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) 8 gluons (W 0 +B) W+W+ W-W- 24 Gauge bosons SU(5) m x  GeV SU(5) gau For symmetry under SU(5), the x and y particles must be massless! Quarks & leptons in same multiplet Left handed d red d green d blue L Georgii & Glashow, Phys. Rev Lett. 32, 438 (1974). ; Gauge invariance e -i  (x,y,t)  L SU(5) is invariant under e - -  rgbrgb

> years 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! D  =   - i g 5 /2  j=1,24 j X i where X i = the 24 gauge bosons SU(5) generators and covariant derivative a) q up = 2/3 ; q d = -1/3 b) sin 2  W  0.23 c) the proton decays! d) baryon number not conserved e) only one coupling constant, g 5 (g 1, g 2, and g 3, are related) Predictions: The = 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 g5g5 BB this matrix _ | _| The term  j =1,2,…,24 j X i /2 can be written: _ | |_. same as SU(3) color same as SU(2) flavor 24

The GUT SU(5) Lagrangian density (1 st generation only) g5g5 Standard Model terms int. SU(5)

+ Note: one coupling constant, g 5 quark to lepton, no color change 3-color vertex 3-color vertex  = 1,2,3 Q = - 4/3 Q= - 1/3 + Hermitian Conjugate (contains X  + and Y  + terms) XX Y Y  Charge conjugation operator T  transpose - -

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

e+e+ e+e+ X -4/3 red d red charge

d red u green u blue X + red e+e+ d red - X red - anti-up blue green blue green X +red proton 0 Decay of proton in SU(5) 3-color vertex

SUPER SYMMETRIC (SUSY) THEORIES: 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 = 2  r L = cm. = Planck Length M planck  GeV/c 2 or See Schwarz, Physics Today, November 1987, p. 33 “Superstrings”.no free parameters The Planck Mass is approximately that mass whose gravitational potential is the same strength as the strong QCD force at r  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: g strong ℏ c /r = GM p M p /r where r = cm (strong force range) and g strong = 1 M p = [g strong ℏ c/G] 1/2 = 1.3 x m proton M Planck  GeV/c 2

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 T he 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- Weak Symmetry breaking Planck region SU(3) Electro weak unification GeV Super- symmetry

Inflation and Spontaneous Symmetry Breaking. At about 10 –36 seconds and an average thermal energy kT  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  GeV). The Higgs keeps its mass (  the thermal energy of the universe, kT  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 R  t 1/2 T  t -1/2 R  e Ht R  t 2/3 T  t -2/ 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 Seconds. GUTSU(2) x U(1) symmetry Planck Era gravity decouples SUSY Supergravity inflation X,Y take on mass W , Z 0 take on mass.. all forces unified bosons  fermions quarks  leptons all particles massless

..... COBE data 2.7K Standard Model W , Z 0 take on mass.. n, p formed nuclei formed atoms formed 100Gev only gluons and photons are massless

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.