Part I 1. Introduction: A Universe of Fermionic Matter bound by Bosonic Fields
4
2
2
2
Th
22
2
natural
00
MKS
units
11
cos
,
197.3
where
fm
2.81fm
4
4
0.511
e
e
ee
e
de
dm
e
c
r
mm
c
c
m
c
σ
θ
α
πε
=+
Ω
≡=
=
×
=×
=
!
!
(I.2)
the “classical electron radius”; the energy
U
of a charged spherical shell of radius
r
is
U
=
½
QV
= ½
e
2
/(4
0
r
) = ½
/
r
. If this is to equal the electron mass energy
m
e
, we find
r
=½
/
m
e
; and
r
=
3
/
5
/
m
e
for a uniformly charged sphere.
(iii)
The muon lifetime formula, calculated later in this course, is:
1
52
3
,
with
105.6584 MeV and
1.16639 10 GeV
192
e
F
e
F
mG
µ
µνν
τ
π
−
−−
→
Γ=
=
=
=
×
(I.3)
where
G
F
is the Fermi constant of the weak interaction. The RHS equals 3.009
×
10
−
19
GeV,
a proper unit of width for an unstable particle – the uncertainty in its mass energy. To con-
vert to the muon’s life time
τ
, simply divide
Γ
µ
into
!
to find 2.19
µ
s.
(iv)
Typical length scales in atomic, nuclear, and particle physics:
the Bohr radius:
a
0
= (
m
e
)
−
1
,
the Compton wavelength:
λ
e
=
(
m
e
)
−
1
, and
the classical electron radius:
r
e
=
/
m
e
, respectively; which all differ by a factor
!
1.2
Forces in Nature
The forces in nature are Gravity, the weak interaction, the electromagnetic interaction (the latter
two now unified as the electroweak interaction, the cornerstone of the Standard Model), and the strong
or color interaction. All modern interaction models are based on the exchange of elementary quanta, in
the simplest case tensor, vector, or scalar bosons. Feynman graphs picture such exchange processes,
but more than that, they associate precise terms for the amplitude describing the process with the vari-
ous items in the graph: vertices (where lines meet) correspond to precisely prescribed vertex factors,
internal lines to propagator terms, and in- and outgoing particle lines to phase-space factors dependent
on particle type.
1.1.2
The Yukawa Interaction
In 1934 Hideki Yukawa proposed the existence of a light mass boson (bosonic field) that would
be responsible for (the mediator of) the short-range nuclear force. The mass of the boson would be
commensurate with the observed mean range of about 2 fm. The conjugate of the range would be the
boson’s mass or energy:
m
=1/
r
=
ħ
c/r
= 197 MeV
⋅
fm / 2 fm
≈
100 MeV. Powell et al. much later (1947)
observed the Yukawa particle, the pion (140 MeV) or pi-meson, in photographic emulsions exposed to
cosmic rays at high-altitude (mountain tops and balloons). Earlier, the muon (105 MeV) had been dis-
covered (at sea level), but was quickly recognized for what it was: a heavy version of the electron (a
charged lepton), which did NOT have the strong interaction, and thus could be hardly its messenger!
In order to explain Yukawa’s approach, we turn to good old electromagnetism. Maxwell’s
equations of electromagnetism can be succinctly described by the electromagnetic vector potential
A
and the scalar potential
A
0
(which simply equals the electric Coulomb potential
V
):