Download Neutrinos and Weak Interactions, Lecture 2

Neutrinos & Weak Interactions
Lecture 2
Early history of the neutrino
Evgueni Goudzovski
School of Physics and Astronomy
University of Birmingham, United Kingdom
[email protected]
Feynman diagrams
Quantum field theories: interactions between particles
proceed discontinuously by exchange of mediator particles.
space
A+B→C+D
A
Amplitude of the process (natural units):
C
ga
x
virtual
“mediator”
particle
gb
B
initial
state
D
“how the
interaction
happened”
final
state
time
• ga, gb: the fundamental
coupling strengths at the vertices;
• Momentum transfer (4-vector):
q2 = (pA − pC)2 = (pD − pB)2;
• 1/(q2 − mx2): the boson propagator.
The reaction rate is
proportional to |A|2
1
Electromagnetic processes (1)
Force carrier: photon (γ), coupling constant:
(α ~ probability assigned to a vertex
of a Feynman diagram)
Example 1: e+e− annihilation to photons
“Perturbative theory”:
very precise predictions
γ
e+
e−
e±
e+
e−
In general,
is process of order
γ
γ
γ
γ
(n≥2)
 Higher-order electromagnetic
processes are suppressed
2
Electromagnetic processes (2)
Example 2: Bremsstrahlung
e−
γ
Allowed in the presence of a nucleus
(energy-momentum is conserved due to nuclear recoil)
Reaction rate:
γ
e−
N
N
(Z: electric charge of the nucleus)
 Practical uses: e.g. X-ray generation in medical imaging
Example 3: e+e− pair production (photon conversion)
is also allowed in the field of a nucleus
Reaction rate:
 The dominant photon interaction at
high energies (E>100MeV);
 Z-dependence is exploited for photon detection
(e.g. Pb calorimeter pre-showers)
e+
γ
γ
N
e−
N
3
The weak charged current
Observable in processes:
 involving neutrinos (no electric/strong charges), or
 quark flavour change (forbidden by other interactions).
Massive force carriers
(mW,Z ≈ 90 GeV/c2):
limited range (2×10−3 fm).
Point (zero-range) approximation
is valid in many cases
(e.g. for weak decays).
The Fermi constant:
Example: electron capture
p
n
n
g
W±
propagator
(g/mW)2
g
e−
p
νe
e−
νe
4
Lepton decays (1)
Electron (e−) is the lightest charged particle:
necessarily stable by conservation of electric charge
Allowed decays of charged leptons
(conserving energy, lepton flavour and lepton number)
µ−
(+conjugate decay modes
for antileptons)
νµ
e−
W−
(Hadronic decays
of the tauon
are also allowed,
e.g. τ−→π−π−π+ντ)
νe
0.5 MeV
106 MeV
1.8 GeV
80 GeV
low energy regime
The decay rate is governed by two
parameters: GF and the initial lepton mass m
5
Lepton decays (2)
Weak decay rate:
dimension: [GeV−4]
Lifetime and the total decay rate or width:
[GeV−1]
[GeV]
(lecture 1)
Therefore,
(“Sargent’s rule”)
Lepton universality:
All leptons have identical coupling strength to the W± bosons, i.e.
K is independent of the mother and daughter lepton types.
For example,
 In agreement with experiment (~0.1% precision)
6
Radioactivity
Deflection of α, β, γ rays by magnetic field
Radioactive decays
(discovered by H. Becquerel, 1896;
Nobel Prize 1903):
spontaneous emission
of ionizing particles
by unstable atomic nuclei.
B
α=4He nucleus
γ=photon
β=electron (e−)
NB: varying curvature
Some types of radioactive decay
α decay:
(A, Z) → (A−4, Z−2) + 4He++
β− decay:
(A, Z) → (A, Z+1) + e−
β+ decay:
(A, Z) → (A, Z−1) + e+
Electron capture:
(A, Z) + e− → (A, Z−1)
Isomeric transition:
(A, Z)* → (A, Z) + γ
Proton or neutron emission.
