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Particle Physics Measurements and Theory Outline Outline 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 Particle Physics Units 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 Natural Natural Units Units 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 Nuclear and Particle Physics ħ = 6.582 ·10-22 MeV s Franz Muheim 2 Particle Physics Measurements 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 Resonances & Widths Allowed/forbidden Decays Conservation laws Force Lifetimes Strong 10-23 -- 10-20 s El.mag. 10-20 -- 10-16 s Weak 10-13 -- 103 s Scattering Elastic scattering e- p → e- p Inelastic annihilation e+ e- → µ+ µCross section total σ Differential dσ/dΩ Luminosity L Particle flux Event rate N Nuclear and Particle Physics Force Cross sections Strong O(10 mb) El.mag. O(10-1 mb) Weak O(10-1 pb) Franz Muheim 3 Relativistic Kinematics Basics ⎛E ⎞ p µ = ⎜ , p x , p y , pz ⎟ ⎝c ⎠ 4-momentum Invariant mass Four-vector notation r 2 p 2 = p µ ⋅ pµ = ( E / c ) − p 2 = m 2 c 2 Useful Lorentz boosts relations set ħ = c = 1 γ = E/mc2 = E/m γβ = pc/mc2 = p/m β = pc/E = p/E invariant mass m2 = E2 – p2 γ = 1/√(1- β2) β = √(1 -1/γ2) 2-body decays P0 → P1 P2 work in P0 rest frame ( r p µ = m0 ,0 ( ) r p1µ = (E1 , p1 ) p22 = p µ − p1µ ) 2 r p2µ = (E2 , p2 ) = p 2 + p12 − 2 p ⋅ p1 m22 = m02 + m12 − 2m0 E1 m02 + m12 − m22 E1 = 2m0 Example: r r p1 = p2 π+→µ+ νµ work in rest frame use mν2 = 0 π+ Nuclear and Particle Physics ( r p µ = mπ ,0 Eµ = ) mπ2 + mµ2 2mπ r r p1µ = (Eµ , pµ ) p2µ = (Ev , pν ) = 109.8 MeV r pµ = E µ2 − mµ2 = 29.8 MeV/c Franz Muheim 4 Lifetimes Decay time distribution Mean lifetime dΓ ⎛ t⎞ = Γ exp⎜ − ⎟ dt τ = <dΓ/dt> ⎝ τ⎠ aka proper time, eigen-time of a particle Γ= 1 τ Lifetime measurements In laboratory frame Decay Length L = γβcτ Example: Bd → π+π- in LHCb experiment <L> ≈ 7 mm Average B meson energy <EB> ≈ 80 GeV Î τ = 1.54 ps <L> π + → µ +ν µ µ + → e +ν eν µ 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 Resonances and Widths Strong Interactions Production and decay of particles Lifetime τ ~ 10-23 s cτ ~ O(10-15 m) unmeasurable Heisenberg’s Heisenberg’s Uncertainty Uncertainty Principle Principle ∆ 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 Scattering Fixed Target Experiments a+b→c+d+… na va nb # of beam particles velocity of beam particles # of target particles per unit area Incident flux F = nava Cross Cross Section Section 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 L N ddσσ 1 1dNdN == ddΩ Ω L LdΩdΩ N dσ σ =∫ dΩ ⇒ N =σL=σ L dΩ dN = na v a nb dσ = Fnb dσ = Ldσ Luminosity Event rate Luminosity Luminosity ⇒ Event Rate N Incident flux times number of targets Depends on your experimental setup 1 barn = 1 b = 10 −24 cm 2 Luminosity [ L] = 10 30...34 cm −2s −1 Event Rate = Luminosity times Cross Section Nuclear and Particle Physics Franz Muheim 7 Scattering Centre-of-Mass Energy a+b→c+d+… Collision of two particles s is invariant quantity Mandelstam 2 r r 2 2 variable s = ( p1µ + p2µ ) = (E1 + E2 ) − ( p1 + p2 ) = p12 + p22 + 2 p1 ⋅ p2 r r = m12 + m22 + 2(E1 E2 − p1 p2 cos θ ) 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 E CoM = s ≥ ∑m j = c ,d ,... j Fixed Target Experiments r p1 = ( E lab , p1 ) µ r p 2 = ( m 2 ,0 ) µ ECoM = s = m12 + m 22 + 2 E lab m 2 ⇒ ECoM ≅ 2 E lab m 2 if E lab >> m i 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 Scattering Collider Experiments Head-on