Speculations on extensions of symmetry and interctions to GUT energies Lecture 16 1 Introduction The use of symmetry, as has previously shown, provides insight to extensions of present physics into physics beyond our present emergy scales. These ideas now begin to influence our thoughts on cosmology and the evloution of our universe. This lecture will outline some of the proposed ways that new physics can be introduced, maintaining present physical laws and symmetries, but extending them to higher energies and interactions. Most, if not all, of these ideas will not survive, at least in their present form. 2 Supersymmetry Supersymmetry (SUSY) is one of many attempts to extend the standard model to address features must be put present theories by hand. For example; the mass hierarchy, certain conservation laws, quantization of charge, etc. A SUSY in which a new set of particles exist at the TeV scale (10 12 ev) can solve the hiearchy problem. The hiearchy problem deals not only with the mass values but also the mass scales of the elementary particles, in particular the large differences in masses between the flavors, and families. It also has features that can be used to unify the strong interaction and electroweak interaction. Finally, SUSY is a natural component of most versions of string theory. String theory adds the possibility of quantizing gravity. 1
One of the more interesting predictions of SUSY is a stable, heavy ( 100 GeV) particle that only interacts weakly with matter. Such a particle (WIMP) is a natural candidate to explain Dark Matter, which is an unexplained requirement of cosmology that we will discuss later. SUSY, in brief, introduces a partner particle to every elementary particle, but partners differ in their spin by 1/2 unit. Thus every fermion has a boson partner and every boson has a fermion partner. There are pairs of related Hamiltonians, where for every eigenstate of one Hamiltonian there is a corresponding partner eigenstate, which without symmetry breakinng, has the same energy. However, SUSY must be broken, although in consideration of our previous discussion of spontaneous symmetry breaking, this can be satisfactorily accomplished. 3 The Vacuum state Vacuum energy is associated with the energy that permeates space. It can be visualized as that sea of particle-antiparticle pairs proposed by Dirac. The vacuum is a complex quantum state that is the lowest energy level of the universe. For example, electromagnetic waves carry the field energy and are quantized as a field of oscillators. The energy of each oscillator is small but are infinite in number. This is after all the concept of renormalization. The vacuum is sometimes discussed in terms of the QED vacuum and the QCD vacuum. The true vacuum is a superposition of all vacuum states. 2
Vacuum energy can be observed in the fluctuations of virtual particles related to the uncertainty principle. Obviously one cannot localize a particle in space which has zero energy. Experimental manifestations of the QED vacuum arise in the Casimir effect, the Lamb Shift, and spontaneous emission. The Casmir effect measures the stress between conducting plates due to modifications on the electromagnetic fluctuations in the vacuum between the plates. The density of zero point electromagnetic fluctuations at the Plank scale 10 33 cm ( 10 43 hz) breaks into a foam of bubbles, corresponds to the situation in the universe at 10 43 s. 4 1/N c expansion Because QCD is non-perturbative at low energies, one cannot expand solutions for a process in powers of the coupling constant as one does in QED. However, a useful expansion can be obtained in powers of 1/N c, where N c is the number of color quantum numbers in QCD. Now we know that N c = 3, but at least some useful information and trends can be observed as we let N c with g 2 N c fixed. Here g is the QCD coupling constant. The calculation proceeds through a series of Feynman diagrams, keeping only the leading order terms. The result predicts observed behavior such as the OZI rule, and the absence of qqqq states. Solutions have the form of solitons, and the predicted spectra has some similarity with string models. 3
Forbidden Allowed Figure 1: The OZI rule 5 Θ Vacuum (QCD Vacuum) As a model of an oscillator consider the classical Hamiltionin for a pendulum. The geometry if shown in the figure. The Hamiltonian is; H = p2 φ 2 + ω 2 [1 cos(φ)] With ω 2 = g/l, and set M = L = 1. To quantize the system choose the angle φ and p φ as conjugate variables, and impose the boundry conditions that ψ(2π) = ψ(0). The specrum of energies is discrete. Choose to assume g 0 so that ω 0 and the solution becomes; ψ = 1 2π e ikφ E = k 2 /2 Here k is an integer so that ψ(2π) = ψ(0). There is no unique direction in space as φ = 0. 4
