LECTURE 4 INTERACTIONS & CROSS SECTIONS PHY492 Nuclear and Elementary Particle Physics

LECTURE 4
INTERACTIONS & CROSS SECTIONS
PHY492 Nuclear and Elementary Particle Physics
Last Lecture
µ- e- Feynman Diagrams
e- + e+ →µ- + µ+ Rules :
1. Initial state (left) and final state (right)
2. Fermions (electrons) are drawn as solid lines
γ e+ µ+ 3. Bosons (photons) are represented by wiggly (broken) lines
4. Arrowhead pointing to the right (left) indicate particles (anti particles)
does not indicate the particle’s direction of motion
5. Lines that end at boundaries are free particles
6. Energy (or momentum) is conserved at a vertex
7. Charge is conserved at a vertex
January 15, 2014 PHY492, Lecture 4 2 Evolution of Forces Through Time
January 15, 2014 PHY492, Lecture 4 3 Virtual Particles
Quantum mechanics allows for a very interesting effect for the lifetime and
behavior of particles.
Consider Heisenberg's uncertainty principle:
ΔEΔt ≥ !
Propagators might not satisfy the normal energy-mass relationship:
2
m2 = E 2 − p2
Any particle that violates this relationship (off mass shell) is referred to as a virtual
particle.
•  Virtual particles may have zero, non-zero or even negative “mass”.
•  Virtual particles don’t manifest in reality, so they can only be internal lines
A consequence: Coulomb forces and magnetic fields exist due to the exchange of
virtual photons.
January 15, 2014 PHY492, Lecture 4 4 Range of Forces
the A + B → A + B reaction
with the exchange of a boson X
A
A
g
From the rest frame of A’ (pA’ =0, EA’=MA’c2),
momentum conservation at the vertex
is given; pA = pX = p
X
B B g
thus EA2 = MA2c4 + pA2c2 = MA2c4 + p2c2
EX2 = MX2c4 + pX2c2 = MX2c4 + p2c2
ΔE = ( EA + EX ) – EA’ = [(MA2c4 + p2c2)1/2 - MAc2] + (MX2c4 + p2c2)1/2
≥ MX 2
≥ 0 c ≥ MXc2 (energy is not conserved at the same time)
From Heisenberg uncertainty principle, this is allowed for January 15, 2014 PHY492, Lecture 4 τ ≈!
ΔE
≤ ! M X c2
5 Range of Forces (2)
r ≤ R = h/MXc2 × c (R: range of the interaction)
= hc / MXc2 useful relation
= 197MeV·fm / MXc2
Electromagnetic interaction
hc = 197 MeV · fm Bosons
Mass
R
γ
0
∞
Weak interaction
W, Z
MWc2 = 80.4 GeV
MZc2 = 91.2 GeV
Strong interaction
(nuclear)
π
Mπ±c2 = 139 MeV
Mπ0c2 = 134 MeV
January 15, 2014 PHY492, Lecture 4 2×10-18 m
1.5×10-15 m
(1.5fm ≈
size of nuclei)
6 Particle Scattering
Consider two particles that interact in some manner.
Eg, Compton scattering of a photon on an electron.
We can calculate the probability that different outcomes can happen.
Sometimes no interaction will occur, other times an interaction will create
non-trivial final-state kinematics.
What about a beam of particles?
January 15, 2014 PHY492, Lecture 4 7 Reaction Rates (Particle Physics)
Cross section (σ) : the probability of a collision of occurring
between two particles (beam and target)
Reaction rate (W = interactions/time)
S ∞
∞
∞
σ
Ntarget : number of targets
illuminated by the beam
Flux J : beam rate per unit area
J = Nb · Vb
Nb Vb Ntarget (Nb : number density of beam particles, Vb: beam velocity)
Luminosity: L = J Ntarget W = σ · Ntarget · J
= σ · L 1 barn = 10-28 m2 = 10-24 cm2
Luminosity has units of Particles / Area / Time.
Usually quoted as cm-2 s-1 or equivalently “inverse barns per second”
January 15, 2014 PHY492, Lecture 4 8 Reaction Rates (Nuclear Physics)
Cross section (σ) : the probability of a collision of occurring
between two particles (beam and target)
Reaction rate (W = interactions / time)
S ∞
∞
∞
σ
Ntarget : number of targets
illuminated by the beam
Flux J : beam rate per unit area
J = Nb · Vb
Vb Nb Ntarget (Nb : number density of beam particles , Vb: beam velocity)
Beam intensity I = J S (S: beam area) W =
=
=
=
(nt: number of target particles/unit volume)
σ · Ntarget · J
(t: thickness of target)
σ · Ntarget · I/S
(ρ: target density)
(NA: Avogadro’s constant, MA: mass)
σ · ( nt · t ) · I
σ · ρ · (NA/MA) · t · I (e.g.) Ibeam = 105 cps, Target 9Be t=0.1cm (ρ=1.85g/cm3), σ =10mb, what is the reaction rate?
