Lecture 3. Experimental Methods & Feynman Diagrams

Transcription

1 Lecture 3 Experimental Methods & Feynman Diagrams Natural Units & the Planck Scale Review of Relativistic Kinematics Cross-Sections, Matrix Elements & Phase Space Decay Rates, Lifetimes & Branching Fractions Feynman Diagrams Particle Wavefunctions 1

2 Natural Units At the Planck scale the natural units have G = 1, h = 1 and c = 1 Planck Mass or Energy: M Pl = Planck length: l Pl = Planck time: t Pl = hg hc G = GeV/c 2 c = m 3 hg c = s 5 I will use h = 1, c = 1. To recover SI units you need to use dimensional analysis to introduce the correct factors of h and c. It is useful to remember the approximate values: h = Js, c = m/s, hc = 197 MeV fm A reminder that 1eV= J and 1fm= m 2

3 Relativistic Kinematics Four momentum components are always conserved Lorentz transformation: p µ = [E, p] p x = γ(p x βe) E = γ(e βp x ) where β = p/e and γ = 1/ 1 β 2 = E/m Products of four-vectors are Lorentz invariants Invariant mass p µ p µ = E 2 p 2 = m 2 E 2 = p 2 + m 2 It is often possible to take one of these limits: Highly relativistic particle: m E, E p, β 1, v c, γ Non-relativistic particle: p m, E = mc mv2, β 0, γ 1 3

4 Two-body Collisions CM frame: p 1 = p 2, p 3 = p 4, p 1 = p 2 = p i, p 3 = p 4 = p f For elastic collisions m 1 = m 3, m 2 = m 4, p i = p f Lorentz Invariant ( mass squared ) s = (E 1 + E 2 ) 2 = (E 3 + E 4 ) 2 Lab frame: p 2 = 0, p 1 = p beam, p 3 = p scatter, p 4 = p recoil s = (p 1 + p 2 ) 2 = (E 1 + m 2 ) 2 p 1 2 = m m E beam m 2 Lorentz transformation from lab to CM frame: β = p beam (E beam + m 2 ) γ = (E beam + m 2 ) s 4

5 Cross-sections & Phase Space Cross-section σ has dimensions m 2 (1 barn = m 2 ). Transition rate between initial and final states W fi is normalized to incident flux, where v i is the relative velocity of the initial state. σ = W fi N i v i = 2π h M fi 2 ρ f N i v i The physics of the underlying interaction is contained in the matrix element, M fi. ρ f is the energy density dn f /de of the final states, known as the phase space. For a two-body collision in the CM frame: ρ f = 1 (2π h) 3 p2 f dp f de 0 dω = 1 (2π h) 3 p 2 f v f dω E 0 = s and v f is the relative velocity of the final state particles. Note that σ, M fi, v i and ρ f are all Lorentz invariants. 5

6 Differential and Polarized Cross-sections Can measure angular distributions dσ/dω, where dω = d(cosθ)dφ In two-body collisions angular dependence comes from M fi (θ, φ) Can measure cross-sections for two-body fermion collisions with specific initial and final state polarizations, e.g. σ( ) may be different from σ( ). Again the spin dependence comes from M fi ( ) and M fi ( ) A lot of information about the matrix element of an interaction comes from angular and polarization measurements To calculate M fi for unpolarized cross-sections: initial state spins are averaged over final state spins are summed over 6

7 Decay Rates & Lifetimes Partial decay width (in GeV) of a particle to a final state f: Γ f = hw fi = 2π M fi 2 ρ f Final state phase space is constrained by four-momentum conservation. Two-body decay of a particle of mass m i at rest has: ρ f = 1 (4π) 2 p f m 2 i There can be several different decay modes. Total decay width and partial branching fraction: Γ = f Γ f B f = Γ f Γ Proper lifetime (in rest frame), and decay length of moving particle τ = h Γ L = γβcτ 7

8 Feynman Diagrams A pictorial representation of fermion and boson interactions designed to help with matrix element calculations. Example of electron-electron (Moeller) scattering: e α e γ α e e N.B. Initial state is on the left, and final state on the right, but the horizontal and vertical axes are not strictly related to space or time! 8

9 Drawing Feynman Diagrams Initial state particles enter from the left. Final state particles exit to the right. A line between two vertices is a virtual particle (virtual particles cannot be observed!) Fermions are solid lines with arrows pointing to right. Antifermions have arrows pointing to left. Photons are represented by wavy lines. Gluons are represented by springs. Heavy bosons (W, Z, H) are dashed lines. 9

10 Four momentum conservation Each particle has a four momentum p µ = [E, p] Need to define these in a frame of reference Initial and final state particles have E 2 p 2 = m 2 where m is the rest mass of the fermion or boson Virtual particles have E 2 p 2 = q 2 where q is not the rest mass of a physical particle. A virtual particle is said to be off the mass shell. Note that q 2 can be positive or negative! Four momentum is conserved at each vertex Four momentum is conserved between initial and final states 10

11 Rules for writing down amplitudes Initial and final state particles have wavefunctions Spin 0 bosons plane waves Spin 1/2 fermions spinors Spin 1 bosons polarizations Vertices have dimensionless coupling constants Electromagnetic interactions α e Strong interactions α s Virtual particles have propagators Virtual photon has propagator 1/q 2 Virtual boson of mass m has propagator 1/(q 2 m 2 ) The sum of the amplitudes of all possible Feynman diagrams gives the Matrix element for the interaction 11

12 Particle Wavefunctions Free particle wavefunction in four-vector notation: ψ = Ne ipµ x µ This contains no charge, spin, colour or flavour information! Normalisation is determined from the density of states in a box: [ ρ = i ψ ψ ] t ψ t ψ ρd 3 x = 2E N = 1 V V When calculating physical cross-sections including phase space, the volume of the box V drops out and can be taken to be one. Probability current associated with particle transition p 1 p 3 : j µ = i[ψ ( µ ψ) ( µ ψ )ψ] = (p 1 + p 3 )e i(p 3 p 1 )x 12

13 A spinless interaction Hypothetical interaction where two spinless particles exchange a massless boson with coupling strength g Particle currents (pi + pf)ei(pf pi)x Propagator gives 1/q 2 Vertex couplings g Matrix element is: M = g 2 (p 1 + p 3 )(p 2 + p 4 ) (p 4 p 2 ) 2 δ 4 (p 3 + p 4 p 1 p 2 ) where the δ function accounts for four-momentum conservation. 13

14 Mandelstam variables Used for the momentum transfer q 2 in two-body collisions Annihilation/creation of pairs of particles is s-channel s = (p 1 + p 2 ) 2 = (p 3 + p 4 ) 2 = 4p 2 Scattering of particles is t or u-channel t = (p 1 p 3 ) 2 = (p 4 p 2 ) 2 = 2p 2 (1 cosθ) u = (p 1 p 4 ) 2 = (p 3 p 2 ) 2 = 2p 2 (1 + cosθ) where p is CM momentum, and θ is CM scattering angle Initial state flux factor and final state phase space: 1 = 1 v i p s ρ f p s ρ f v i 1 s 14

15 Identical particles The Mandelstam variables can be used to express the spinless cross-section more compactly: 2 (u s) M = g t If the spinless particles are identical then there are two diagrams: p 1 p 3, p 2 p 4 OR p 1 p 4, p 2 p 3 The matrix element becomes: M = g 2 [ (u s) t + ] (t s) u The two diagrams are related by an exchange symmetry t u 15