Beyond the God Particle (46 page)

We can ask: “What would happen if we somehow changed the phase of the wave function
without changing the observable probability at any point in space and time
?” We keep the probability of finding the electron at any point in space the same. We call this a “
gauge transformation
.” But, in making this change, there is apparently nothing invariant here. This would affect the derivatives of the wave function with respect to t and x, and those determine the energy and momentum. This is evidently not a symmetry of the original quantum state, but rather it seems to produce a new quantum state with different observable energy and momentum.

Let us now suppose that there is
some other quantum particle wave
that modifies the derivatives with respect to t and x. And let us further suppose that when we change the electron's wavelength or frequency,
we are simultaneously required to modify the new field in such a way as to keep the derivatives with respect to t and x the same
. The net effect is that we have kept the probability, the energy and the momentum invariant under our transformation.
Together with the gauge particle, we can maintain both the original incoming total energy and the momentum, even though we scramble the unobservable phase of our electron's wave function. Thus, the term “gauge” means that the actual determination of the physical momentum of the electron requires the presence of the calibrating “gauge” field. Only the electron wave function, together with the “gauge” field, yields a physically meaningful description of the electron. The presence of the new gauge field in the derivatives causes the interaction of the photon with the electron (see
note 11
).

The gauge theory asserts that, if the electron is given a physical kick, if an electron is
accelerated
, then the gauge field is actually shaken off—it is emitted as an independent particle wave with a physical momentum of its own, and the electron recoils to conserve energy and momentum. The gauge field becomes a true physical entity and is radiated out into space. From the point of view of a distant observer, an accelerated electron has radiated a new particle, the
photon
.

Light is emitted from accelerated charges. This occurs in countless physical processes, such as the
scattering
of an electron off of an atomic nucleus, or an atom, or another electron. It can be observed readily in the laboratory. At very low energies, it is the way in which electrons emit the photons from a campfire. Accelerated electrons radiate the microwaves that heat our coffee in a microwave oven, or transmit the evening news into our living rooms, or cause the sun to shine.

We can graphically represent a physical process by a set of Feynman diagrams that represents the quantum computation. These diagrams tell us precisely how to compute the quantum outcome, the probability of a given process, provided that the strengths of the interactions are known and are not too large. We can often visualize a process through Feynman diagrams even when we cannot compute the result. A graduate student, writing from Cornell University where Feynman developed this technique, commented, “At Cornell, even the janitors use Feynman diagrams.”
¹²
With the full machinery of Feynman diagrams we can compute the scattering rate for two beams of electrons to arbitrary precision, including many diagrams that represent detailed quantum corrections to the basic result. The experimentalist can compare the calculations with the results measured in the lab, and these are found to agree to extremely high precision.

YANG–MILLS GAUGE THEORY

The modern era of gauge theories began with a remarkable paper of Chen Ning Yang and Robert Mills in 1954.
¹³
These authors asked a straightforward question: “What happens if we extend the gauge symmetry of the electron to larger symmetries?” The symmetry of electrodynamics involves, as we have seen, the phase of the electron wave function. This is called “U(1) symmetry.”

Yang and Mills turned to the next more complicated symmetry, “SU(2),” the symmetry of the rotations sphere in three real dimensions (or the symmetry of the rotations of two particles, such as (
u
,
d
) quarks or (
v
_e
, e
⁻
), that is, rotations in 2 complex dimensions). It turns out that this symmetry leads to a more general form of a quantum gauge theory called a “Yang–Mills theory.” SU(2) has three gauge fields, hence three photon-like objects, and now the gauge fields themselves carry charges, unlike the case of electrodynamics in which the photon carries no electric charge. Moreover, the Yang–Mills construction works for any symmetry. Symmetry thus becomes partially fundamental to the basic structure of a quantum theory of forces.

In the Standard Model electroweak theory, the symmetry is the “product group” of SU(2) × U(1), with 4 gauge fields, W
⁺
, W
^–
, Z
⁰
, and
γ
fully and accurately described by the Yang–Mills theory. The Higgs boson, as we have seen, causes the W
⁺
, W
^–
, and Z
⁰
to become heavy, while the photon
γ
remains massless. Likewise, as we've seen, the quarks carry 3 “colors,” and the resulting SU(3) gauge theory has 8 gauge bosons, known as the gluons.

Indeed, all known forces are based upon gauge theories. Yet, there are four completely different structures, or
styles
, of gauge invariance. Einstein's theory of gravity contains a coordinate system invariance, that is, it doesn't matter what coordinate system you use, or how you choose to move, inertially or non-inertially through space and time, to describe nature. This leads to gravity as a bending and reshaping of geometry, governed by the presence of energy (equivalent to mass) and matter. Particles must then emit and absorb
gravitons
, which are the gauge fields, or the “quanta,” of gravity. The Newtonian gravitational theory is recovered only as an approximation at low energies (slow systems, without too much mass). The description of the remaining nongravitational forces in nature is based upon the Yang–Mills theory of SU(3) × SU(2) × U(1) as codified by the Standard Model.

