ParticleParticle PhysicsPhysics -- MeasurementsMeasurements andand TheoryTheory

OutlineOutline

Natural Units Relativistic Kinematics Particle Physics Measurements Lifetimes Resonances and Widths Scattering Cross section Collider and Fixed Target Experiments Conservation Laws Charge, Lepton and Baryon number, Parity, Quark flavours Theoretical Concepts Quantum Field Theory Klein-Gordon Equation Anti-particles Yukawa Potential Scattering Amplitude - Fermi’s Golden Rule Matrix elements

Nuclear and Particle Physics Franz Muheim 1 ParticleParticle PhysicsPhysics UnitsUnits

Particle Physics is relativistic and quantum mechanical Î c = 299 792 458 m/s Î ħ = h/2π = 1.055·10-34 Js Length size of proton: 1 fm = 10-15 m Lifetimes as short as 10-23 s Charge 1 e = -1.60·10-19 C Energy Units: 1 GeV = 109 eV -- 1 eV = 1.60·10-19 J use also MeV, keV Mass in GeV/c2, rest mass is E = mc2

NaturalNaturalNatural UnitsUnitsUnits Set ħ = c = 1 Î Mass [GeV/c2], energy [GeV] and momentum [GeV/c] in GeV Î Time [(GeV/ħ)-1], Length [(GeV/ħc)-1] in 1/GeV area [(GeV/ħc)-2] Useful relations ħc = 197 MeV fm ħ = 6.582 ·10-22 MeV s

Nuclear and Particle Physics Franz Muheim 2 ParticleParticle PhysicsPhysics MeasurementsMeasurements How do we measure particle properties and interaction strengths? Static properties Mass How do you weigh an electron? Magnetic moment couples to magnetic field Spin, Parity Particle decays Lifetimes Force Lifetimes Resonances & Widths Strong 10-23 -- 10-20 s Allowed/forbidden El.mag. 10-20 -- 10-16 s Decays Weak 10-13 -- 103 s Conservation laws Scattering Elastic scattering e- p → e- p Inelastic annihilation e+ e- → µ+ µ- Cross section total σ Force Cross sections Differential dσ/dΩ Strong O(10 mb) Luminosity L El.mag. O(10-1 mb) Particle flux Weak O(10-1 pb) Event rate N

Nuclear and Particle Physics Franz Muheim 3 RelativisticRelativistic KinematicsKinematics

Basics µ ⎛ E ⎞ 4-momentum p = ⎜ , px , p y , pz ⎟ ⎝ c ⎠ Invariant mass 2 µ 2 r 2 2 2 p = p ⋅ pµ = ()E / c − p = m c Four-vector notation Useful Lorentz boosts relations set ħ = c = 1 invariant mass γ = E/mc2 = E/m m2 = E2 –p2 γβ = pc/mc2 = p/m γ = 1/√(1- β2) β = pc/E = p/E β = √(1 -1/γ2)

2-body decays

P0 → P1 P2

work in P0 rest frame µ r µ r µ r p = (m0 ,0) p1 = ()E1, p1 p2 = (E2 , p2 ) 2 µ µ 2 2 2 p2 = ()p − p1 = p + p1 − 2 p ⋅ p1 2 2 2 m2 = m0 + m1 − 2m0 E1 2 2 2 m0 + m1 − m2 r r E1 = p1 = p2 2m0 r + + p µ = (m ,0) p µ = (E , pr ) p µ = (E , pr ) Example: π →µ νµ π 1 µ µ 2 v ν 2 2 + m + m work in π rest frame E = π µ =109.8 MeV µ 2m 2 π use mν = 0 pr = E 2 − m2 = 29.8 MeV/c µ µ µ

Nuclear and Particle Physics Franz Muheim 4 LifetimesLifetimes

Decay time distribution Mean lifetime dΓ ⎛ t ⎞ 1 = Γ exp⎜ − ⎟ Γ = τ = dt ⎝ τ ⎠ τ aka proper time, eigen-time of a particle Lifetime measurements In laboratory frame Decay Length L = γβcτ + - Example: Bd → π π in LHCb experiment ≈ 7 mm Average B meson energy

≈ 80 GeV + + + + B π → µ ν µ µ → e ν eν µ Î τ = 1.54 ps

Example: π+ discovery Decay sequence + + + + π → µ ν µ µ → e ν eν µ Emulsions exposed to Cosmic rays

Nuclear and Particle Physics Franz Muheim 5 ResonancesResonances andand WidthsWidths