(A: mass number; Z: atomic or proton number)
7
Two-body decays
Energy-momentum conservation (in terms of 4-momenta):
In rest frame of A (
),
pC
pB
A
Momentum
conservation:
“back-to-back”
decay in the
A rest frame
4 equations with 4 unknowns
a unique solution:
if mB≪mA, mC
and mA≈mC
(relevant for α-decays)
≈ mA − mC
(the expression for EC is similar)
8
Three-body decays
Various ways to satisfy energy-momentum conservation:
momenta of daughters are not uniquely defined.
Allowed kinematic range (in the A rest frame):
pC
A
pD
pB
Momentum
conservation:
coplanar momenta
in A rest frame
B is at rest:
C and D are back-to-back
pD
A,B
pc
“Endpoint”:
another back-to-back
configuration
pC
pD
A
pB
For mD=0, the endpoints of the B and C energy spectra are
equal to the fixed energy of a 2-body decay A  B + C
9
Typical α- and β-decay spectra
210Bi
Arbitrary units
Arbitrary units
α-decay spectra
Two-body: (A, Z) → (A−4, Z−2) + 4He++
β-decay spectrum
(τ1/2=5 days)
(J. Chadwick,
1914)
Powerful research tool
for early atom exploration
(e.g. Rutherford exp’t)
4
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
α energy, MeV
Characteristic discrete lines,
as expected for a 2-body decay
Electron kinetic energy, MeV
Continuous spectra observed
for all beta-emitters
β-spectrum was measured with two methods:
(1) by the deflection in magnetic field; (2) by the produced ionization.
Hypothesis 1 (Rutherford & Meitner): electrons from β decay lose energy
via secondary interactions with the material of the source.
Hypothesis 2: a third unseen particle is emitted.
10
Ellis & Wooster experiment
+?
210Bi
β-decay spectrum
Ethermal
Emean=0.39MeV
(Cambridge, 1927)
210Bi
210Po
= 83p + 127n + 83e
= 84p + 126n + 84e
Assume a two-body decay.
Convert from the atomic mass
to the mass of a nucleus:
Ee = (m(210Bi)−83me) − (m(210Po)−84me)
210Bi) − m(210Po) + m ,
=
m(
e
Emax=1.16MeV
Eekin = Ee − me =
= m(210Bi) − m(210Po) = 1.16 MeV.
Electron kinetic energy, MeV
A calorimetric measurement by Ellis and Wooster:
average thermal energy emitted per
210Bi
decay is Ethermal = (0.34±0.03) MeV.
“a striking confirmation of the hypothesis that the continuous spectrum
is emitted as such from the nucleus” [Nature 119 (1927) 563]
Eendpoint ≈ E2-body: the third daughter, if any, is light.
11
The postulation of the neutrino
Problems with β-decay in 1920s:
 Continuous rather than discrete beta-spectrum (the “energy crisis”):
Niels Bohr’s proposal:
energy conservation is valid only statistically (!)
 Angular momentum non-conservation:
β-decay never changes nuclear spin by ½.
At the same time, spins of n, p, e− were known to be ½.
½ → ½ + ½ (?)
n → p + e−:
Proposal by Wolfgang Pauli (4th December 1930):
a new light spin-½ particle produced with the electron, escaping detection.
210Bi
→
210Po
+ e− + νe
n → p + e− + νe
(called “neutron” by W.Pauli in 1930; renamed “neutrino” by E.Fermi in 1933)
12
Further evidence for the neutrino
(Powell, Fowler, Perkins; 1949)
e±
“kink”
Discoveries made by the Bristol group
(Powell et al.) in the late 1940s
by photographic emulsion method:
apparent momentum non-conservation
(A)
µ±
(B)
π±
“kink”:
escaping neutrino
(now known to be a neutrino in case A,
and two neutrinos in case B)
13
Nuclear reactors as νe sources
Nuclear fission: splitting of a nucleus into smaller nuclei.
