Background

It is often said that Higg's analysis seems so ad hoc and contrived. Well, it is and there is a good reason why this so. Prior to Higg's analysis the only thing that Particle Physics theory had concerning the mass of particles was the Goldstone Theorem. What Goldstone showed was that using a conventional model of particle physics based on conventional assumptions the common particles should have no mass. It was a crisis of theoretical particle physics. What was needed was some way around the Goldstone Theorem no matter how contrived. Physicists had to find a loophole in Goldstone's analysis and they did, but it necessarily had to deviate from the conventional analysis of Goldstone. Higgs was not the only physicist (or even the first) to find a way to justify the oserved masses of the electrons (postitive and negative) and the protons and neutrons. Higgs, however in contrast to the others, placed special emphasis on the existence of a boson for the universal field. To explain Higgs' analysis it is convenient to consider electric fields and gravitational fields.

An Electric Field

A charge of Q establishes a field of intensity E at point Z given by'

E(Z) = kQ/s²

where k is a constant and s is the separation distance between the point Z and the center of the charge.

The relationship is shown below in which the units are completely arbitrary.

The total energy of the electric field created by a charge Q distributed over a sphere of radius R is

E_{total = kQ²/R}

This is the energy that would be required to bring infinitesimal bits of charge from an infinite distance to build up the charge at R against the repulsion of the charge that is already there.

The energy density of the field at point Z is E(Z)²/8π. If there is also a magnetic field of intensity B at Z the energy density would be

(E(Z)²+B(Z)²)/8π

A Gravitational Field

The energy density at a point Z due to a mass M is given by

E(Z) = GM/s²

where G is the gravitational constant and s is the separation distance between the point Z and the center of the mass.

The display for the gravitational field differs from that for the electric field in that there is no negative mass.

The total energy of the field due to a mass M distriuted over a spherical ball of radius r is approximately

E_{total = −GM²/R}

It is negative because no work has to be done to bring bits of mass from infinity to the radius R. Instead energy is released due to the attraction of bits of mass for each other.

The Higgs Field

Higgs hypothesized that the energy density due to the universal field would have the form

E = κH² + λH⁴

where H is a charge parameter for the field. The quadratic coefficient κ Higgs took to be negative as in the case of the gravitational field but in contrast to the electric field. The fourth power parameter λ Higgs assumed to be positive. Under these assumptions the energy density as a function of H could take the following form.

The units in the graph are completely arbitary.

Now the lowest energy state does not occur for H=0, but instead for ±H*. According to the analysis the interaction of the fields of electrons and such with the universal field gives rise to masses for them. The interaction of the particles themselves with the universal field takes place through the interaction of the particles with bosons that are the vibrations of the universal field.

Higgs believed the quadratic coefficient parameter κ should be a squared term, say μ².That meant that μ would have to be an imaginary number.

The rationale for the negativity of μ² is that it depends on the scale of analysis. At the Planck scale of 10⁻³⁵ meters it is positive but at the scale of the Standard Model of particle physics 10⁻¹⁷ meters it is negative, as shown in the following display.

The Planck scale is considered to be somehow the appropriate scale for particle physics.

(To be continued.)