Decays and Scattering Decay Rates Cross Sections Calculating Decays Scattering Lifetime of Particles 1
Decay Rates There are THREE experimental probes of Elementary Particle Interactions - bound states - decay of particles - scattering of particles Decay of a particle - is of course a statistical process - represented by the average (mean) life time Remember: - elementary particles have no memory - probability of a given muon decaying in the next microsecond is independent of time of creation of muon (humans are VERY different!) Decay Rate Γ: probability per unit time that particle will disintegrate
Decay Rates Consider large collection of, for example, muons: N(t) Decay rate dn = ΓNdt N( t) = N(0) exp ( Γt) Mean lifetime τ = 1/Γ Most particles will decay by several routes - total decay rate Γ tot = Σ Γ i ; τ = 1/Γ tot Branching ratio for ith decay mode - BR(i) = Γ i / Γ tot 3
Decay Width Γ Example: decay width of the J/ψ Shown is the original measurement at the SPEAR e + e - Collider; the line width is larger (~ 3.4. MeV) due to energy spread in the beams Natural line width is 93 kev Lifetime is ΔE Δ t h/ π = 6.6 10-16 ev Δ t ~ 7*10-1 s We will see later within a discussion of the quark model why this lifetime is relatively long 4
Decay: Examples Derive τ = 1/Γ - Hint: what fraction of original sample decays between t and t+dt? - With this result: what is the initial probability p(t) dt of any particle decaying between t and t+dt? - Mean lifetime τ = t( p) dt 0 Starting with 10 6 muons at rest, how many will survive 10-5 seconds? - τ (muon) =, 10-6 A muon with momentum p=10gev/c is produced in a cosmic ray collision with an oxygen nucleus of the atmosphere at 10km altitude. What is the probability of the muon reaching the earth s surface? 5
Scattering: Cross sections What do we measure? What do we calculate? Classical analog: - aiming at target - what counts is the size of the target Elementary particle scattering - size ( cross-sectional area ) of particle is relevant - however: size is not rigidly defined; size, area is fuzzy - effect of collision depends on distance between proectile and scattering center - we can define an effective cross section Cross sections depend on - nature of proectile (electrons scatter off a proton more sharply than a neutrino) - nature of scattering interaction (elastic, inelastic many different possibilities) 6
Cross Sections Elastic scattering: - Kinetic energy of scattering system remains unchanged; only directions of scatters are changed in the interaction (with a particle, with a potential) Inelastic scattering: kinetic energy is not conserved (lost or gained in the collision) In particle collisions: typically, several final states - e + p e + p + γ ; e + p + π 0 ;. - each channel has its exclusive scattering cross section - total ( inclusive ) cross section σ tot = Σ σ i ; Cross section depends in general on velocity of proectile - naively: σ proportional to time in vicinity of target σ ~ 1/v - strong modification, if incident particle has energy to form a resonance (or an excitation): a short-lived, quasi-bound state - Frequently used procedure to discover short-lived particles 7
Hard-Sphere Scattering Particle scatters elastically on sphere with radius R b = R sin α α + θ = π sin α = sin(π/ θ/) = cos (θ/) b = R cos (θ/) θ = cos -1 (b/r) for b = 0 θ = π If particle impinges with impact parameter between b and b + db will emerge under angle θ and θ + dθ 8
Scattering into Solid Angle dω More generally, if particle passes through an infinitesimal area dσ at impact parameter b, b+db will scatter into corresponding solid angle dω Differential cross section D(θ): - differential with respect a certain parameter: solid angle; momentum of secondary particle dσ = D(θ) dω dσ = b db dφ, dω = sinθ dθdφ D(θ) = dσ dω = b sin θ ( ) db dθ (areas, solid angles are intrinsically positive, hence absolute value signs) 9
Cross Sections Suppose particle (e.g. electron) scatters of a potential (e.g. Coulomb potential of stationary proton) Scattering angle θ is function of impact parameter b, i.e. the distance by which the incoming particle would have missed the scattering center 10
Scattering: Some examples Hard-sphere scattering db dθ = R sin ( θ ) D(θ) = Rb sin ( θ / ) sin θ = R ( ) cos( θ / )sin θ / sin θ = R 4 Total cross section σ = dσ = D(θ) dω σ total (hard-sphere) = R dω = π R 4 - for hard-sphere: πr obviously presents the scattering area Concept of cross section applies also to soft targets 11
Rutherford Scattering Particle with charge q 1 and kinetic energy E scatter off a stationary particle with charge q Classically: b q q E q1q ( / ) bmin E 1 = cot θ = Differential cross section: D ( θ ) = ( ) q q 4Esin 1 ( θ / ) Total cross section: σ = π q 1q 4E ( ) 1 sinθ dθ = ( ) sin θ / infinite, because Coulomb potential has infinite range Remember: in calculation of energy loss of a charged particle one imposes a cut off (b max ), beyond which no atomic excitations are energetically possible. π 0 1