THE WEAK FORCE AS A GAUGE THEORY

Let's now consider in a little more detail how the
weak interactions
are described by a gauge symmetry that unifies them together with the electromagnetic force. Taken together, the quarks, leptons, and the gauge symmetries (including Einstein's general relativity) provide a complete accounting of all observed laboratory physics to date and define what is called the “Standard Model.”
¹⁴

Recall that, subsequent to Becquerel et al., yet more than 65 years ago, Enrico Fermi wrote down the first descriptive quantum theory of the “weak interactions.” Fermi had to introduce a new fundamental constant into physics to specify the overall strength of the weak interactions, called
G
_F
, and it represents a fundamental unit of mass, which sets the scale of the weak forces, about 175 GeV.

In the 1960s the weak forces were found to involve a gauge symmetry based upon the symmetries SU(2) × U(1) (by Sheldon Glashow, Abdus Salam, and Steven Weinberg, and this was perfected as a quantum theory by Gerhard ‘t Hooft and Martinus Veltman). Let us now describe the gauge symmetry of the weak interactions.

We see that, within each generation, the quarks and leptons are paired. That is, the red up quark is paired with the red down quark, the electron neutrino with the electron, the charm quark with the strange, the top with the bottom, and so forth. We thus imagine that the electron and its neutrino are a single entity that lives in a two-dimensional space, with one axis meaning “electron” and the other “electron neutrino.” The quantum state is an arrow in this space that can point in any direction. When the arrow points along the electron axis, we have an electron. Rotating the arrow, we have a neutrino. The rotations we can do on the arrow form the symmetry group, called SU(2).

So we now imagine an electron neutrino particle wave with a given momentum and energy. Then we perform a
gauge transformation
that rotates this into an electron, which has negative charge, and also scrambles the electron momentum and energy. To make this into a symmetry, we need to introduce a gauge field, the W
⁺
that can restore the total energy and momentum, and rotate the quantum arrow back to its original electrically neutral “electron neutrino” direction. In a sense, the gauge field rotates the coordinate axes, so the arrow is now pointing back in the original direction,
relative to the coordinate system, and we get back the original neutrino we started with. This is completely analogous to what we do with quark color, where the gauge rotation from one color to another is compensated by the gluon field. This requires a total of three new gauge fields, W
⁺
, W
^–
, Z
⁰
, in addition to the photon
γ
. In fact, electrodynamics and the weak interactions now become blended together into one combined entity called the “electroweak interactions.”

There is, however, an enormous difference between the photon and these three new gauge fields. The photon is a massless particle, while the W
⁺
, W
^–
, Z
⁰
are very heavy particles. The forces that are produced by the quantum exchange of W particles between quarks and leptons give rise to the weak force that Fermi was describing 65 years ago. As we've seen, the Higgs field in the vacuum causes the W
⁺
, W
^–
, Z
⁰
to become massive.

The strength of the Higgs field in the vacuum is already determined by Fermi's theory to be 175 GeV. The field implies the existence of a new particle, the Higgs boson, the necessary quantum of the Higgs field. All the matter particles, and the W
⁺
, W
^–
, Z
⁰
, get their masses by interacting with the vacuum-filling Higgs field (unlike a superconductor, however, the photon does not interact with this particular field and remains massless). The Higgs field is “felt” by the various particles through their “coupling strengths.” For example, the electron has a coupling strength with the Higgs field,
ge
. Therefore, the electron mass is determined to be
me
=
ge
× (175 GeV). Since we know
me
= 0.0005 GeV, we see that
ge
= 0.0005/175 = 0.0000029. This is an extremely feeble coupling strength, so the electron is a very-low mass particle. Other particles, like the top quark, which has a mass
mtop =
175 GeV, has a coupling strength almost identically equal to one (suggesting that the top quark is playing a special role in the dynamics of the Higgs field). Still other particles, like neutrinos, have nearly zero masses and therefore nearly zero coupling strengths.

All of this sounds like a spectacular success, and it is, but there is a slight shortcoming—there is, at present,
no theory for the origin of the coupling constants
, such as that of the electron
g
_e
. These appear only as input parameters in the Standard Model. We learn almost nothing about the electron mass, swapping the known experimental value, 0.511 MeV, for the new number,
ge
= 0.0000029. Furthermore, we are clueless as to what generates the mass of the Higgs boson itself.

The Standard Model did successfully predict the coupling strength of the W
⁺
, W
^–
, Z
⁰
particles to the Higgs field. These coupling strengths are determined from the known value of the electric charge and another quantity, called the weak mixing angle, measured in neutrino scattering experiments. So the masses of W and Z, M
_W
and M
_Z
(note that the W
⁺
and W
^–
are particle and antiparticle of each other and must have the same identical masses; the
Z
⁰
is its own antiparticle), are predicted (correctly) by the theory. The W
⁺
and W
^–
have a mass of about 80 GeV, and the Z
⁰
has a mass of about 90 GeV. These have been measured to very high precision in experiments at CERN, SLAC, and Fermilab.

Symmetry and its spontaneous breaking through of the Higgs particle, therefore, completely controls the mass generation of all the particles in the universe. And it appears that it is the Higgs boson, the quantum of the Higgs field, that was discovered in the two experiments, ATLAS and CMS, at CERN on July 4, 2012, with a mass of m
_h
= 126 GeV. “What generates the Higgs boson mass?” is now the most important scientific question of our time.