Discovered by Otto Hahn in 1938 (Nobel Prize for Chemistry 1945).
n
An example of a neutron-induced fission reaction:
+ 200 MeV
92p+144n
98p+138n
~106 times greater than
in chemical reactions
Neutron production: chain reaction
First nuclear reactor: December 1942, University of Chicago.
First atomic bomb: July 1945, New Mexico, US.
Effectively, 6 neutron to proton conversions:
140Ce
94Zr
Antineutrino production rate in a reactor:
(typical modern commercial reactor power is ~1 GW)
n
Fν / Pth ~ 6 / 200 MeV = 6 / (200×1.602×10−13 J) ~ 1020 s−1GW−1.
n
Nuclear reactors and bombs: first artificial high intensity antineutrino sources
14
Neutrinos or anti-neutrinos?
Chart of nuclides
238U
208Pb
Neutron-to-proton ratio grows
as a function of atomic number.
Reason: mutual Coulomb repulsion of
the protons.
Examples:
16O =
8p + 8n;
94Zr = 40p + 54n;
208Pb = 82p + 126n;
238U = 92p + 146n;
94Zr
N(n)/N(p)=1
N(n)/N(p)=1.35
N(n)/N(p)=1.54
N(n)/N(p)=1.59
Nuclear fission necessarily leads
to neutron-to-proton conversion.
By electric charge and
lepton flavour conservation,
anti-neutrino production:
8O
15
Inverse beta-decay
Beta decays:
(mn > np: free neutron is unstable, τ = 882 s)
(proton is stable; decay occurs within nuclei)
Neutrinos are created in beta-decays.
Could they be absorbed in the reverse process?
(recall bremsstrahlung
and photon conversion
)
Crossing symmetry:
 antiparticles are equivalent to particles going backwards in time.
Inverse beta decay (IBD): related to beta-decays by crossing symmetry
IBD: production of charged particles (e±).
A possible tool for indirect (anti)neutrino detection.
16
Reactor νe detection via IBD
Energy threshold for
:
(see lecture 1)
Antineutrinos/fission
Reactor νe energy spectra
Detection rate
Flux
Eth
Antineutrino energy, MeV
Cross-section
Antineutrino energy, MeV
17
Interaction cross-section
Number of “interaction centres” in a volume:
n: density of the “interaction centres” [cm–3]
Area S
The fraction of area covered by interaction centres:
(=probability for an incoming particle to interact)
Thickness ∆L
σ: geometrical cross-section of an interaction centre [cm2]
Fluence
: the number of particles that intersect a unit area
[cm−2]
The number of reactions per interaction centre:
Differential equation for the fluence Φ:
18
Beam attenuation
;
Beam attenuation law:
Mean free path:
[ (cm–3×cm2)−1 = cm ]
Physical meaning of the mean free path:
 A beam is attenuated by a factor of e=2.71 over the mean free path 19
Neutrino interaction cross-section
Elastic scattering:
Dimensional estimate assuming
:
(ECM is the only available Lorentz-invariant scale parameter)
Dimensional analysis:
Energies in the CM frame (ECM) and the lab frame (Eν)
Therefore,
[see lecture 1]:
20
The numerical result
Natural units:
[unit: GeV−2]
Practical units:
[Unit: GeV−2 × (GeV×cm)2 = cm2]
The cross-section has a linear energy-dependence
Numerically,
21
scattering cross-section
We expect:
σ(1 MeV) ~ 10−43 cm2, σ(1 GeV) ~ 10−40 cm2
 in agreement with data!
10−37 cm2
10−40 cm2
10−43 cm2
1 MeV
1 GeV
arXiv:1305.7513
Neutrino energy, eV
22
Neutrino mean free path
Mean free path of a typical reactor/solar (~1 MeV) (anti)neutrino in rock:
n: density of protons [cm−3].
ρ: density of matter [g cm−3].
About half of the nucleons are protons.
(~ distance to α Canis Minoris)
Consider a ~1 MeV neutrino produced in the Solar core.
Probability of interaction before leaving Sun:
(average Solar density = 1.4 g/cm3)
 Low energy neutrinos are direct probes into
the Sun’s (and Earth’s) interior (but not into neutron stars)
23