Strong Interactions Production and decay of particles Lifetime τ ~ 10-23 scτ ~ O(10-15 m) unmeasurable Heisenberg’sHeisenberg’s UncertaintyUncertainty PrinciplePrinciple ∆E∆t ≈ h Time and energy measurements are related Natural width Energy width Γ and lifetime τ of a particle Γ = ħ/τ → Width Γ = O(100 MeV) measurable Example - Delta(1232) Resonance Production π + p → ∆++ → π + p Peak at Energy E = 1.23 GeV (Centre-of-Mass) Width Γ = 120 MeV Lifetime τ = ħ/Γ ≈ 5·10-24 s

Nuclear and Particle Physics Franz Muheim 6 ScatteringScattering

Fixed Target Experiments a + b → c + d + …

na # of beam particles va velocity of beam particles nb # of target particles per unit area

Incident flux F = nava CrossCross SectionSection effective area of any scattering happening normalised per unit of incident flux depends on underlying physics What you want to study dN # of scattered particles in solid angle dΩ dσ/dΩ differential cross section in solid angle dΩ σ total cross section ddσσ 1 1dNdN dN = nava nbdσ = Fnbdσ = Ldσ ⇒ == dΩ L dΩ L Luminosity dΩ L dΩ dσ N N Event rate σ = dΩ ⇒ N =σL=σ ∫ dΩ L Event Rate N LuminosityLuminosity Incident flux times number of targets Depends on your experimental setup 1 barn = 1 b = 10−24 cm2 Luminosity [L] = 1030...34 cm−2s−1 Event Rate = Luminosity times Cross Section

Nuclear and Particle Physics Franz Muheim 7 ScatteringScattering

Centre-of-Mass Energy a + b → c + d + … Collision of two particles s is invariant quantity Mandelstam µ µ 2 2 r r 2 s = ()p1 + p2 = (E1 + E2 ) − (p1 + p2 ) variable 2 2 = p1 + p2 + 2 p1 ⋅ p2 2 2 r r = m1 + m2 + 2()E1E2 − p1 p2 cosθ

ECoM = s centre-of-mass energy Total available energy in centre-of-mass frame

ECoM is invariant in any frame, e.g. laboratory Energy Threshold

for particle production ECoM = s ≥ ∑ m j j=c,d ,...

Fixed Target Experiments

µ r µ r p1 = (Elab , p1 ) p2 = (m2 ,0)

2 2 ECoM = s = m1 + m2 + 2Elabm2 ⇒ ECoM ≅ 2Elabm2 if Elab >> mi Example: 100 GeV proton onto proton at rest

ECoM = √s = √(2Epmp) = 14 GeV Most of beam energy goes into CoM momentum and is not available for interactions

Nuclear and Particle Physics Franz Muheim 8 ScatteringScattering

Collider Experiments

Head-on collisions of two particles

θ = 1800 2 2 r r s = m1 + m2 + 2(E1 E2 − p1 p2 cosθ )

2 2 r r ECoM = m1 + m2 + 2()E1 E2 + p1 p2 ⇒ ECoM ≅ 4E1 E2 if Ei >> mi All of beam energy available for particle production Example LEP - Large Electron Positron Collider at CERN 100 GeV e- onto 100 GeV e+ Centre-of-mass energy

ECoM = √s = 2E = 200 GeV

Cross section σ(e+ e- → µ+ µ-) = 2.2 pb Luminosity ∫Ldt = 400 pb-1 Number of recorded events N = σ ∫Ldt = 870

Nuclear and Particle Physics Franz Muheim 9 ConservationConservation LawsLaws

Noether’s Theorem Every symmetry has associated with it a conservation law and vice-versa Energy and Momentum, Angular Momentum conserved in all interactions Symmetries – translations in space and time, rotations in space Charge conservation Well established -21 |qp + qe| < 1.60·10 e Valid for all processes Symmetry – gauge transformation Lepton and Baryon number (L and B) |L+B| conservation = matter conservation Proton decay not observed (B violation)

Lepton family numbers Le, Lµ, Lτ conserved Symmetry – mystery Quark Flavours, Isospin, Parity conserved in strong and electromagn processes Violated in weak interactions Symmetry – unknown

Nuclear and Particle Physics Franz Muheim 10 TheoreticalTheoretical ConceptsConcepts

StandStandardStandardard Model ModelModel of Particle ofof ParticleParticle Physics PhysicsPhysics

Quantum Field Theory (QFT) Describes fundamental interactions of Elementary particles Combines quantum mechanics and special relativity Very small ∆x ∆p ≈ħc Classical Quantum Physics mechanics Very fast Special Quantum v → c relativity field theory

Natural explanation for antiparticles and for Pauli exclusion principle Full QFT is beyond scope of this course Introduction to Major QFT concepts Transition Rate Matrix elements Feynman Diagrams Force mediated by exchange of bosons

Nuclear and Particle Physics Franz Muheim 11 KleinKlein--GordonGordon EquationEquation

