A question of mass
Jeremy Bernstei
Received 7 May 2010; accepted 17 August 2010
We present a pedagogical discussion of spontaneous symmetry breaking, the Goldstone theorem,
and the Higgs mechanism. If the Higgs boson is found, it might provide an explanation of the origin
of mass. ©
2011 American Association of Physics Teachers.
DOI: 10.1119/1.3487939
“The quantity of any matter is the measure of it by its
density and volume conjointly
…
. This quantity is what I
shall understand by the term
mass
or
body
in the discussions
to follow. It is ascertainable from the weight of the body in
question. For I have found by pendulum experiments of high
precision, that the mass of a body is proportional to its
weight; as will hereafter be shown.” Isaac Newto
When I took freshman physics as a sophomore at Harvard
in 1948, this definition of mass was still used in our text-
book. As it happens, the previous year I had taken a philo-
sophically oriented course in modern physics given by the
philosopher-physicist Philipp Frank. He introduced us to the
work of his fellow Austrian philosopher-physicist Ernst
Mach. Therefore I knew of Mach’s devastating critique in his
book,
The Science of Mechanics
and how it had influenced
Einstein. Newton’s definition of mass is circular. What is
density? How does it is apply to the photon, which has no
mass?
When I began writing my Ph.D. thesis in the early 1950s,
I would have described myself as an “elementary particle”
theorist rather than as a nuclear theorist. Elementary particles
were considered to not consist of anything else, whereas the
atomic nucleus consists of neutrons and protons, which were
taken to be elementary particles. There were not that many
elementary particles known at the time. In addition to the
neutron and the proton, there was the photon, the electron,
and the neutrinos. The positron was known, and most physi-
cists, Feynman being a notable exception, believed that the
antiproton would be found once accelerators were suffi-
ciently energetic. A few years earlier, a “heavy electron” had
been discovered that had some of the properties of the elec-
tron, including its weak and electromagnetic interactions, ex-
cept that it was about two hundred times more massive and
unstable. For various reasons that no longer make any sense,
it was first called the mu meson and then eventually the
muon. It seemed to serve no purpose and when I. I. Rabi
heard of it, he asked, “Who ordered that?”
Rabi’s pique was understandable. In the 1930s, a theory of
the nuclear force had been proposed. It had to account for
satisfying two conditions. First, the nuclear force was very
short ranged and acted only when the neutrons and protons
were practically on top of each other. Second, it had to be
much stronger than the electrical force; otherwise, the posi-
tively charged protons, which repel each other, would tear
the nucleus apart. As it is, heavy nuclei with many protons
tend to fission spontaneously. Both of these conditions could
be satisfied if a fairly massive particle was exchanged be-
tween the neutrons and protons and among themselves. The
strength of this interaction was postulated to be large com-
pared to the electrostatic force. It was also shown that the
range
r
of this force was related to the mass
m
of the particle
being exchanged. From the uncertainty principle for energy
and time, with the energy uncertainty equal to
mc
2
, we have
mc
2
c
/
r
, and thus its mass was predicted to be about 400
times larger than that of the electron. This mass, unlike the
mass of the muon, seemed to have a connection to the dy-
namics. The muon had something like the correct mass, but it
only interacted electrically and weakly, and thus it was the
wrong particle. The right particle was called the pi-meson or
the pion. Why did it not show up in cosmic rays rather than
the muon? The answer turned out to be very simple. The
pion, when it was not absorbed in the atmosphere, decayed
into a muon and neutrinos. When accelerators became suffi-
ciently powerful, they produced pions in droves. At the time
I was writing my thesis, pion physics was flourishing. But
then, the roof fell in.
Particles that no one anticipated also began to show up in
droves in cosmic rays. They became known as “strange par-
ticles” because they were. There was the K meson, which
came in a charged and neutral variety, and hyperons, which
had masses greater than the proton or neutron. The latter
category includes a lambda particle, which is neutral, a
sigma particle with charges plus, minus, and zero, and a xi
particle with charges zero and minus. These were the lowest
mass particles, which were repeated in higher mass replicas.
In short, it was a particle zoo. It took strong nerves to see any
pattern, a pattern that might reflect an underlying symmetry.
At the time, there was a clear idea of how such a symme-
try might appear. The neutron and proton were prime ex-
amples. They had many properties in common including
their spins, which were identical, and their masses, which did
not differ by much. The neutron was a bit heavier and de-
cayed into a proton, an electron, and an antineutrino. Sup-
pose we imagined a world in which electromagnetism was
switched off. In this world, the neutron and proton would
have the same mass and would collapse into a doublet. The
three pi mesons, plus, minus, and zero charge, would col-
lapse into a triplet. A symmetry would emerge, which was
called “isotopic spin,” which is invariance under the group
SU 2 . The predictions were reasonable, and hence isotopic
spin was a useful approximate symmetry. Maybe something
analogous could be found for the strange particles.
The trouble was that the mass differences were too great.
Although the mass difference of the neutron and proton was
only a fraction of a percent of the mass of either particle, the
K mesons had nearly four times the mass of the pions. It took
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Am. J. Phys.
79
1 , January 2011
© 2011 American Association of Physics Teachers
