A x U x U x U xµ µ µ

+ +

! + "

matrix 1U U+

=

Gauge invariant kinetic term ( ) ( )i x D xµ

µ! " !

[ , ] [ , ]D D G A A g A A! µ µ !µ!µ ! µ != = " # " + ( ) ( ) ( ) ( )G x U x G x U xµ! µ!

+

"

Gauge field kinetic term 1

4 Tr G G

µ!

µ!" Field strength tensor

( ) ( )jii jx U x! !+

"

Lagrangian of the SMLagrangian of the SM

gauge Yukawa HiggsL L L L= + +

1 1 14 4 4

†( ) ( )

a a i i

gaugeL 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

YukawaL y L E H y Q D H y Q U H!!" " !" " !" "! != + +

2 † † 2

2( )HiggsL V m H H H H!= " = "

†

2H i H!=

(3) (2) (1)c L Y

SU SU U! !

α,β=1,2,3 - generation index

FermionFermion Masses in the SM Masses in the SM

0L RL R

! ! ! != = L RR L

! ! ! !+

cc

L LL L! ! ! !+

Direct mass terms are forbidden due to SU(2)L invariance !

SU(2) doublet SU(2) singlet

Dirac Spinors

5 5

0 0 2 *1 1, , , ,

2 2

c

L RC i

! !" " " " " " " ! " ! " ! "+# +

= = = = =

left right Dirac conjugated Charge conjugated

Lorenz invariant Mass termsSUL(2)

cc

R RR R! ! ! !+

SUL(2) & UY(1) UY(1)

Unless Q=0, Y=0

c

RR

! !

Majorana mass term

( )

0

exp( )v2

2

H H i H H S! "!# =$

% &' () )* = +++* =' (+' (, -

rrr r

Spontaneous Symmetry BreakingSpontaneous Symmetry Breaking(3) (2) (1) (3) (1)c L Y c EM

SU SU U SU U! ! " !

0

HH

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 minimumAt the minimum

0

0

exp( )v2v

22

HH

H i SS iPH

!"+

+# $ # $

# $ % & % &= = =+% & % & % &++ % &' ( % &' (' (

rr

scalar

pseudoscalar

v.e.v.

Gauge transformation Higgs boson

h

0

vH

! "= # $% &

3 - 3 -

14 + 3 + 3

gW ' 2gW gW ' 2gW 0(0 v)

v2gW -gW ' 2gW -gW '

g B g B

g B g B

µ µ µ µ µ µ

µ µ µ µ µ µ

! "! "+ + ! "# $# $% # $# $# $+ + & '& '& '

The Higgs MechanismThe Higgs Mechanism

( )1e e e ea a a a a a a a

i i i i

gW W

! " ! " ! " ! "µ µ µ

# #$ + %

a a! "= #

Q: What happens with missing d.o.f. (massless goldstone bosons P,H+ or ξ ) ?

A: They become longitudinal d.o.f. of the gauge bosons Wµi, i=1,2,3

Gauge transformation

Longitudinal components

Higgs field kinetic term22

'

2 2

g gD H H W H B Hµµµ µ= ! " "

22 2 3 21v v ( ' )

2 4

gW W gW g Bµ µ µ µ

+ !" + ! +

1 2

3

3

2

Z sin cos

cos sin

W W

W W

W WW

B W

B W

µ µ

µ

µ µ µ

µ µ µ

! !

" ! !

± =

= # +

= +

m

2 2 212

2 2 2 212

v

( +g' )v

W

Z

M g

M g

=

=

tan /

0

Wg g

M !

" #=

=

The Higgs Boson and The Higgs Boson and FermionFermionMassesMasses

2 † † 2

2( )V m H H H H

!= " +0

v2

H h

! "# $=# $+# $% &

4

2 2 3 4v vv

2 82V h h h

! ! !!= " + + +

( )v, ( )v, ( )vu u d d l l

i i iM Diag y M Diag y M Diag y!" !" !"= = =

2 2 vhm m != =

2 2v /m !=

E D U

YukawaL y L E H y Q D H y Q U H!!" " !" " !" "! != + +

α, β =1,2,3 - generation indexDirac fermion mass

( )vN N

iy L N H M Diag y

!""# # "#$ = Dirac neutrino mass

The Running CouplingsThe Running Couplings

( ) ( / ) ( )Bare R

Z! µ ! µ" = "

2 2( / ) 1 ( / ) ...Z b Logµ ! µ" = # " +

2 2 2 2 2 2( / ) ( / ) ( / )Log p Log Log p! ! µ ! µ" # " =

Radiative Corrections ~α (log Λ2/p2 +fin.part) UV divergence

Renormalization operation

UV cutoff Renormalization scale

Renormalization constant

Subtraction of UV div

Finite( )

R! µ 2 2

2 2( ), ( ) ( / )

d dLog Z

d d

!µ " ! " ! µ µ

µ µ= = # $

Running coupling

Renormalization GroupRenormalization Group2 2 2 2 2( , ( )) (1 ( ) ( ))PTR Q b Log Q Oµ ! µ ! ! µ != + +

2 2 2

2 2 2( ) 0

d dR R

d d

!µ µ µ

µ µ µ !

" "= + =

" "

2 2 2 2

2

2

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

( )

RG PTR Q R Q

dQdQ

µ ! µ ! µ !

!" !

=

=

Observable

RG Eq.

Solution to RG eq.

Effective coupling

( )! "

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

2 2

2 2(1, ) , (1 ( ) ...)

1 ( )PTR b Log Q

b Log Q

!! ! ! ! ! µ

! µ= = " + +

#

Asymptotic Freedom andAsymptotic Freedom andInfrared SlaveryInfrared Slavery

2( ) b! " "=

One-loop order 4/3 nf QED

-11+2/3 nf QCD

2 2

1 ( )b Log Q

!!

! µ=

"

QED QCD

α_

α_

UV Pole IR Pole

Comparison with ExperimentComparison with ExperimentGlobal Fit to Data Higgs Mass Constraint

Remarkable agreement of ALL the datawith the SM predictions - precision testsof radiative corrections and the SM

Though the values of sin ϑw extractedfrom different experiments are in goodagreement, two most precisemeasurements from hadron and leptonasymmetries disagree by 3σ

•• Inconsistency at high energies due to Landau polesInconsistency 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•• FlavourFlavour 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 BeyondThe SM and BeyondThe 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, EDGUT, SUSY, String, ED

•• NEW fields with NEW NEW fields with NEW interactions interactions

Compositeness, Compositeness, TechnicolourTechnicolour,, preonspreons

We like elegant solutionsWe like elegant solutions