Grand Unified Theory, Running
Coupling Constants and
the Story of our Universe
These next theories are in a less rigorous state and we shall talk about
them, keeping in mind that they are at the ‘”edge” of what is understood
today. Nevertheless, they represent a qualitative view of our universe,
from the perspective of particle physics and cosmology.
GUT -- Grand Unified Theories – symmetry between quarks and leptons; decay
of the proton.
Running coupling constants: it’s possible that at one time in the development
of the universe all the forces had the same strength
The Early Universe: a big bang, cooling and expanding, phase transitions
and broken symmetries
We have incorporated into the Lagrangian density invariance under
rotations in U(1)XSU(2)flavor space and SU(3)color space, but these were
not really unified. That is, the gauge bosons, (photon, W, and Z, and
gluons) were not manifestations of the same force field. If one were
to “unify” these fields, how might it occur? The attempts to do so are
called Grand Unified Theories.
Grand Unified Theory (GUT)
GUT includes invariance under U(1) X SU(2)flavor space and SU(3)color
and invariance under the following transformations:
quarks
leptons
 leptons
 quarks
Grand Unified Theory - SU(5)
SU(5)
Georgii & Glashow, Phys. Rev Lett. 32, 438 (1974).
d red
dgreen
d blue
e-
rgb
L
-
SU(5)
gau
Quarks
& leptons
in same
multiplet
;
mx  1015GeV
8
gluons
24
Gauge
bosons
(W 0+B)
W-
Left handed
Gauge invariance
For symmetry under SU(5), the
L SU(5)
W+
(W 0 +B)
is invariant under
e-i(x,y,t)
x and y particles must be massless!
SU(5) generators and covariant derivative
The
52 -1 = 24 generators of SU(5) are the
i
do not commute. SU(5) is a non-abelian local gauge theory.
24 components: i(x,y,t) =
5x5 matrices which
i(x,y,t) has
all real, continuous functions
D = - i g5/2 jXi
where
j=1,24
Xi = the 24 gauge bosons
This includes the Standard Model covariant derivative (couplings are different).

Predictions:
a) qup = 2/3 ; qd = -1/3
b) sin2W  0.23
c) the proton decays!
> 1034 years
d) baryon number not conserved
e) only one coupling constant, g5 (g1, g2, and g3, are related)
So far, there is no evidence that the proton decays. But note that the
lifetime of the universe is 14 billion years. The probability of detecting
a decaying proton depends a large sample of protons!
“Particle Physics and Cosmology”,
P.D. B. Collins, A. D. Martin and E. J. Squires,
Wiley, NY, page 169
The term

j =1,2,…,24
jXi /2
can be written:
same as SU(3)color
_
|
|
|
|_
B
.
same as SU(2)flavor
_
|
|
|
_|
24
this matrix
X comes in 3 color states with |Q|
= 4/3
y comes in 3 color states with |Q|
= 1/3
g5
The GUT SU(5) Lagrangian density (1st generation only)
Standard Model terms
int.
SU(5)
g5
-
X
+
 = 1,2,3
quark to lepton, no color change
3-color
vertex
Q = - 4/3
Y-
 = 1,2,3
quark to lepton, no color change
3-color
vertex
Q= - 1/3
+ Hermitian Conjugate (contains X+ and Y+ terms)
Charge conjugation
operator
T  transpose
Note: one coupling constant,
g5

