NuclearNuclear andand ParticleParticle PhysicsPhysics 11 (PHY211/281)(PHY211/281) HerbstsemesterHerbstsemester 20192019

OlafOlaf SteinkampSteinkamp

36-J-05 [email protected] 044 63 55763 Units

We are dealing with the extremely small, e.g. typical length scale is “femtometer” 1 fm = 10–15 m

KT1 HS19 Basics (2/21) O. Steinkamp ElectronVolt (eV)

SI units not very well suited, e.g.

→ Electron charge e = 1.6 × 10–19 C

–27 → Proton rest mass mp = 1.67 × 10 kg

2 –10 → Proton rest energy Ep = mp c = 1.5 × 10 J

Define “electronVolt” (eV) as our basic unit of energy: energy gained by an electron moved across a potential of 1 V

1 eV = 1.6 × 10–19 J

→ Binding energy of electrons in atoms few eV → Proton rest mass 938 MeV (MeV = 106 eV)

KT1 HS19 Basics (3/21) O. Steinkamp Natural Units

Derived units

Observable Unit in SI units

Energy eV 1.60 × 10–19 J Mass E = mc2 eV / c2 1.78 × 10–36 kg Momentum E = pc eV / c 5.34 × 10–28 kg m/s Time ΔE Δt ≥ ℏ ℏ / eV 6.58 × 10–16 s Distance Δx Δp ≥ ℏ ℏc / eV 1.97 × 10–7 m

KT1 HS19 Basics (4/21) O. Steinkamp Natural Units

Often use “Natural units”, where c ≡ 1 and ℏ = h ≡ 1 2 π

Natural Observable Unit in SI units unit Energy eV eV 1.60 × 10–19 J Mass E = mc2 eV / c2 eV 1.78 × 10–36 kg Momentum E = pc eV / c eV 5.34 × 10–28 kg m/s Time ΔE Δt ≥ ℏ ℏ / eV 1 / eV 6.58 × 10–16 s Distance Δx Δp ≥ ℏ ℏc / eV 1 / eV 1.97 × 10–7 m

→ Energy, mass and momentum have the same unit → Time and distance have the same unit

KT1 HS19 Basics (5/21) O. Steinkamp Natural Units

Often use “Natural units”, where c ≡ 1 and ℏ = h ≡ 1 2 π

Natural Observable Unit in SI units unit Energy eV eV 1.60 × 10–19 J Mass E = mc2 eV / c2 eV 1.78 × 10–36 kg Momentum E = pc eV / c eV 5.34 × 10–28 kg m/s Time ΔE Δt ≥ ℏ ℏ / eV 1 / eV 6.58 × 10–16 s Distance Δx Δp ≥ ℏ ℏc / eV 1 / eV 1.97 × 10–7 m

→ Express time and distance through units of energy, e.g. 1 fm ≈ 1 = 5 GeV−1 200MeV

KT1 HS19 Basics (6/21) O. Steinkamp Natural Units

Often use “Natural units”, where c ≡ 1 and ℏ = h ≡ 1 2 π

Natural Observable Unit in SI units unit Energy eV eV 1.60 × 10–19 J Mass E = mc2 eV / c2 eV 1.78 × 10–36 kg Momentum E = pc eV / c eV 5.34 × 10–28 kg m/s Time ΔE Δt ≥ ℏ ℏ / eV 1 / eV 6.58 × 10–16 s Distance Δx Δp ≥ ℏ ℏc / eV 1 / eV 1.97 × 10–7 m

→ useful relation: ℏ c ≈ 200MeV⋅fm

KT1 HS19 Basics (7/21) O. Steinkamp Electric charge

Electric charge defined through Coulomb force

e2 F C = 2 4 π ϵ0 r Use “Heaviside-Lorentz” units with

ϵ0 ≡ 1 → Dimension of electric charge [q] = [F⋅r 2]1 /2 = [E⋅r ]1/2 = √ ℏ c → Dimensionless in natural units Fine-structure constant

2 2 α ≡ e = e = 1 4 π ℏ c 4 π 137

KT1 HS19 Basics (8/21) O. Steinkamp Some more numbers

Binding energy of nucleons in nuclei few MeV (MeV = 106eV) Binding energy of quarks in nucleons few GeV (GeV = 109eV) Neutrino rest masses < 0.1 eV Electron rest mass 511 keV Rest mass of top quark 173 GeV Mass of 208Pb nucleus/atom 194 GeV Rest mass of W Boson 80 GeV Rest mass of Z Boson 91 GeV Rest mass of Higgs Boson 125 GeV Collision energy in LHC 13 TeV (TeV = 1012eV) Highest energy cosmic ray measured so far 300 EeV (EeV = 1018eV)

KT1 HS19 Basics (9/21) O. Steinkamp Prefixes ] l m t h . s e x i f e r p / s t i n U / u u c / v o g . t s i n . s c i s y h p / / : s p t t h [

KT1 HS19 Basics (10/21) O. Steinkamp Radioactive Decays

Can happen when the sum of masses of produced particles is smaller than mass of decaying particle Some examples:

Nuclear decays Decays of instable particles – – X* → X + γ π → μ νμ A A – π0 → γγ Z X → Z+1 Y + e + νe ρ0 → π+ π– A A + Z X → Z-1 Y + e + νe

A A-4 4 Z X → Z-2 Y + He

There are other rules besides energy conservation, we’ll discuss some of these in later lectures ...

