Cross sections

1
Cross sections
A cross section s is a measure of interaction
probability. A number of interactions is
Nint Ntarget s F
where F is a flux ( cm-2) of the projectile
particles. Hence s is measured in cm2. It can be
seen as an effective surface of a target. For
strong interactions s is 1barn10-24
cm2 Differential cross sections
distributions of energies and angles
of secondary particles
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
2
Neutrino interactions
A neutrino of energy E and wavelength lh/E can
interact with
an electron on the
atomic orbit a nucleus as a whole (when
l is comparable to the nucleus size)
a free proton or a nucleon bound in a
nucleus (when l is comparable to the
nucleon size) a quark (when l is much
smaller than the proton size)
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
3
Neutrino cross sections
From Fermi theory the cross section for a CC
interaction
are the relative velocities in the initial and
final states
is
q is momentum in cms Fermi constant
For reaction
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
4
Neutrino cross sections
For a CC interaction
is
For reaction
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
5
Neutrino interaction length
or mean free path between collisions
For neutrinos of 1 GeV pathing Earth lets take
Energy needed for l to become of the size of
Earth
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
6
n interactions with nucleons

• Fermi theory did not involve intermediate
  bosons.
• G constant was determined from experiments
  (neutron lifetime)
• Modern theory of electro-weak interactions
  provides the same
• cross sections at not too large energies

At low energies only anti neutrinos on free
protons neutrinos only on neutrons,
(but those are always bound in nuclei
extra energy needed to compensate for
nuclear binding
7
Neutrino scattering on electrons
NC
CC
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
8
ne cross sections low energies
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
9
nm cross sections low energies
m mass 106 MeV
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
10
nm cross sections high energies
QE (quasi-elastic) Single Pion (resonances)
DIS (deep inelastic scattering)
at high energies
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
11
Angular distributions in n e- scattering
Lets look at Lorentz transformation from CMS to
LAB
independently of qcms
both neutrino and electron mostly forward
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
12
Angular distributions in n e- scattering
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
13
Angular distributions in n N scatteringat low
energies
From neutrinos to cosmic sources, D.
Kielczewska and E. Rondio
14
Quark mixing in Standard Model
u c t d s b
States partcipating in strong interactions with
well defined masses (mass matrix eigenstates)
u c t d s b
Staes participating in weak interactions
Quark mixing
From neutrinos to cosmic sources, lecture 5,
2003
15
Neutrino mixing NOT in Standard Model
IF neutrinos are massive States with well
defined masses (mass matrix eigenstates)
Staes participating in weak interactions
Lepton mixing
From neutrinos to cosmic sources, lecture 5,
2003
16
Neutrino oscillation 2 flavors
mixing angle
mass states
q
are defined as different proportions of n1 n2
states
n1 n2 states have different masses
different velocities
changes during propagation, hence
From neutrinos to cosmic sources, lecture 5,
2003
17
Oscillation probability 2 flavors (part 1)
A state of mass mk, energy and momentum Ek,pk
propagates
with phase
Lets assume an initial state
From neutrinos to cosmic sources, lecture 5,
2003
18
Oscillation probability 2 flavors (part 2)
During propagation the contribution of n1,n2
components changes
A probability that after t,x the state a is still
in its initial a state
Finally
From neutrinos to cosmic sources, lecture 5,
2003
19
Oscillation probability 2 flavors (part 3)
Probability of transition from a state a to a
state b
oscillation parameters

• m - mass (in eV)
• - mixing angle

En neutrino energy (in GeV) L - distance from a
neutrino source to detector (km)
experimental conditions
Oscillation length
From neutrinos to cosmic sources, lecture 5,
2003
20
Neutrino oscillations
21
Mixing of 3 flavors
For 3 flavors we need 3x3 matrix. In quark case
the corresponding matrix is called CKM
(Cabibo-Kobayashi-Maskava).
From neutrinos to cosmic sources, lecture 5,
2003
22
Mixing of 3 flavors (part 2)
The 3x3 matrix has 4 independent real parameters
where
4 independent parameters
Current experiments are not sensitive to f. Its
assumed f0
From neutrinos to cosmic sources, lecture 5,
2003
23
Mixing of 3 flavors (part 3)
The mixing matrix can be rewritten
f0
rotation by
rotation by
rotation by
From neutrinos to cosmic sources, lecture 5,
2003
24
Oscillations with 3 flavors
With 3 generations there are 3 Dm2s but only
two are independent.
From neutrinos to cosmic sources, lecture 5 2003