Elementary Particles and the World of Planck Scale

Gauge Theory and the Standard Model

The modern theory of elementary particles depends heavily on the concept of gauge invariance, which was used to describe some changes that do not have any effect on observation. For example, if the electric voltage throughout the circuit is raised uniformly by the same amount, there would not be any observable effect. Such instantaneous global change seems to be somewhat un-natural as it takes a definite time to signal the change. Theory with local gauge invariance is more

Figure 15-06a Global and Local Transformations [view large image]

realistic, but much more stringent. The difference between a global and local transformation is illustrated pictorially in Figure 15-06a (just for visualization purpose - it's not the gauge variety). Since local gauge symmetry can change in a

different way at every point, the only way in which theory can be kept invariant under such general changes is for certain forces to constrain the allowed motions. That something turns out to be the gauge bosons mentioned earlier in Figure 15-03. For example, in the electromagnetic interaction a local disturbance can be considered as a two dimensional rotation of the quantum field (which is usually a complex function²) in an

"internal space". The photon (the gauge boson) is the response to restore the appearance (see Figure 15-06b), which signifies mathematically the invariance of the "Action" under this internal rotation. Theoretical physicists are fond of putting similar objects together called a group. For the case of electromagnetic interaction, there is only one kind of objects -- the two dimensional internal rotation. The different rotational displacements form a group, this particular group is called U(1). The symbol U indicates that the transformation (the internal rotation) is unitary, which preserves the normalization (probability). This U(1) group has the property that the internal rotation operations are commutative -- mathematicians call such kind of group an Abelian group. Similar gauge invariance exists for the strong and weak interactions, the "internal rotation" depends on more than one parameter in these cases. Group of objects can be formed from these generalized "rotational displacements". However, these elements are no longer commutative. Such groups are called non-Abelian. The gauge theory for the U(1) is called Quantum Electrodynamics (QED).

Figure 15-06b Local Gauge Field Invariance [view large image]

The non-Abelian group called SU(2) is applicable to the case of weak interaction. The internal rotation is generalized to three parameters corresponding to three different gauge bosons -- W⁺, W^-, and Z⁰. The participating particles are the left-handed³ pair of leptons. In the Weinberg-Salam model, the left-handed leptons can undergo both electromagnetic and weak interactions while the right-handed electron can only participate in electronmagnetic interaction. Thus the model unifies these two interactions. This asymmetry in chirality is related to the phenomenon of parity violation⁴ in weak interaction, and has been verified conclusively in the 1950s. The electroweak unification occurs at energy above 10² Gev. A complication arises with regards to the mass of gauge bosons for which, the original Yang-Mills theory⁴ failed to account for.

In superconductivity, it is known that the Meissner effect, where the magnetic field is expelled from the superconducting material, can be interpreted as the electromagnetic field (the photon) acquiring mass while trying to disrupt the ordered structure of the Copper pairs (see Figure 15-06c). But the gauge bosons in weak interaction have mass even in empty space, and no form of mass generating substance has been identified so far for such boson mass. We are therefore led to postualate that there is a new form of matter doing the job. Accordingly, what we perceive as empty space is in

Figure 15-06c Higgs Condensate
[view large image]

fact filled with an exotic, suitably aligned substrate: the Higgs condensate.

In the mathematical formalism, these Higgs fields are added to the "Action". At the transition temperature (early in the Big Bang),

these Higgs fields⁵ move to more stable states in lower energy level. However, unlike the electromagnetic field, which has its minimum energy at zero field strength; the Higgs field, in contrast, has its minimum energy at a nonzero field strength (see Figure 15-06d, in this diagram the "ball" represents the preferred state of the universe). Thus, the universe, in its natural lowest energy state, is permeated by that nonzero value of the Higgs field. Once this happens, all the particles (both bosons and fermions) would acquire mass by interacting with the Higgs field. There is now a convincing consensus of experimental results supporting this electroweak theory.

Figure 15-06d Higgs Field [view large image]

When the internal rotation is generalized to SU(3), The gauge theory can be applied to the case of strong interaction. There are eight parameters for this group corresponding to eight gauge bosons called gluons. The participating particles are the quarks with 3 different colour charges -- red (r), green (g), and blue (b). Three quarks with different colour charges combine to form a baryon. Each quark can carry different colour charges at different time, provided the colour combination is "white". Unlike the case of the U(1) group where the gauge boson (the photon) does not carry charge, the gluons do themselves carry the colour charges. Such difference produces phenomenon such as asymptotic freedom and quark confinement. The gauge theory for the SU(3) group is called Quantum Chromodynamics (QCD). The formulism for QCD and electroweak interaction together is known as the Standard Model (SM), which describes all the phenomena associated with leptons and quarks.

See 2025 update on SM

²In general, the function of a field F is complex, which can be decomposed into the form: F = F_R + iF_I similar to the complex number c = a + ib. The real part F_R and the imaginary part F_I correspond to particle with negative and positive charge respectively.

³It is found that only fermions with left-handed chirality participates in weak interaction. The chirality in elementary particle is related to the property of spin. A right-handed particle has its spin oriented along the particle's direction of motion, while the spin of a left-handed particle points the other way. All neutrinos are left-handed, and all antineutrinos are right-handed (if the neutrino mass is strictly zero). Other particles can exist in either state.

⁴In the summer of 1953 C. N. Yang and Robert Mills (a graduate student at that time) invented the SU(2) gauge theory that has become synonymous with their names. They did not immediately publish their results because they were aware of the difficult problems posed by the gauge-field masses and renormalization. After studying these problems for some time and realizing that they would not be solved in the short term, they sent their paper for publication in the spring of 1954. The problems were eventually resolved twenty years later in a modern version called the Standard Model.

⁵The transition is similar to water frozen to ice, the Higgs fields move away from a state with higher symmetry to a state without this symmetry but in lower energy. This is called spontaneous symmetry breaking. It is related to the fact that although the system has certain symmetry as portrayed in the "Action", the field itself needs not to possess the same kind of symmetry. According to GUT (Grand Unified Theory), the transition occurred at about 10^-37 sec after the Big Bang when temperature was 10²⁹ ^oK corresponding to 10¹⁶ Gev.