grand unification
Post on 13-Jan-2016
91 Views
Preview:
DESCRIPTION
TRANSCRIPT
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%
top related