1. Scattering Cross Sections (light projectile off heavy target)
a + b  c + d
counting number
of final states
counting number
of initial states
“flux”
from (E)=dN/dE
Note: If a and b are unpolarized (randomly polarized)
the experiment cannot distinguish between
or separate out the contributions from
different possibilities, but measures the
scattered total of all possible spin combinations:
averaging over all possible (equally probable) initial states

2. mfi2 or m2fi2 fi fi or fi fi g fi fj fk
We will find that everything is in fact derivable from a comprehensive
Lagrangian (better yet a Lagrangian density, where
As a preview assume L describes some generic fields: fi, fj, fk
wave functions…of matter fields
…or interaction fields (potentials)
Invariance principles (symmetries) will guarantee
that schematically we will find 3 basic type of terms:
mfi2 or m2fi2 1
mass terms:
Fermion
Boson 2
kinetic energy terms:
fi fi or fi fi 3
interaction terms:
g fi fj fk
may also contain derivatives
or >3 fields…or both

3. g Cross sections or decay rates are theoretically computed/predicted
from the Matrix Elements (transition probabilities)
pick out the relevant terms in L
can be expanded in a series of approximations
[the coefficients of each term being
powers of a coupling constant, g]
Griffiths outlines the Feynmann rules that translate L terms into M factors
propagator
mass & kinetic terms
vertex g
interaction terms g

4. Vertices get hooked together with propagators, with each
vertex contributing one power of coupling to the calculation
of the matrix element for the process. g g

5. You know from Quantum Mechanics: Amplitudes are, in general,
COMPLEX NUMBERS
In e+e- collisions,
can’t distinguish
All diagrams with
the same initial and final states
must be added,
then squared. e- e- g e+ e+ + e- e-
The cross terms introduced by squaring
describe interference between the diagrams
(sometimes suppressing rates!) g e+ e+

6. kinematic constraints
d = [flux] × | M |2 × (E) × 4
initial
state
properties
statistical factor
counting the number
of ways final state
produced
kinematic constraints
on 4-momentum
will be a Lorentz
invariant phase space
dp3 2E
Matrix elements get squared
electromagnetic 1
137 e
e2 
Basic vertex of any interaction
introduces a coupling factor, g g
weak g
g2 GF10-5GeV-2
g2 is the minimum factor associated
with any process…usually the
expansion parameter (coefficient)
of any series approximation for
the matrix element W±
strong gs
gs2  s  0.1 g

7. Schrödinger’s equation
For “free” particles (unbounded in the “continuum”)
the solutions to
Schrödinger’s equation
with no potential
Sorry!…this V is
a volume appearing
for normalization V

8. q q pi q = ki - kf =(pi-pf )/ħ pi momentum transfer the momentum
given up (lost)
by the scattered
particle pi
q = ki - kf =(pi-pf )/ħ pi

9. Proton-proton (strong interaction) cross sections
The strong force has a very short effective range (unlike the coulomb force)
If assume a simple “black disk” model with fixed geometric cross section:
typical hadron size
rp~1fm = 10-13cm
(1 barn=10-24 cm2)
This ignores the dependence on E
or resonances,
but from 1 to several 1000 GeV of beam energy
its approximately correct!

10. 35-40 mb pp collisions pp collisions Note: Elastic scattering
1/E dependence

11. Letting one cat out of the bag:
Protons, anti-protons, neutrons are each composed of 3 quarks
The (lighter) mesons (+, 0, -, K+, K0, K-, …) … 2 quarks
Might predict:
~38mb
~42mb
and
±p~ 25mb
K±p~ 20mb

13. n  p + e- + e -  e- + e +  Ne*  Ne +  N  C + e + + e
The transition rate, W (the “Golden Rule”) of initialfinal
is also invoked to understand
ab+c (+  )
decays
Some observed decays
n  p + e- + e
-  e- + e + 
Fundamental particle decays
Ne*  Ne + 
N  C + e + + e
Pu  U +  20 10 20 10 13 7 13 6
Nuclear decays
236 94
232 92
How do you calculate an “overlap” between ???

17. J conserved. Any decay that’s possible will happen!
It almost seems a self-evident statement:
Any decay that’s possible will happen!
What makes it possible?
What sort of conditions must be satisfied?
Total charge q conserved.
J conserved.

18. probability of surviving
to time t
mean lifetime  = 1/
For any free particle (separation of space-time components)
Such an expression CANNOT describe an unstable particle
since
Instead mathematically introduce the exponential factor:

19. a decaying probability
then
a decaying probability
of surviving
Note: G=għ
Also notice: effectively introduces an imaginary part to E

20. Applying a Fourier transform:
What’s this represent?
E distribution of
the unstable state
still complex!

21. Expect
some constant
Breit-Wigner Resonance Curve

22. 1.0 
MAX
0.5
 = FWHM E Eo
When SPIN of the resonant state is included:

23. 130-eV neutron resonances
Transmission
130-eV neutron resonances
scattering from 59Co
-ray yield for
neutron radiative
capture

24. +p elastic scattering cross-section in the region of the Δ++ resonance.
The central mass is 1232 MeV with a width =120 MeV

25. e+e- anything near the Z0 resonance plotted against cms energy
Cross-section for the reaction
e+e- anything
near the Z0 resonance
plotted against
cms energy

27. In general: cross sections for free body decays (not resonances)
are built exactly the same way as scattering cross sections.
except for how the “flux” factor has to be defined
DECAYS (2-body example)
(2-body) SCATTERING
enforces conservation
of energy/momentum
when integrating
over final states
in C.O.M.
in Lab frame:
Now the relativistic
invariant phase space
of both recoiling
target and
scattered projectile

28. Number scattered per unit time = (FLUX) × N × total density (a rate)
of targets
(a rate)
/cm2·sec
A concentration
focused into a
small spot and
small time interval
size of
each
target
Notice:
is a
function
of flux!