Feynmann Diagrams
M. Cobal, PIF 2006/7
Feynman Diagrams
M. Cobal, PIF 2006/7
Feynman Diagrams
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Vertices
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Virtual processes
M. Cobal, PIF 2006/7
M. Cobal, PIF 2006/7
Real processes
- For a real process there must be energy conservation
it has to be a combination of virtual processes.
Electron-electron scattering, single g exchange
- Any real process receives contributions from
all possible virtual processes.
M. Cobal, PIF 2006/7
- Points at which 3 or more particles meet are called vertices
Each vertex corresponds to a term in the transition matrix element
which includes the structure and strength of the interaction.
-The nr. of vertices in a diagram is called order
-Each vertex has an associated probability proportional to a coupling
constant, usually denoted as a. In the em processes this is:
a em 
1
137
-For the real processes, a diagram of order n gives a contribution
of order an
-Providing that a is small enough, higher order contributions to
many real processes can be neglected
M. Cobal, PIF 2006/7
-Diagrams which differ only by time-ordering are usually implied by
drawing only one of them
-This kind of process:
-Implies 3!=6 different time orderings
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Annihilation diagrams
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Exchange diagrams
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Virtual particles
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-From the order of diagrams the ratio of rates of processes can be
estimated:
R
Rate(e  e   ggg )
 
Rate(e e  gg )
 O(a )
For example, this ratio measured appears to be: R = 0.9 x 10-3,
smaller than aem (but estimate is only a first order prediction)
-For nucleus, coupling is proportional to Z2a, hence the rate for this
process is of order Z2a3
M. Cobal, PIF 2006/7
Exchange of a massive boson



-In the rest frame of particle A: A( E , p)  A( E ' , p' )  X ( E x , p' )

where: E M A , p  (0,0,0),
E' 
p '2  M A2 , E x 
p '2  M X2
From this one can estimate the max distance
'
over which X can propagate: E  E X  E  M A  M X
This energy violation can be only for t   / E
 The interaction range is: r  c / M X
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- For a massless exchanged particle, the
interaction has an infinite range (e.g: em)
-If the exchanged particle is very heavy (like the W boson in the
weak interaction), the interaction can be approiximated by a
zero-range or point interaction
18
197
.
3

10
18
RW  c
 c


2
x
10
m
2
MW
(80.4 GeV / c )
80.4
-If one consider the particle X as an electrostatic potential V(r),
then the Klein-Gordon equation looks like
1 
V 
2
 2V (r )  2  r 2
  M X V (r )
r r  r 
M. Cobal, PIF 2006/7
Yukawa Potential (1935)
-Integration of previous equation gives:
g 2 r / R
V (r )  
e
4pr
-g is an integration constant, interpreted as the strength of the
coupling of particle X to particle A and B (strong nuclear charge)
-Yukawa theory described the short-range nature of force between
n and p in nucleus: interaction is due to massive quanta
-In Yukawa theory, g is analogous to the electric charge in QED,
and the analogue of aem is: ax = g2/4p
ax characterizes the interaction strength at distances: r < R
M. Cobal, PIF 2006/7
- A particle is scattered by this potential, thus receivig a momentum
transfer q .
-The potential has an amplitude (which is its Fourier transform):


 iq

f ( q )   V ( x )e x d 3 x

- Using polar coordinates: V ( x )  V (r )
- and assuming: d 3 x  r 2 sin ddrd, the amplitude is:
sin( qr ) 2
 g2

f (q )  4pg  V (r )
r dr  2
qr
q  M x2
-For the point interaction, M2X >> q2 hence :
 4pa X

f ( q )  G 
M X2
The point-interaction is characterized not only by aX,
but also by MX
M. Cobal, PIF 2006/7
For nuclear forces with a range R ~ 10-15 m, Yukawa hypothesis
predicted a spinless quantum of mass:
Mc2 = hc/2pR  100 MeV.
The pion observed in 1947 had M = 140 MeV, spin 0 and strong
nuclear interactions.
Nowadays: pion exchange still accounted for the longer-range part
of nuclear potential.
However, full details of interaction are more complicated
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Electromagnetism
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Strong Interactions
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Ew Interactions
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Use of Feynmann Diagrams
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Exercises
1) Knowing the dimensions of neutron and protons, you can estimate
the mass of the particle, which is responsible for the interaction
between the nucleus
Hint:
 The interaction between charged particles is carried by
photons. The range of this interaction is infinite, the rest mass of
photon has to be zero. The interaction between nucleons is limited
to a range of about 10-15 m.

The Heisemberg uncertainty relation allows fluctuation of energy
for a very short time, so that ”virtual” particles can be created
which are responsible for the interaction
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Solution:
r =c.t
t = r/c = 3.34.10-24 s
Et  h/4p
E = h/(4pt) ~ 100 MeV
The uncertainty relation for a distance from 10-15 m allows a max
energy deviation of about 100 MeV.
A particle which is responsible for the interaction of two nucleons
should have a mass in the order of 100 MeV but can only exist for
about 10-24 s
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