Grand Unified Theory RunningGrand Unified Theory, Running Coupling Constants and h S f U ithe Story of our Universe

These next theories are in a less rigorous state and we shall talk aboutThese 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:  at one time in the development of the universeall the forces had the same strength

Story of our universe:  a big bang, cooling and expanding, phase transitionsand 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 

ll ifi d Th i h b ( h W d Z dnot 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)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       leptonsl kleptons      quarks 

Grand Unified Theory ‐ SU(5)

SU(5)mx 1015GeVQuarks

& leptons

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

8gluons

24Gaugebosons

in samemultipletd red

dgreen

d blL ;rgb

(W 0+B)

(W 0 +B)

W+

W‐

bosons

Left handed

d blueLe‐

‐

SU(5)     Gauge invariance  L SU(5) is  invariant undergau

e‐i(x,y,t)SU(5)

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

SU(5) generators and covariant derivative

Th 52 1 24 t f SU(5) th 5 5 t i hi hThe      52 ‐1   =   24   generators  of SU(5) are the   i24 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.    

D = ‐ i g5/2j 1 24jXi whereXi = the 24 gauge bosons

p i( ,y, )

D   i g5/2j=1,24jX where X  the 24 gauge bosons    

a)    qup =  2/3 ;    qd = ‐1/3    Predictions:

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

>  1034 yearsb)   sin2W ‐.23c)  the proton decays!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 thelifetime of the universe is 14 billion years.  The probability of detectinga decaying proton depends a large sample of protons!

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

Xi /2

_|

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

_|

same as SU(3)color

B

|||_|

||||_

this matrix

. same as SU(2)flavor24

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

g5

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

g5

Standard Model terms

i tg5int.

SU(5)

+quark to lepton, no color change 3‐color

vertex = 1,2,3 Q = ‐ 4/3

X‐

, , /

Y ‐

quark to lepton, no color change3‐colorvertex = 1,2,3 Q= ‐ 1/3

( d )+ Hermitian Conjugate (contains X+ and Y+ terms)

Note: one coupling constant, g5T transposeCharge  conjugation 

operator

T  transpose

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

X‐4/3red

dred

charge

e+ e+

Decay of proton in SU(5)Decay of proton in SU(5)   

d red

u

d red

d‐

anti‐up

green0

u green

u blue X+ red

e+

d redXred‐

g

blue

3‐color e

blue

greenX +redprotonvertex

SUPER SYMMETRIC (SUSY) THEORIES:SUSY t i i i f th L i d it d ti hi h hSUSYs  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

SupergravitySupergravity

Supersymmetric String Theories

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

.no free

L = 2r

or parameters

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

See Schwarz, Physics Today, November 1987, p. 33“S t i ”“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 minisculeblack hole whose Schwarzchild radius is equal to the Planck Length.   

A quick way to estimate the Planck mass is as follows: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

V l iVery early universeAll ideas concerning the very early universe are speculative. As of earlytoday, no accelerator experiments probe energies of sufficientmagnitude to provide any experimental insight into the behavior ofg p y p gmatter at the energy levels that prevailed during this period.

Planck epochPlanck epochUp to 10 – 43 seconds after the Big Bang

At the energy levels that prevailed during the Planck epoch the fourAt the energy levels that prevailed during the Planck epoch the fourfundamental forces— electromagnetism U(1) , gravitation, weakSU(2), and the strong SU(3) color — are assumed to all have thesame strength, and “unified” in one fundamental force.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 describingthis era.

Grand unification epoch: GUT

Between 10–43 seconds and 10–36 seconds after the Big Bangg g

The universe expands and cools from the Planck epoch. After about 10–43seconds the gravitational interactions are no longer unified with theg gelectromagnetic 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 candecay, 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 andelectromagnetic fields are unified.

Running Coupling Constants

Planckregion

Electro weak unification

Electro‐WeakSymmetry SU(3)

Super‐symmetry

GUT

Symmetrybreaking

SU(3)

electroweak

GeV

Inflation and Spontaneous  Symmetry Breaking. 

At about 10–36 seconds and an average thermal energy kT 1015GeV, a phase transition is believed to have taken place.

In this phase transition, the vacuum state undergoesspontaneous symmetry breaking.

Spontaneous symmetry breaking:Consider a system in which all the spins can be up, or all can bedown with each configuration having the same energy Theredown – with each configuration having the same energy. Thereis perfect symmetry between the two states and one could, intheory, transform the system from one state to the otherwithout altering the energy But when the system actuallywithout altering the energy. But, when the system actuallyselects a configuration where all the spins are up, the symmetryis “spontaneously” broken.

Higgs Mechanism

When the phase transition takes place the vacuum state transformsinto a Higgs particle (with mass) and so‐called Goldstone bosonswith no mass. The Goldstone bosons “give up” their mass to thegauge particles (X and Y gain masses 1015 GeV) The Higgs keepsgauge particles (X and Y gain masses 10 GeV). The Higgs keepsits mass ( the thermal energy of the universe, kT 1015 GeV). ThisHiggs particle has too large a mass to be seen in accelerators.

What causes the inflation?

h “f ll ” l ll b hThe 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.p p , ( gy)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“fl t” d h i b t d Th i i“flat” and homogeneous universe we observe today. The expansion isexponential in time.

Schematic of Inflation 

T (GeV/k)R(t) m

1019

Rt1/2

Rt2/3

T t‐1/2

1014 R eHt

Rt1/2T t‐1/2

Tt‐2/3

T=2.7K

10‐43 10‐34 10‐31 10

T t

10‐13

time (sec)

Electroweak epoch 

B t 10 36 d d 10 12 d ft th Bi BBetween 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. Theonly surviving symmetries are: SU(3) separately, and U(1)X SU(2). The W and Z

lare massless.

A second phase transition takes place at about 10–12 seconds at kT = 100 GeV. Inthis phase transition, a second Higgs particle is generated with mass close to 100p gg p gGeV; 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 particleIt is the search for this second Higgs particle that is taking place in the particleaccelerators at the present time.

After the Big Bang:  the first 10‐6 Seconds

inflation W , Z0 

.Planck Era gravitydecouples

SUSY

inflationX,Y take on mass

take  on mass

GUT SU(2) x U(1) symmetrySupergravity

.

.all forces unifiedbosons  fermions quarks  leptons

all particles massless

. W , Z0 take  on 

.COBE data

2.7KStandard Model

mass

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 densityby including scalar fields for the vacuum state When the scalar fieldsby 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. Thesymmetry is broken.