January 15, 2014 PHY492, Lecture 4 9 Differential cross section
dW = J ·Ntarget ·
dσ(θ,φ)
· dΩ
dΩ
dθ
(differential cross section)
sinθ
θ
dφ
σ
=
dσ
dΩ , dΩ = dθ · sinθ · dφ
dΩ
2π
=
Ring Area
2πsinθdθ
π
dσ
dφ sinθ · dθ · dΩ
0 0 January 15, 2014 PHY492, Lecture 4 10 The Matrix Element
Recall our notion of Feynman Diagrams as tools for calculations.
We can use the information encoded in the Feynman diagram to
calculate things like the reaction’s cross section, the properties of
the decay of particles and resonance phenomena.
Cross sec&on depends on the square of M 2
dσ
1 qf
2
=
M
dΩ 4π 2 vi v f
Par&cle 4-­‐momentum Propagator info Decay rates depend on the square of M 1
2
dΓ(A → a +!n) =
M dQ
2mA
January 15, 2014 Matrix Element for a given reacDon ( M ) is encoded in the Feynman diagram. Also known as the Sca8ering Amplitude. Interac&on Strength PHY492, Lecture 4 11 The Matrix Element
Recall our notion of Feynman Diagrams as tools for calculations.
We can use the information encoded in the Feynman diagram to
calculate things like the reaction’s cross section, the properties of
the decay of particles and resonance phenomena.
A & B: Incoming par&cle wavefunc&ons. P: Propagator wavefunc&on C & D: Outgoing par&cle wavefunc&ons. I1 & I2: Interac&on strengths M ∝ ( A×I1×B ) ( P ) ( C×I2×D ) P A C I1 B January 15, 2014 PHY492, Lecture 4 I2 D 12 The Matrix Element
Recall our notion of Feynman Diagrams as tools for calculations.
We can use the information encoded in the Feynman diagram to
calculate things like the reaction’s cross section, the properties of
the decay of particles and resonance phenomena.
If there are N Feynman diagrams for the same final state, then the matrix elements sum linearly: e+
e+
e-
e-
e+
e+
e-
e-
Mtot = M1 + M2 + …. + Mn The matrix element can be nega&ve, giving rise to interference effects. For example: dσ
2
2
2
∝ M tot ∝ M1 + M 2 ± 2 M1M 2
dΩ
Interference can be construcDve or destrucDve January 15, 2014 PHY492, Lecture 4 13 Particle Decays
In general, all particles are unstable and can decay to other particles. Rough rules:
1)  You cannot decay to a higher energy (mass) final state. A W boson cannot create real
(ie, not virtual) W→tb decays because MW < MTop + MBottom
2)  If there is not a lower energy/mass state, the particle cannot decay. Electrons do not
decay because there is no lighter charged lepton.
3)  Must satisfy conservation rules. Eg, charge conservation, lepton/quark number, etc.
Each par&cle decay is associated with a life&me τ and a “natural decay width” Γ Γ= 1
τ
Par&cle abundance has exponen&al behavior: N(t) = N 0 e−Γt
Decay rates depend on the square of M dΓ(A → a +!n) =
January 15, 2014 1
2
M dQ
2mA
PHY492, Lecture 4 14 Particle Decays
In general, all particles are unstable and can decay to other particles. Rough rules:
1)  You cannot decay to a higher energy (mass) final state. A W boson cannot create real
(ie, not virtual) W→tb decays because MW < MTop + MBottom
2)  If there is not a lower energy/mass state, the particle cannot decay. Electrons do not
decay because there is no lighter charged lepton.
3)  Must satisfy conservation rules. Eg, charge conservation, lepton/quark number, etc.
If a par&cle can exhibit many different decays (eg, Z→e
+e-­‐, Z→µ+µ-­‐, etc) then the width is the sum of the par&al widths. Γ = ∑ Γi
This natural width can be observed in experiments that produce par&cles “on resonance”. Breit-­‐Wigner distribu&on has a width related to the natural width: 1
Γ
P(E) =
2π (E − M )2 + Γ 2 4
January 15, 2014 P: probability to produce the parDcle E: center-­‐of-­‐mass energy of the collision M: mass of the parDcle being produced PHY492, Lecture 4 15 Units
-  Range of forces : R = hc/Mxc2 = 197MeV·fm / Mxc2
strong interaction
weak interaction
Mπc2 = 134MeV → R=1.5fm (fm=10-15m)
MZc2 = 90GeV → R=2×10-18m ≈ 10-3 fm
-  Cross section : b (barn) = 10-28 m2 = 10-24 cm2, mb = 10-3 b
-  Energy : 1 eV = energy required to raise the electric potential of
an electron by one volte
= 1.6×10-19(C) × 1(V) = 1.6×10-19 (J, in S.I. unit)
1 keV = 103 eV, 1 MeV = 106 eV, 1 GeV = 109 eV
-  Mass : 1 MeV/c2 = 1.78×10-30 (kg), (S.I. unit)
-  Mass unit (nuclear physics) : 1 u = Mass(12C)/12 = 931.5 MeV/c2
- Natural units (particle physics) : h = c = 1 (for simplicity)
January 15, 2014 PHY492, Lecture 4 16