Grand Unified Theory, Running Coupling Constants and the Story of our UUiniverse
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: at one time in the development of the universe all the forces had the same strength
Story of our 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 unifie d. That is, the gauge bosons, (h(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). 15 Quarks mx 10 GeV & leptons in same 8 d red multiplet rgb gluons d 24 green Gauge ; bosons L d blue 0 + e‐ (W +B) W
‐ 0 ‐ Left handed W (W +B)
SU(5) Gauge invariance L SU(5) is invariant under gau e‐i(x,y,t)
For symmetry under SU(5), the x and y particles must be massless! SU(5) generators and covariant derivative
2 The 5 ‐1 = 24 generators of SU(5) are the i 555x5 matitrices whic h do not commute. SU(5) is a non‐abelian local gauge theory. i(x,y,t) has
24 components: i(x,y,t) = all real, continuous functions
/2 i where i = the 24 gauge bosons D = ‐ i g5 j124j=1,24jX X This includes the Standard Model covariant derivative (couplings are different).
Predictions: a) qup = 2/3 ; qd = ‐1/3 2 b) sin W ‐.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”, PDP.D. B. Collins, A. D. Martin and E. J. Squires, Wiley, NY, page 169 i The term j =1,2,…,24jX /2 can be written: same as SU(3)color _ _ | | | | B | | |_ _|
24 . same as SU(2)flavor
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 g itint. 5 SU(5) ‐ X + quark to lepton, no color change 3‐color = 1,2,3 vertex Q = ‐ 4/3
‐ Y
3‐color = 1,2,3 quark to lepton, no color change vertex Q= ‐ 1/3
+ + + Hermitian Conjugate (contains X and Y terms)
Note: one coupling constant, g5 Charge conjugation T transpose operator 31 a great failure of SU(5)! proton, SU(5) 10 years ‐‐ a great failure for SU(5) charge
‐4/3 dred X red e+ e+ Decay of proton in SU(5)
d d red red 0 anti‐up ‐ green u green d red ‐red u X blue X+ red blue 3‐color + vertex e proton X +red green blue SUPER SYMMETRIC (SUSY) THEORIES: SUSYs conttiain iiinvariance of the LiLagrangian ditdensity under operations whic h 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 .no free or parameters
L = 2r
‐33 19 2 L = 10 cm. = Planck Length Mplanck 10 GeV/c
See Schwarz, Physics Today, November 1987, p. 33 “Supers tr ings ”
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
‐15 where r = 10 cm (strong force range) and gstrong = 1
1/2 Mp = [gstrong ℏc/G]
19 = 1.3 x 10 mproton
19 2 MPlanck 10 GeV/c 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 –43seconds 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 —areassumed 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 Biggg 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 Planck region
Electro‐ Super‐ Weak symmetry Symmetry SU(3) breaking 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 “fa llsinto” a low energy state, oscillates about the minimum (giving rise to the masses) and then expands rapidly.
When the phase transition takes pp,lace, latent heat ((gy)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 “fla t” and homogeneous universeweobserve tdtoday. Theexpansion is exponential in time. Schematic of Inflation
2/3 R(t) m 1019 T (GeV/k) Rt
Rt1/2 T t‐1/2
1014 R eHt
T t‐1/2 1/2 Rt Tt‐2/3 10‐13 T=2.7K 10‐43 10‐34 10‐31 time (sec) 10 Electroweak epoch
BtBetween 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
0 inflation W , Z Planck Era gravity X,Y. take take on decouples on mass mass SUSY Supergravity GUT SU(2) x U(1) symmetry
.
all forces unified bosons fermions quarks leptons . all particles massless W , Z0 . take on mass
COBE data . Standard Model 2.7K
100Gev . . . only gluons and photons are massless
. atoms formed n, p formed nuclei 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.