collisions of two particles θ = 1800 r r s = m12 + m 22 + 2(E1 E 2 − p1 p2 cos θ ) r r E CoM = m12 + m 22 + 2(E1 E 2 + p1 p2 ) ⇒ E CoM ≅ 4 E1 E 2 if E i >> m i 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 Conservation Laws 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 |qp + qe| < 1.60·10-21 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 Theoretical Concepts Standard Model of Particle Physics Standard Model of Particle Physics Standard Model of Particle Physics Quantum Field Theory (QFT) Describes fundamental interactions of Elementary particles Combines quantum mechanics and special relativity Very small ∆x ∆p ≈ ħc Very fast v→c Classical Physics Quantum mechanics Special relativity Quantum 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 Klein-Gordon Equation Schroedinger Equation For free particle non-relativistic 1st order in time derivative 2nd order in space derivatives not Lorentz-invariant pˆ 2 ψ = Eˆ ψ 2m − h2 2 ∂ ∇ ψ = ih ψ ∂t 2m Klein-Gordon (K-G) Equation Start with relativistic equation ∂ E → ih E2 = p2 + m2 (ħ = c = 1) ∂t Apply quantum mechanical operators ⎛ ∂2 r 2 ⎞ ⎜⎜ − 2 + ∇ ⎟⎟ψ = m 2ψ ⎠ ⎝ ∂t r r p → − ih ∇ ⎞ ⎛ ∂2 r 2 or ⎜⎜ 2 − ∇ + m 2 ⎟⎟ψ = 0 ⎠ ⎝ ∂t 2nd order in space and time derivatives Lorentz invariant Plane wave solutions of K-G equation ψ ( x µ ) = N exp(− ipν xν ) ⇒ E = ± p 2 + m 2 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 Klein-Gordon Equation 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 = m ψ Solution is spherically symmetric g2 ψ (r ) = − exp(− mr ) 4π r Interpretation - Potential analogous to Coulomb potential Force is mediated by exchange of massive bosons Yukawa Potential YukawaPotential Potential Introduced to explain nuclear force g2 ⎛ r⎞ exp⎜ − ⎟ V (r ) = − 4π r ⎝ R⎠ R= h 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 Antiparticles 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 r r exp[− i (( − E )( − t ) − ( − p ) ⋅ ( − x ))] r r = exp[− i (( Et − p ⋅ x )] Nuclear and Particle Physics Franz Muheim 14 Scattering Amplitude Transition Rate W Scattering reaction a + b → c + d W=σF Interaction rate per target particle related to physics of reaction Fermi’s Fermi’s Rule Fermi’s Golden Golden Rule Rule W = 2 2π M fi ρ f h Matrix Element Mfi scattering amplitude Density ρf # of possible final states “phase space” non-relativistic 1st order time-dependent perturbation theory see e.g. Halzen&Martin, p. 80, Quantum Physics Matrix Matrix Element Element 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 Matrix Element Scattering in Potential Example: e- p → e- p Incoming and outgoing plane waves Matrix element r r r q Momentum transfer = pi − p f rr r * 3r M M fifi == ∫ ψ *ffV V ((rr ))ψ iidd 3rr ( ) r r r r r 3r 1 ( i p r V r i p exp ( ) exp − ⋅ f i ⋅ r )d r N2 ∫ r r r r r r r 1 q = pi − p f = 2 ∫ exp(iq ⋅ r )V ( r )d 3 r N = Mfi (q) is Fourier transform of Potential V(r) Scattering in Yukawa Potential V (r ) = − g2 exp(− mr ) 4π r r g 2 ∞ π 2π exp(− mr ) 2 ( ) M fi = − i q r r dr sin θdθdφ exp cos θ r 4π ∫0 ∫0 ∫0 r r g2 ∞ = − r ∫ (exp(i q r ) − exp(− i q r ))exp(− mr )dr 2i q 0 g 22 g =− − 2 r2 M fi = m2 + + qqr 2 m ( Cross section ) Propagator term in Mfi 1 dσ 2 ∝ M ∝ r2 dΩ m2 + q ( ) 2 1/(m2 +q2) ⇒ dσ 1 ∝ 4 dΩ q m=0 Result still holds relativistically 4-momentum transfer q µ = (E i − E f , pr i − pr f ) Nuclear and Particle Physics Franz Muheim 16
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