L φ M Figure 2: Geometry of the Pendulum example Now suppose ψ(φ + 2π) = e iθ ψ(φ). Here φ is a rotation angle and θ is a phase. Then; ψ(ψ) = e i(k+θ/2π)φ E k = (k + θ/2π)2 2 Therefore introduce a U(1) symmetry so that the symmetry operator is; Uψ(φ) = ψ(φ + 2π) This rotates the pendulum by one turn clockwise. The operator commutes with the Hamiltonian. The energy spectrum as a function of θ is shown in the figure. The energy is discontinuious at θ = π(2k + 1). When ω 0 the spectrum is more complicated but similar. Now one can set this up 5
E k = 1 k = 0 k = 1 2π π π 2 π θ Figure 3: Ground state energy of the vacuum to find a solution in a Lattice gauge calculation. Each path involves an integral over all possible trajectories with the boundry conditions of the problem. The most probable trajectories are found to be tunneling between the states. This forms a soloton-type solution. The QCD vacuum is then obtained from the expectation value ; θ e HT θ = n,m e i(n m)θ V n e HT V m which is equilivant to a Lagrangian density for the QCD vacuum of the form; L θ = L + θg2 32π 2 F µνf a a µν For a non-zero choice of, θ, the QCD vacuum violates parity and time reversal. One can introduce a combined QCD and electroweak global symmetry, but spontaneous symmetry breaking requires the existence of a boson, the axion, which has not been experimentally observed. The problem of why the choice of the θ parameter is essentially zero is called the strong CP problem. 6
6 The strong CP problem As has been mentioned in class, the simultaneous operation of charge conjugation and parity is violated in the weak interaction. Because CPT must be a good symmetry for any Lorentz invarient field theory, the violation of time symmetry is inferred. However experimentally, it appears that CP is a good symmetry for the strong interaction. QCD does not violate CP at the same level as the electroweak theory because the gauge fields in QCD couple to vector currents while in the electroweak theory they couple to chiral currents. However, there are terms in QCD which naturally violate CP and these must be suppressed because CP violation in the strong interaction is observed to be very small. The requirement that CP is suppressed by some mechanism in QCD is called the strong CP problem. See the note on the Θ vacuum above. Two of the possible solutions to this problem suggest; 1) that CP is conserved so that a new current is required which is mediated by a new particle, the axion, and 2) that space-time has extra dimensions beyond the 4-components of relativity so that Lorentz invarience is violated. 7 Quark-gluon plasma Normal nuclear densities are obtained from the empirical equation A = r 0 A 1/3, with A the atomic number of a nucleus (number of 7
nucleons) and r 0 the experimentally determined value of r 0 = 1.07 fm. Then the volume of the nucleus is; V = 4πR 3 /3 = 4πr 3 0A A/V = 3 4πr 3 0 = 1.95 10 38 nucleons/cm 3 Now note that in each momentum state one can place 4 nucleons, 2 protons and 2 neutrons, each with one spin up and one spin down, to satisfy Fermi statistics. Then the nuclear density is; A/V = 4 (2π) 3 k F 0 d 3 k F = 2k3 F 3π 2 Equating these values of A/V ; k F = [ 9π 8r0 3 ] 1/3 = 1.42 fm 1 A/V = 0.195 nucleons/fm 3 The above represents nuclear densities at essentially 0 temperature. The quark-gluon plasma (QGP) is a phase of QCD in which the quarks and gluons are presumed to be (almost) free. The plasma exists at high energies (temperatures) and/or high densities. Remember a property of QCD is asymptotic freedom. In the QGP quarks were supposed to interact weakly, and thus form a gaseous state. More recent experiments, indicate that the hadrons melt into a fluid of quarks and gluons which are deconfined from their hadrons, but remain confined within the extent of the plasma. Observations 8
Critical Point Figure 4: The thermodynamic diagram of QCD show that the mean free path of quarks and gluons in the QGP is comparable to the interparticle spacing, indicating a fluid-like behavior. In the QCP the color charge is screened similar to electric charge in a plasma of electrons and ions. Transition to this state of matter is studied experimentally by relativistic heavy ion collisions and theoretically by Lattice gauge calculations. Theory predicts that at a temperature of 175 MeV per particle the hadrons melt by a phase transition into the QGP phase. Note that the concept of temperature requires an equilibrum condition of the system. The plasma is 9
presumed to be created in the collision of relativistic, heavy atomic nuclei (i.e. Gold on Gold). A temperature of 175 Mev per particle corresponds to an energy density of 1 GeV/fm 3. Relativistic Heavy Ions E > 10 GeV/Nucleon Relativistic Heavy Ions At Rest Outward Flow Central Collision Relativistic Heavy Ions QGP Relativistic Heavy Ions Figure 5: A schematic of a relativistic heavy ion collision 10