KT1 HS19 Basics (11/21) O. Steinkamp Radioactive Decays

Constant probability λ for a single nucleus/particle to decay For an ensemble of N particles:

d N (t ) = −λ N (t ) dt

N(t ) = N (t=0) e−λ t

Mean lifetime Halflife

τ = 1/ λ t 1/2 = τ ⋅ln2

Intensity of a given source = mean number of decays per time

dN (t ) I(t) = − = N (t=0) λ e−λ t = I (t=0) e−λ t dt

KT1 HS19 Basics (12/21) O. Steinkamp Radioactive Decays

The number of decays measured in a fixed time interval Δt is a random number that follows a … Poisson distribution μ n P(n ∣μ) = e−μ n! with expectation value μ = λ ⋅Δt

Standard deviation σ of the Poission distribution (= uncertainty on the measurement) σ = √ μ ≈ √ n

KT1 HS19 Basics (13/21) O. Steinkamp Lifetime and Decay Width

Stable particles have a well-defined mass m , e.g.

me = 511 keV But for short-lived particles with finite lifetime τ ΔE Δt ≥ ℏ implies that the mass of the particle has an uncertainty Γ = ℏ / τ The probability distribution for the mass is described by a relativistic Breit-Wigner distribution

f (m) ∝ 1 short-lived particles (m2−M 2)2 + M2 Γ2 are also called ”resonances” M is the mean mass of the particle Γ is called the decay width

KT1 HS19 Basics (14/21) O. Steinkamp Lifetime and Decay Width

Plays a role only for very short-lived particles, e.g.

for Z boson: for muon: τ ≈ 3 × 10–25 sec τ ≈ 2.2 × 10–6 sec Γ ≈ 2.5 GeV Γ ≈ 3 × 10–19 GeV ( Γ / m ≈ 2 % ) ( Γ / m ≈ 3 × 10–18 )

Usually quote decay width for very short-lived particles and quote lifetime for longer-lived particles, e.g. from Particle Data Group:

KT1 HS19 Basics (15/21) O. Steinkamp Z boson decay width

Determination of the number of neutrino species at LEP:

Measure the width of the Z mass distribution → width depends on lifetime → lifetime depends on the number of different possible decay modes → number of decay modes depends on the number of neutrino species

Measured width agrees well with three types of neutrinos, i.e. three generations of elementary particles

KT1 HS19 Basics (16/21) O. Steinkamp Special Relativity: Four-Vectors

(Contravariant) four-vector

aμ ≡ (a0 ; a1 , a2 , a3 )

Example: space-time

xμ = (t ; x , y , z) = (t ; x⃗ )

Example: energy-momentum

μ p = (E ; p x , p y , p z) = (E ; p⃗ )

( using natural units with c ≡ 1 )

KT1 HS19 Basics (17/21) O. Steinkamp Special Relativity: Four-Vectors

Covariant four-vector

ν 0 1 2 3 aμ ≡ (a0 ; a1 , a2 , a3 ) = gμ ν a = (a ;−a ,−a ,−a )

with the metric tensor

1 0 0 0 g ≡ 0 −1 0 0 μ ν 0 0 −1 0 ( 0 0 0 −1 )

Using “Einstein notation”, summation over repeated indices is implied

3 ν ν gμ ν a ≡ ∑ gμ ν a ν=0

KT1 HS19 Basics (18/21) O. Steinkamp Special Relativity: Four-Vectors

Scalar product of two four-vectors

μ μ ν 0 0 1 1 2 2 3 3 a bμ = a gμ ν b = (a b −a b −a b −a b )

μ μ aμ b = a bμ

Norm of a four-vector

2 μ μ 0 0 1 1 2 2 3 3 | a| = a aμ = aμ a = (a a −a a −a a −a a )

KT1 HS19 Basics (19/21) O. Steinkamp Special Relativity: Derivative

Contravariant derivative

μ ∂ ≡ ∂ = ∂ ; − ∂ , − ∂ , − ∂ = ∂ ; −∇⃗ ∂ x μ ( ∂ t ∂ x ∂ y ∂ z ) ( ∂ t ) Covariant derivative

∂ ∂ ∂ ∂ ∂ ∂ ⃗ ∂μ ≡ = ; + , + , + = ; ∇ ∂ x μ ( ∂ t ∂ x ∂ y ∂ z ) ( ∂ t ) Derivative of a four-vector aμ

0 μ ∂ a ∂ a = + ∇⃗⋅a⃗ μ ∂ t In particular:

2 2 2 2 2 μ ∂ ∂ ∂ ∂ ∂ ⃗ 2 μ ∂μ ∂ = − − − = −∇ = ∂ ∂μ ( ∂ t 2 ∂ x 2 ∂ y 2 ∂ z2 ) ∂ t 2

KT1 HS19 Basics (20/21) O. Steinkamp Lorentz Transformation

Consider a reference frame that moves at relative speed β = v/c along the x-axis Four-vectors transform from the rest frame to this moving reference frame as μ μ ν x ' = Λ ν x with

γ −β γ 0 0 Λμ ≡ −β γ γ 0 0 ν 0 0 1 0 ( 0 0 0 1 ) and

γ = 1 √ 1−β2

KT1 HS19 Basics (21/21) O. Steinkamp