problems:

three unrelated coupling constants?

quantization of electric charges?

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

subgroup

of a

semisimple group

Leptons as a fourth color

ddde

uuue

ddde

uuue

numberleptonL

numberbaryonB

LBTTQ RLEM

:

:

)(2

133

Grand Unification

SU(3) x SU(2) x U(1)subgroup of a

simple group

rank U(1) = 1 rank SU(n)=n-1

rank SU(2)=1 rank SU(3)=2

rank SU(3) x SU(2) x U(1) =4

rank SU(5) = 4

SU(3) x SU(2) x U(1) < SU(5)

fermions of one family in

two different representations:

)5()10(

)2,1()1,3(5

)2,1()1,3(5

)1,1()2,3()1,3(10

e

d

d

d

b

g

r

5

0

0

0

0

0

)10(

e

du

duu

duuu

bb

ggr

rrgb

20

)1,1()2,1()1,3(2)2,3(105

umdm

tmbm

em

cm

u

d

m

wM

HM2m

1m

sun

atm

choozl

2

3

e e

5SUrd

gd

bd

guburu

bd

gd

rd

e

ru

gu

bu

)1,1()2,1()1,3(2)2,3(105

Gauge bosons in SU(5) > SU(3) x SU(2) x U(1):

adjoint representation ~ (24)

5 x 5* = 24 + 1

24 = (8,1) + (1,3) + (1,1) + (3,2) + (3*,2)

Interesting features of SU(5):

ed QQ

trQ

3

1

0

Grand unification takes placeabove a very high unified mass scale M. There is only one unified coupling constant g. The coupling constants for QCD and for the SU(2) x U(1) theory are the same, apart from Clebsch-Gordan coefficients.

The different values of the three coupling constants at low energies are due to renormalization

effects.

30

)1()3(

)1()2()3(

)5(

USU

USUSU

SU

31

s

s

GUT

12

)())((

)()(4

222

1

22

HTrHTr

HTrmHV

The convergence of the three coupling constants at M implies a condition at low energies:

tan2

1 g

g

2

232sin

trQ

trT

This formula is independent of the details of the grand unified theory, e.g the gauge group, but only depends on the spectrum of the fermions, described in the Standard Model.

2

232sin

trQ

trT

8

3

13)3/1()3/2(

)13()2/1(2sin 222

22

2

232sin

trQ

trT

)/log(46

11

)(

cos

5

3

)(

4

)/log()224(6

11

)(

sin

)(

4

)/log()334(6

11

)(

1

)(

4

:

2

21

2

22

23

MFg

MFg

MFg

ationrenormaliz

GUT

GUT

GUTs

)1sin6(10

3 2

s

)/log(116

11sin 2

M

s

M

FGUT

log223

32

6

11

3

81

3

8GUT

No convergence of coupling constants

067.0

128/1

2315.0sin

)1sin6(10

3

2

2

s

W

s

convergence in case of

supersymmetry

Each particle of the Standard Model has a supersymmetric partner:

quark squark ( spin 0)lepton slepton ( spin 0)photon photino ( spin ½)gluon gluino ( spin ½)W – boson wino ( spin ½)

Grand unification

n complex numbers can be described by

(2n) real numbers.

SU(3) x SU(2) x U(1) < SU(4) x SU(2,L) x SU(2,R)

~ SO(6) x SO(4)

=> SO(10)

51

1det1

1010

:)10(

OOO

Omatricesxreal

SO

T

Representations of SO(10):16-dimensional spinor representation

( => Pauli spinor ~ SO(3) )

SO(10)

ee uuuuuu ::: edddddde :::

RR

SO(10):Due to two intermediate energy scales there is no problem with the convergence of the coupling constants at a grand unified mass scale.

RLLRD

LRM

R

L

RRL D

D

0

,

R

R

m

Dm

2

21 /

R

)126()120()10(1616 FF

)126()120()10(1616 FF

Grand Unified Theories

beyond

SU(5) - SO(10)

RLC SUSUSUE )3()3()3()6(

)3,3,1()3,1,3()1,3,3()27(

RL SUSU )3()3(

Fermions and bosons are both in a 248 representation

supersymmetry ?

2700038751))248()248(( S

             

kgGeV 819 102.21022.1

Energy scale of grand unification only 4 orders of magnitude smaller than Planck mass

point string

11 dimensions of space-time

7 space dimensions curled up:

Experiments in Mozumi Mine near Kamioka south of Toyama in Japan ( zinc, silver, )

ddde

uuue

ep

or

p

4

:

3

e

d

d

d

b

g

r

5

52

4

)(pGUT

GUT

m

Mp

proton lifetime:

ep 0

ep 0

Proton decay in SO(10):

like in SU(5), but proton lifetimedepends on unification energy, which is larger

yearsp _1034

Problem:

If proton has a lifetime largerthan 10^34 years, the decaycannot be detected – neutrino background from cosmicrays

1,00,1

)(

:..

LBLB

udd

ge

Proton decay in SO(10):

like in SU(5), but proton lifetimedepends on unification energy, which is larger

yearsp _1034

L

quarks > antiquarks 10 billions +1 10 billions

1 second later

1 minute after bang He: 24%