Here I have used Eq. (4.22) for the wave-vector exchange q = 2k sin ✓/2, and the fact that
~k = p = mv is the momentum of the incoming particle. This last expression is Rutherford’s
scattering. It is not surprising that we obtained something like Rutherford scattering in
the q ˜
1 limit since in that case the Yukawa potential, Eq. (4.47) reduces exactly to
the Coulomb interaction. It is, however, somewhat surprising that the first order Born
approximation result agrees precisely with the classical result, Eq. (2.80). This is not true
for other power-law potentials, such as 1/r2 for example.
The other interesting limit is the low energy limit when the momentum exchange is much
smaller than the inverse interaction length q ⌧ ˜ 1 . In that case f ! 2m↵ ˜ 2 /~2 and we
have
d
4m2 ↵2 ˜ 4
!
d⌦
~4
(4.51)
This agrees precisely with the result obtained in the previous subsection for the case of a
short range potential, Eq. (4.39).
4.7
Pions, muons, and nuclear forces
With the discovery of the proton and neutron (see the reactions in Eqs. (2.16) and (2.20))
in 1918 and 1932 respectively, it became clear that the nucleus is not a point particle, but
rather a composite of neutrons and protons. It was also apparent that some new force has to
be present to hold these together. It was clear that this force has to be short range since it is
not observed at long distances and it must also be much stronger than the electric repulsion
between protons.
People also observed di↵erent radiations associated with nuclei. By 1903 three types were
identified and classified according to their penetration depth and their ability to ionize,
↵-radiation
-radiation
-radiation
only weakly penetrating
can penetrate a few millimeters of aluminum
strongly penetrating
This simple characterization is of course rather simplistic, and we now know much more
about these di↵erent rays. As we have already seen, ↵-rays are helium nuclei. They get
easily stopped in matter because of the strong interaction between nuclei.
-rays were
identified early on through mass-to-charge (m/e) ratio measurements as electrons. -rays
are understood to be electromagnetic emissions with energies above an MeV from nuclear
transitions.
The radioactive processes associated with -emissions were particularly puzzling. One of the
first reactions of this type to be studied was the transmutation of carbon-14 (an unstable, but
67
important isotope of carbon) into nitrogen-14 with the emission of a -ray. It was puzzling
because experiments showed that the particle emerged with a continuous energy. As we will
study in detail when we discuss relativistic kinematics, if the decay was really 146 C ! 147 N+ ,
then we expect the energy of the electron to be monochromatic. Experimentally it was not
an easy measurement, and at first they did see characteristic lines in the emission spectrum.
But, with better apparatus and more careful measurement, it slowly (a couple of decades —
progress is slow) became clear that the electron has a continuous energy spectrum. This was
a great puzzle because it seemed to imply the violation of energy conservation. The history of
this subject is fascinating, but beyond the scope of this brief account. It eventually dawned
on Pauli that if the radioactive decay process involve another particle in the final state, then
energy can be conserved. This particle must be neutral, very weakly interacting, and very
light. It was thus christened “neutrino” by Fermi. The radioactive beta-decay of a carbon-14
was then understood to be a three-body decay
14
6C
!
14
7N
+ e + ⌫¯e .
(4.52)
The subscript e on the neutrino is because we now know there are three types of neutrinos,
and the bar on the neutrino is to denote the anti-particle, since by convention it is the antiparticle that is emitted in this reaction. In 1934 Fermi proposed his model of beta decay,
which involves the basic reaction,
n ! p + e + ⌫¯e
(4.53)
It is not at all clear how to describe a process like this with the tools we currently have since
it involves the transmutation of one particle type into other particles. We will learn more
about it later in this course.
At this point in history only few particles were known: the neutron and the protons, these
were the heavy particles, or baryons (from the Greek word for “heavy”); then there were
the light particles, the electron and the hypothetical neutrino of Pauli, these were the light
particles, or leptons (from the Greek word for “small”); and there was the photon, the quanta
of light. It was not clear what was responsible for the nuclear force, which was evident from
the existence of nuclei and most clearly seen in the case of the deuteron (recall it is a bound
state of a proton and neutron - see section 3.7). It was also suggested that these forces
can transmute a neutron into a proton and vice-versa. People attempted to explain the
force between the neutron and proton through the same interaction that leads to beta decay,
Eq. (4.53). But it was evident very quickly that it is far too weak to explain the strong
nuclear force between nucleons.
In 1935 a young Japanese physicist named Hideki Yukawa realized that the finite range
of the nuclear force can be understood with a Yukawa type potential, Eq. (4.47), hence
the name (of the potential, not the person of course - causality is an important feature of
reality). Empirical data on the size of nuclei and in particular the deuteron suggest that the
characteristic length scale is,
˜ ⇠ 10 12 10 13 cm
(4.54)
68
Experiment also suggests that the strength of the coupling ↵ is several times that of electromagnetic interactions.
Yukawa went further. He reasoned that just like the photon is the quanta associated with
the electromagnetic field, there should be some quanta associated with this nuclear force
with a mass related to the range of the interaction. In other words, just as the photon
mediates the electromagnetic interaction, there should be some particle that mediates the
strong interactions. The Compton relation between the mass of a particle and its wavelength
yields10 ,
m⇡ =
h
⇡ 100 MeV/c2
˜c
(4.55)
Thus, he predicted the existence of a charged particle with a mass some 200 times heavier
than that of the electron. He considered charge particles because he was attempting to
explain the forces that transmute a neutron into a proton and vice-versa. These particles
came to be known as ⇡-mesons. Mesons from the Greek word for “intermediate”, since their
mass was in an intermediate range between the light leptons (electrons and neutrinos) and
the heavy baryons (protons and neutrons).