Schroedinger Equation

For free particle pˆ 2 ψ = Eˆψ non-relativistic 2m 2 st − ∂ 1 order in time derivative h ∇ 2ψ = i ψ 2m h ∂t 2nd order in space derivatives not Lorentz-invariant Klein-Gordon (K-G) Equation Start with relativistic equation ∂ r E2 = p2 + m2 (ħ = c = 1) E → i pr → −i ∇ h ∂t h Apply quantum mechanical operators

⎛ ∂ 2 r ⎞ ⎛ ∂ 2 r ⎞ ⎜ − + ∇ 2 ⎟ψ = m 2ψ or ⎜ − ∇ 2 + m 2 ⎟ψ = 0 ⎜ 2 ⎟ ⎜ 2 ⎟ ⎝ ∂t ⎠ ⎝ ∂t ⎠

2nd order in space and time derivatives Lorentz invariant Plane wave solutions of K-G equation µ ν 2 2 ψ ()x = N exp(− ipν x )⇒ E = ± p + m Î negative energies (E < 0) also negative probability densities (|ψ|2 < 0) Negative Energy solutions Î Dirac Equation, but –ve energies remain Î Antimatter Nuclear and Particle Physics Franz Muheim 12 KleinKlein--GordonGordon EquationEquation

Interpretation K-G Equation is for spinless particles Solutions are wave-functions for bosons Time-Independent Solution Consider static case, i.e. no time derivative ∇ 2ψ = mψ Solution is spherically symmetric g 2 ψ (r) = − exp()− mr 4π r Interpretation - Potential analogous to Coulomb potential Force is mediated by exchange of massive bosons

YukawaYukawaPotentialPotentialPotential Introduced to explain nuclear force g 2 ⎛ r ⎞ V (r) = − exp⎜ − ⎟ R = h 4π r ⎝ R ⎠ mc g strength of force – “strong nuclear charge” m mass of boson R Range of force see also nuclear physics For m = 0 and g = e → Coulomb Potential

Nuclear and Particle Physics Franz Muheim 13 AntiparticlesAntiparticles

Klein-Gordon & Dirac Equations predict negative energy solutions Interpretation - Dirac Vacuum filled with E < 0 electrons 2 electrons with opposite spins per energy state - “Dirac Sea” Hole of E < 0, -ve charge in Dirac sea -> antiparticle E > 0, +ve charge -> positron, e+ discovery (1931) Predicts e+e- pair production and annihilation

Modern Interpretation – Feynman-Stueckelberg E < 0 solutions: Negative energy particle moving backwards in space and time correspond to Antiparticles Positive energy, opposite charge moving forward in space and time

exp[− i((−E)(−t) − (− pr)⋅(− xr))] = exp[]− i()(Et − pr ⋅ xr

Nuclear and Particle Physics Franz Muheim 14 ScatteringScattering AmplitudeAmplitude

Transition Rate W Scattering reaction a + b → c + d W = σ F Interaction rate per target particle related to physics of reaction Fermi’sFermi’sFermi’s GGoldenolden RuleRuleRule Matrix Element Mfi scattering amplitude 2π 2 Density ρf W = M fi ρ f h # of possible final states “phase space” non-relativistic 1st order time-dependent perturbation theory see e.g. Halzen&Martin, p. 80, Quantum Physics

MatrixMatrix ElementElement Contains all physics of the interaction ) M fi = ψ f H ψ i

Hamiltonian H is perturbation – 1st order Incoming and outgoing plane waves works if perturbation is small Born Approximation Nuclear and Particle Physics Franz Muheim 15 MatrixMatrix ElementElement

Scattering in Potential Example: e- p → e- p Incoming and outgoing plane waves Matrix element qr = pr − pr Momentum transfer i f

** rr 33rr M fi = ψ fVV((rr))ψψidd rr fi ∫ f i 1 = exp()− ipr ⋅ r V (r )exp()ipr ⋅ rr d 3r N 2 ∫ f i 1 = exp()iqr ⋅ r V (rr)d 3r qr = pr − pr N 2 ∫ i f

Mfi (q) is Fourier transform of Potential V(r) g 2 Scattering in Yukawa Potential V (r) = − exp()− mr 4π r 2 g ∞ ππ2 exp(− mr ) M = − exp()i qr r cosθ r 2dr sinθdθdφ fi 4π ∫∫00∫0 r 2 g ∞ = − ()exp()i qr r − exp()− i qr r exp()− mr dr 2i qr ∫0

2 g 2 = − g M = − 2 2 Propagator fi m 2 + qr 2 ()m + q 2 2) term in Mfi 1/(m +q Cross section dσ 2 1 dσ 1 ∝ M ∝ 2 ⇒ ∝ 4 m = 0 dΩ ()m 2 + qr 2 dΩ q Result still holds relativistically µ r r 4-momentum transfer q = (Ei − E f , pi − p f )

Nuclear and Particle Physics Franz Muheim 16