The Higgs Boson

In 1964 Peter Higgs (at the University of Edinburgh) and several other theoretical physicists presented an hypothesis to explain why some of Nature’s elementary particles have mass and others (such as the photon) don’t. The hypothesis asserts that space is filled with a scalar field, somewhat in the manner of the old-fashioned aether, and that particles possess mass in accordance with how strongly they couple with that field. Physicists now refer to that field as the Higgs field and the Higgs boson is the particle corresponding to the minimal wave disturbance of that field. I have adapted what follows from Chapter 11 of "Introduction to Elementary Particles" by David Griffiths (1987, John Wiley & Sons) with the addition of some information from other sources.

Analysis of the Higgs field and its associated particles starts with a Lagrangian density that looks a little like the one associated with the Klein-Gordon equation, which applies to particles with zero spin,

(Eq’n 1)

In that equation the i-indices run over all four dimensions of spacetime and
∂_{i}=d/dx^{i}.
The field description is a two-component entity,

(Eq’n 2)

whose square equals the sum of the squares of the components. We can thus plot values of the field strength on an analogue of the complex plane.

The first term on the right side of Equation 1, involving the derivatives of the field, corresponds to the kinetic energy term in a classical Lagrangian function. The second and third terms correspond to the potential energy term in the classical Lagrangian, though they have the wrong form to match up with the actual Klein-Gordon Lagrangian. We can fix that error by noticing that the function

(Eq’n 3)

has its minimum value away from the origin of the
ϕ_{1}-ϕ_{2}
plane. If we plot U
as an altitude above that plane, we get a bowl-shaped figure with a hill at its
center: the minimum value of U,
calculated through the usual dU/dϕ=0,
lies on a circle described by

(Eq’n 4)

Particles emerge as quantized vibrations of the ϕ-field from the ground state of the field, the state in which U has its minimum value. To gain a description of such particles we want to shift the center of our Lagrangian onto the ground state. Toward the achievement of that end we define two new fields

(Eq’ns 5)

Substituting those equations into Equation 1 and working through the algebra gives us

(Eq’n 6)

The expression enclosed in the first pair of square brackets depicts the free Klein-Gordon Lagrangian for a particle of mass

(Eq’n 7)

The expression enclosed in the second pair of square brackets depicts the free Klein-Gordon Lagrangian for a massless particle. The expression enclosed in curly brackets represents five couplings of the equivalent particles (essentially interactions) in the Feynman diagrams. And the last term is merely a constant, which gets eliminated when we apply the Euler-Lagrange operator to transform the Lagrangian density into an equation of field evolution.

The disappointment here lies in the fact that the massless
particle is a scalar (spin-0) boson, a Goldstone boson. As far as physicists
know or suspect, no such particle exists. So we must take one more step to rid
our Lagrangian density of that particle. To achieve that riddance we must
eliminate the ξ-field.
And to achieve that elimination we must transform the
ϕ_{1}-ϕ_{2}
coordinates.

We use the Greek letter theta to denote the angle between
a straight line drawn across the ϕ_{1}-ϕ_{2}
plane and the ϕ_{1}-axis.
We can use theta, then, to indicate a point on the circle where
U
achieves its minimum value: for devising the Lagrangian density of Equation 6 we
implicitly made θ=0.
We can thus transform ϕ
by writing

(Eq’n 8)

If theta has a constant value across all of space, then the Lagrangian density remains invariant under the transformation; that is,

(Eq’n 9)

If, on the other hand, the value of theta is a function of spatial position, θ(x), then we must replace the partial derivatives in the Lagrangian density with the covariant derivative,

(Eq’n 10)

In that equation A_{i} represents a massless gauge field and q
represents the charge that couples to it to generate a force. Because
ξ
represents the imaginary part of our
ϕ-field,
we eliminate it by making ϕ’
real; that is, we set the value of theta at

(Eq’n 11)

With that transformation in place, the Lagrangian density becomes

(Eq’n 12)

The first square brackets in that expression enclose the free Klein-Gordon Lagrangian density for the Higgs field and its particle, the Higgs boson. The second square brackets enclose what looks like the Proca Lagrangian for a particle of mass

(Eq’n 13)

The curley brackets enclose terms describing various interactions. And the last term is the constant that we ignored before.

Of course, that theory must be proven and verified so that we can be certain that we understand how that part of Reality works. We can’t observe the Higgs boson in Nature, so we have to contrive the observation as we do for all subatomic particles, by smashing particles together in a synchro-cyclotron.

That procedure gives researchers the data that they want, because the vacuum is filled with virtual particles, the ghosts of real particles, which can be made real by any phenomenon that can supply the necessary properties, especially energy. Head-on collisions between bunches of protons traveling close to the speed of light provide the energy. The realized particles’ other properties are provided ex nihilo by each particle being realized along with its antimatter counterpart: the properties held by the pair, such as electric charge, cancel each other out.

At the CERN high-energy physics laboratory near Geneva,
Switzerland, physicists used the Large Hadron Collider to create Higgs bosons.
In 2012 they had gained enough data of the right kind to announce that they had
good evidence for the existence of the particle. Because the particle is so
massive, it doesn’t last long enough for anyone to detect it and make
measurements on it; instead, the physicists detect and measure the lesser
particles into which it decays. They looked for signs of a particle that had,
for example, decayed into a Z^{0}-boson and its antiparticle, which
particles then decayed into electron-positron pairs, muon-antimuon pairs, or
pairs of high-energy photons (gamma rays). By studying the patterns of those
decays, the physicists worked backward to trace them to their origin in a
particle that matched the expected properties of the Higgs boson.

So now we know that the Higgs boson ponders a mass
equivalent to 125 Gev of energy (1 Gev = one billion (10^{9}) electron
volts); it possesses zero spin (which makes it a boson); and it possesses zero
electric charge.

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