Grand Unified Theory, Running Coupling Constants and

the Story of our UniverseThese next theories are in a less rigorous state and we shall talk aboutthem, keeping in mind that they are at the ‘”edge” of what is understoodtoday. 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)

8gluons

(W 0+B)

(W 0 +B)

W+

W-

24Gaugebosons

SU(5)mx 1015GeV

SU(5) gau

For symmetry under SU(5), the x and y particles must be massless!

Quarks& leptonsin same multiplet

Left handed

d red

dgreen

d blueL

Georgii & Glashow, Phys. Rev Lett. 32, 438 (1974).

;

Gauge invariance

e-i(x,y,t)

L SU(5) is invariant under

e-

-

rgb

> 1034 years

So far, there is no evidence that the proton decays. But note that thelifetime of the universe is 14 billion years. The probability of detectinga decaying proton depends a large sample of protons!

D = - i g5/2j=1,24jXi where Xi = the 24 gauge bosons

SU(5) generators and covariant derivative

a) qup = 2/3 ; qd = -1/3

b) sin2W 0.23 c) the proton decays! d) baryon number not conserved

e) only one coupling constant, g5 (g1, g2, and g3, are related)

Predictions:

The 52 -1 = 24 generators of SU(5) are the

i

24 components: i(x,y,t) =

i(x,y,t) has all real, continuous functions

5x5 matrices which

do not commute. SU(5) is a non-abelian local gauge theory.

This includes the Standard Model covariant derivative (couplings are different).

“Particle Physics and Cosmology”,P.D. B. Collins, A. D. Martin and E. J. Squires,Wiley, NY, page 169

X comes in 3 color states with |Q| = 4/3 y comes in 3 color states with |Q| = 1/3

g5

B

this matrix

_ | | |_|

The term j =1,2,…,24jXi /2 can be written:

_||||_

.

same as SU(3)color

same as SU(2)flavor24

The GUT SU(5) Lagrangian density (1st generation only)

g5

Standard Model terms

int.SU(5)

+

Note: one coupling constant, g5

quark to lepton, no color change

quark to lepton, no color change

3-colorvertex

3-colorvertex = 1,2,3

= 1,2,3 Q = - 4/3

Q= - 1/3

+ Hermitian Conjugate (contains X+ and Y

+ terms)

X

Y

Charge conjugation

operator T transpose

-

-

a great failure of SU(5)!proton, SU(5) 1031 years -- a great failure for SU(5)

e+ e+

X-4/3red

dred

charge

d red

u green

u blue X+ red

e+

d red

d red-

Xred- anti-up

blue

green

blue

greenX +redproton

0

Decay of proton in SU(5)

3-color vertex

SUPER SYMMETRIC (SUSY) THEORIES:

http://www.pha.jhu.edu/~gbruhn/IntroSUSY.html

Supergravity

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

Supersymmetric String Theories

Elementary particles are one-dimensional strings: closed strings open strings

L = 2r

L = 10-33 cm. = Planck Length Mplanck 1019 GeV/c2

or

See Schwarz, Physics Today, November 1987, p. 33 “Superstrings”

.no freeparameters

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 minisculeblack 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

= 1.3 x 1019 mproton

MPlanck 1019 GeV/c2

Particle Physics and the Development of the Universe

Very early universeAll 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 epochUp 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

GUT

electroweak

Electro-WeakSymmetrybreaking

Planckregion

SU(3)

Electro weak unification

GeV

Super-symmetry

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.

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.

Higgs Mechanism

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

T (GeV/k)R(t) m

T=2.7K

10-43 10-34 10-31

1019

10

1014

Rt1/2

Rt1/2

T t-1/2

T t-1/2

R eHt

Rt2/3

Tt-2/3

10-13

time (sec)

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

.

GUT SU(2) x U(1) symmetry

Planck Era

gravitydecouples

SUSYSupergravity

inflationX,Y take on mass

W , Z0

take on mass

.

.

all forces unifiedbosons fermions quarks leptons

all particles massless

.

.

.

. .

COBE data

2.7K Standard Model

W , Z0

take on mass

.

.

n, p formed nuclei formedatoms formed

100Gev

only gluons and photons are massless

Field theoretic treatment of the Higgs mechanism

One can incorporate the Higgs mechanism into the Lagrangian densityby including scalar fields for the vacuum state. When the scalar fieldsundergo a gauge transformation, they generate the particle masses. The Lagrangian density is then no longer gauge invariant. The symmetry is broken.