The theory very quickly received experimental support when in 1936 a particle with a negative charge and a mass of about 100 MeV/c2 were observed by Anderson and Neddermeyer
in cosmic rays, again by measuring their mass-to-charge ratio with a magnetic field. But
this early success soon proved a mirage. The particle discovered, now know as the muon, did
not have the correct properties to be the ⇡-meson of Yukawa. First, it does not participate
in strong nuclear reactions as the pions are supposed to do. Second the muon was found
to carry spin-1/2, hence it was a fermion not a boson, which the ⇡-mesons had to be if
they were to mediate transitions between two known fermions, namely the proton and the
neutron. For a while people refer to this particle as a µ-meson, because of its mass, but
this name fell out of favor since its properties were determined to be very di↵erent compared
with all the other mesons (including the ⇡-mesons) that were later discovered. We now know
that the muon in fact has nothing to do with all the other mesons, rather it is a heavy and
unstable version of the electron. Over the years its basic properties were determined to be,
10
There are di↵erent ways to reason for this relation, some involve the uncertainty principle, and these
di↵erent ways di↵er by a factors of 2⇡, depending on whether h or ~ enters the estimate. So this prediction
should be understood more as an order of magnitude estimate for the approximate mass and not as a precise
relation.
69
symbol:
µ±
mass = 105 MeV/c2
spin = 12
electric charge = ±e
mean lifetime = 2.2 ⇥ 10 6 sec
The muon decays through the electroweak force, similar to the neutron decay.
It took another decade before the ⇡-meson was finally identified, again in cosmic rays. In
1947 the charged pion was discovered and its basic properties are now established,
symbol: ⇡ ±
mass = 139.6 MeV/c2
spin = 0
electric charge = ±e
mean lifetime = 2.6 ⇥ 10 8 sec
In 1950, what is now known as the neutral pion, ⇡ 0 , was discovered. It also participates in
the strong nuclear force, it is also a boson with a mass similar to that of the charged pion,
but it is neutral and its lifetime is far shorter than that of the charged pion,
symbol: ⇡ 0
mass = 135.0 MeV/c2
spin = 0
electric charge = 0
mean lifetime = 8.5 ⇥ 10 17 sec
This is very curious. On the one hand the ⇡ ± and ⇡ 0 are very close in mass. Indeed, as we
shall see later when we discuss isospin, there is a symmetry that relates them. However, their
lifetimes are vastly di↵erent. In fact, the ⇡ ± has a lifetime closer to that of the muon and
indeed both decay through their electroweak interactions. The neutral pion decays through
a very di↵erent process and has a much shorter lifetime.
Since these early days it also became clear that pion exchange cannot be the whole story
for the strong nuclear force between nucleons for various reasons, in fact it is not even close
70
to being the full story. First, nuclear forces exhibit a strong repulsion at sufficiently short
distance scales . 10 13 cm or about 1 fermi. The Yukawa force is only attractive (or only
repulsive, depending on the sign). Second, nuclear forces depend on the orientation of the
spins of the interacting nucleons. We know that already from the deuteron: if the force that
binds a proton and a neutron to form a deuteron was spin-independent, then the deuteron
(which is the ground state) would be at an S = 0 state with the spin of the neutron antialigned with that of the proton; but, the deuteron is known to have spin-1 with zero angular
momentum ` = 0 and so in fact the spins are aligned. More generally, people originally
thought nuclear forces would be
• Two-Body only — acting between one nucleon and another.
• Central — depends only on the radial vector between two nucleons.
• Static — depends only on the distance between the nucleons, not on their momenta.
• Short ranged.
• Spin-independent.
• Purely attractive.
• Charge independence — the nuclear force between two neutrons is the same as between
two protons or a neutron and a proton.
Nuclear forces are now known to violate pretty much all of these assumptions, with the exception of the last one of charge independence. They are complicated, almost as complicated
as they could be.
Despite the fact that nuclear forces are not as simple as Yukawa envisioned, Yukawa’s intuition led him to a concrete prediction for the existence of a new particle, the pion. In some
sense he was a bit lucky, but then again you need luck to make big discoveries11 .
4.8
Resolving structure
Before moving on, I want to make some more general statements about the Yukawa potential
we met in section 4.6 and the lessons we can learn from it. There is a deep significance to this
simple lesson because it applies very broadly in particle physics. In order to probe structures
11
By the way, it is not very well-known, but in his original paper Yukawa also tried to use the pions as
the mediators of beta decay. He even worked out the necessary coupling of pions to electrons and neutrinos
to yield the experimentally measured neutron lifetime. He was pretty wrong about that too, beta decay is
mediated by the charged W ± bosons associated with the electroweak force not by pions.
71
that emerge at shorter and shorter distance scales, we need beams of higher and higher
momentum or higher and higher energy. As we saw above, when the momentum transfer is
much smaller than the characteristic inverse length scale of the interaction, q ⌧ ˜ 1 , we are
only sensitive to a contact potential. This makes sense because at such low momentum the
de-Broglie wavelength of the beam is much larger than the characteristic length scale ˜ , and
we shouldn’t be able to resolve structures smaller than the wavelength. In that case we can
only determine the combination,
at low energies can measure ↵ ˜ 2
see Eq. (4.51).
(4.56)
This combination is essentially the coupling constant ↵ times the characteristic length scale
squared. We are only sensitive to the product. It’s only when we go to sufficiently high
momentum transfers q & ˜ 1 that we can start probing the structure of the interaction.
Then we can separate the coupling ↵ from the characteristic length scale ˜ and indeed at
sufficiently high momentum transfer ˜ is irrelevant and we can determine,
at high energies can measure ↵
see Eq. (4.50).
(4.57)
This is precisely the behavior seen in the case of the electroweak force and we will see concrete
examples later in the course.
72