proton, SU(5)
 1031 years
--aagreat
greatfailure
failureof
forSU(5)!
SU(5)
charge
X-4/3red
e+
dred
e+
Decay of proton in SU(5)
d red
d red
-
u green
d red
u blue
proton
-
green
Xred
X+ red
3-color
vertex
anti-up
0
blue
e+
X +red
green
blue
SUPER SYMMETRIC (SUSY) THEORIES:
SUSYs contain invariance of the Lagrangian density under operations which change
bosons (spin = 01,2,..)
fermions (spin = ½, 3/2 …).
SUSY  unifies E&M, weak, strong (SU(3) and gravity fields.
usually includes invariance under local transformations
http://www.pha.jhu.edu/~gbruhn/IntroSUSY.html
Supergravity
Supersymmetric String Theories
Elementary particles are one-dimensional strings:
open strings
closed strings
or
.no free
parameters
L = 2r
L = 10-33 cm. = Planck Length
Mplanck  1019 GeV/c2
See Schwarz, Physics Today, November 1987, p. 33
“Superstrings”
The Planck Mass is approximately that mass whose gravitational potential is the
same strength as the strong QCD force at r  10-15 cm.
An alternate definition is the mass of the Planck Particle, a hypothetical miniscule
black hole whose Schwarzchild radius is equal to the Planck Length.
A quick way to estimate the Planck mass is as follows:
gstrong ℏc/r = GMpMp/r
where r = 10-15cm (strong force range) and gstrong = 1
Mp = [gstrong ℏc/G]1/2
=
MPlanck 
1.3 x 1019 mproton
1019 GeV/c2
Particle Physics and the Development of the Universe
Very early universe
All ideas concerning the very early universe are speculative. As of early
today, no accelerator experiments probe energies of sufficient
magnitude to provide any experimental insight into the behavior of
matter at the energy levels that prevailed during this period.
Planck epoch
Up to 10 – 43 seconds after the Big Bang
At the energy levels that prevailed during the Planck epoch the four
fundamental forces— electromagnetism U(1) , gravitation, weak
SU(2), and the strong SU(3) color — are assumed to all have the
same strength, and “unified” in one fundamental force.
Little is known about this epoch. Theories of supergravity/
supersymmetry, such as string theory, are candidates for describing
this era.
Grand unification epoch: GUT
Between 10–43 seconds and 10–36 seconds after the Big Bang
The universe expands and cools from the Planck epoch. After about 10–43
seconds the gravitational interactions are no longer unified with the
electromagnetic U(1) , weak SU(2), and the strong SU(3) color interactions.
Supersymmetry/Supergravity symmetires are roken.
After 10–43 seconds the universe enters the Grand Unified Theory (GUT)
epoch. A candidate for GUT is SU(5) symmetry. In this realm the proton can
decay, quarks are changed into leptons and all the gauge particles (X,Y, W, Z,
gluons and photons), quarks and leptons are massless. The strong, weak and
electromagnetic fields are unified.
Running Coupling Constants
Electro weak
unification
ElectroWeak
Symmetry
breaking
Planck
region
Supersymmetry
SU(3)
GUT
electroweak
GeV
Inflation and Spontaneous Symmetry Breaking.
At about 10–36 seconds and an average thermal energy kT  1015
GeV, a phase transition is believed to have taken place.
In this phase transition, the vacuum state undergoes
spontaneous symmetry breaking.
Spontaneous symmetry breaking:
Consider a system in which all the spins can be up, or all can be
down – with each configuration having the same energy. There
is perfect symmetry between the two states and one could, in
theory, transform the system from one state to the other
without altering the energy. But, when the system actually
selects a configuration where all the spins are up, the symmetry
is “spontaneously” broken.
Higgs Mechanism
When the phase transition takes place the vacuum state transforms
into a Higgs particle (with mass) and so-called Goldstone bosons
with no mass. The Goldstone bosons “give up” their mass to the
gauge particles (X and Y gain masses 1015 GeV). The Higgs keeps
its mass ( the thermal energy of the universe, kT 1015 GeV). This
Higgs particle has too large a mass to be seen in accelerators.
What causes the inflation?
The universe “falls into” a low energy state, oscillates about the minimum
(giving rise to the masses) and then expands rapidly.
When the phase transition takes place, latent heat (energy) is released.
The X and Y decay into ordinary particles, giving off energy.
It is this rapid expansion that results in the inflation and gives rise to the
“flat” and homogeneous universe we observe today. The expansion is
exponential in time.
Schematic of Inflation
1019
R(t) m
Rt2/3
T (GeV/k)
Rt1/2
T t-1/2
R eHt
1014
T t-1/2
Rt1/2
Tt-2/3
T=2.7K
10-13
10-43
10-34
10-31
time (sec)
10
Electroweak epoch
Between 10–36 seconds and 10–12 seconds after the Big Bang
The SU(3) color force is no longer unified with the U(1)x SU(2) weak force. The
only surviving symmetries are: SU(3) separately, and U(1)X SU(2). The W and Z
are massless.
A second phase transition takes place at about 10–12 seconds at kT = 100 GeV. In
this phase transition, a second Higgs particle is generated with mass close to 100
GeV; the Goldstone bosons give up their mass to the W, Z and the particles
(quarks and leptons).
It is the search for this second Higgs particle that is taking place in the particle
accelerators at the present time.
After the Big Bang: the first 10-6 Seconds
Planck Era
SUSY
Supergravity
inflation
gravity
.X,Y take
decouples on mass
GUT
W , Z0
take on
mass
SU(2) x U(1) symmetry
.
all forces unified
bosons  fermions
quarks  leptons
all particles massless
.
.
W , Z0
take on
mass
COBE data
.
2.7K
Standard Model
100Gev
.
.
.
.
.
only gluons and photons are massless
n, p formed nuclei formed
atoms formed
Field theoretic treatment of the Higgs mechanism
One can incorporate the Higgs mechanism into the Lagrangian density
by including scalar fields for the vacuum state. When the scalar fields
undergo a gauge transformation, they generate the particle masses.
The Lagrangian density is then no longer gauge invariant. The
symmetry is broken.