Cowan, Reines Experiment of Neutrino Discovery: estimate of event rate neutrino flux: 5* 10 13 neutrinos/ cm s Tanks was located 11 m from reactor core, 1 m blow surface (why?) ν absorbed in water (00 l); effective area was 3000 cm; depth ~ 70 cm Cross section for absorption (inverse β-decay σ = 6 * 10-44 cm Energy threshold for reaction: 1.8 MeV 3% of the reactor neutrinos are above threshold 13
Cowan, Reines Experiment of Neutrino Discovery: estimate of event rate N a = 6.0 * 10 3 / mol...18 g H O Number of scatterers N S / cm N S = N A L ρ / 18 =.3 * 10 4 Probability for interaction = Effective cross section / cm = P = N S σ =.3 * 10 4 * 6* 10-44 L= 70 cm σ= 6* 10-44 cm Water: molecular weight = 18 ρ = 1 g / cm 3 = 1.4 * 10-19 Rate= P * Flux *Detector Area * Threshold = 6.3 * 10-4 Rate is ~ events/ hour was actually observed! 14
Luminosity Luminosity L : number of particles per unit time and unit area: [ L ] = cm - s -1 dn = L dσ. number of particles per unit time passing through area dσ and b, b+db, which is also number per unit time scattered into solid angle dω : dn = L dσ = L D(θ) dω Setting up a scattering experiment with a detector covering a solid angle dω and counting the number of particles per unit time dn Measurement of differential cross section If dω 1 N = σl At LHC: L = 10 34 cm - s -1, σ = 100 mb= 10-5 cm N = 10 9 coll/s 15
Luminosity vs. Collision Rate Collision rate: - Probability P for interaction = Effective cross section / cm = P = N S (Number of scatters/ cm ) σ Collision Rate Rate = P * Flux *Detector Area Luminosity L : dimension cm - sec -1 - Groups these parameters into one Parameter, Luminosity - In colliders: includes the details about the shpe and paricle density of the colliding beams - Defined as the proportionality between the rate R i of process i with with a specific cross section σ i R i = L σ i Time- Integrated Luminosity: - In 011: LHC delivered 5 fb -1-1 fb -1 = 10 4 *10 15 cm - - Total number of collisions: N= L σ i = 5 * 10 39 * 10-5 = 5*10 14
Calculating Decay Rates and Cross Sections Calculation needs two ingredients - the amplitude M for the process (dynamics) - the phase space available (kinematics) M depends on the physics, type of interaction, reaction, Phase space: room to maneuver - heavy particle decaying into light particles involves a large phase space factor - neutron decaying into p + e + ν e has a very small phase space factor Fermi Golden Rule - transition rate = (phase space) amplitude - Non-relativistic version formulated with perturbation theory - can be derived with relativistic quantum field theory 17
Golden Rule for Decays Particle 1 (at rest) decaying into, 3, 4,. n particles 1 + 3 + 4 +. + n Decay rate Γ m, p is mass, four-momentum of ith particle ; S is statistical factor to account for identical particles in final state; (if no identical particles in final state S = 1) M is amplitude squared; contains the dynamics The rest is Phase Space 18 ( ) ( ) ( ) ( ) ( ) 4 4 0 3 1 4 4 M π θ πδ δ π p d n n m S p c m p p p p p = Γ =
Phase Space Integration over all outgoing four-momenta subect to three kinematical constraints p = m c - each outgoing particle lies on its mass shell: i.e. - each outgoing energy is positive: p 0 = E /c > 0 ; this is ensured by the θ function (θ(x) = 0 for x<0; θ(x) = 1 for x>0) - energy and momentum is conserved 1 3 ; this is ensured by delta function 4 Practical points: - - δ d 4 E p = dp p 0 c d 3 = p m c 4 ( p m ), enforcedby δ c δ p = p + p + p ( p ) 1 0 0 ( p m c ) = δ[ ( p ) p m c ] p integrals doable,usingδ function Γ = n S m M 1 3 n = 3 4 4 1 d p ( π ) δ ( p p p p ) ( ) 3 1 p + m c p n p n π 19
Two-Particle Decays For two particles in final state, previous expression reduces to Γ = ( p p p ) 4 S δ 1 3 3 3 M d p 3 d p m π 3 1 p + m c p + m c After several pages of algebra and integration 3 3 Γ = S 8π p1 m c M Surprisingly simple: phase space integration can be carried out without knowing functional form of M two-body decays are kinematically determined: particles have to emerge back-to-back with opposite three-momenta no longer true for decays into more than two particles 0
Golden Rule for Scattering Particles 1 and collide, producing 3,4,. n particles 1 + 3 + 4 + n : after performing p 0 integrals σ = 4 ( p same definitions, procedures as for decay Two-body scattering in CM frame 1+ 3+4; p = p1 c S M p f, p or outgoing particle 1 S p ) ( m m c ( ) 8 π ( E + E ) p i dσ = dω 1 1 ) M with 3 4 4 1 d p ( π ) δ ( p + p p p ) ( ) 3 p i f = 3 the magnitude of either incoming Dimensions and units decay rate Γ = 1/τ [sec -1 ]; cross section: area [cm ], 1 barn = 10-4 cm ds/dω barns/steradian M units depend on total number of particles involved [M] = (mc) 4-n 1 1 3 n n p + m c π