Lecture 7
39
Nov 13, 2002
6
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.
6.1
The Yukawa Interaction
In 1934 Hideki Yukawa proposed the existence of a light mass boson (bosonic field) that would be re-
sponsible for (the mediator of) the short-range nuclear force.
2
The mass of the boson would be com-
mensurate 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 ex-
posed to cosmic rays at high-altitude (mountain tops and balloons).
3
Earlier, the muon (105 MeV) had
been discovered (at sea level),
4
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!
An appropriate way of deriving this is to start with the relativistic mass-energy relationship, and “quan-
tify” it by making the momentum and energy into operators:
2
222
22
2
;
using
, and
,
this becomes:
( , )
0
pmE
Ei
i
m
t
tt
ψ
∂∂
∂
+=
→
→
−
−+ +
=
∂
pr
r
∇
(I.94)
This equation is the famous
Klein-Gordon equation
, the relativistic version of the Schrödinger equa-
tion or the massive version of the equation of motion for the electromagnetic field. For spherically
symmetric solutions
(
r
,
t
) =
(
r
,
t
) the three-dimensional space derivative can be written purely in
terms of radius
r
(the derivatives with respect to polar and azimuthal angles
θ
and
φ
have nothing to
operate on!):
2
2
2
2
2
2
2
2
2
()
2
1
1
1
co
t
sin
rr
r
r
r
ψψ
ψθ
θθ
∂
∂ ∂ ∂
∂
∂
≡=
+
+ + +
=
∂∂∂∂
∂
∂
r
∇
r
(I.95)
In the stationary case, the time derivative also vanishes. For a point-like nucleon at rest, equation (I.94)
has a stationary, spherically symmetric solution
4
mr
e
r
r
α
π
−
=
, the
Yukawa potential
. Here
r
=|
r
|;
is
a proportionality “constant” (which could be a function of something other than
r
, and is in fact a func-
tion dependent on the nucleon spin as well); and the exponential has the particle (boson) mass
m
in it
(in the proper units!).