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Subatomic Physics: Physics Lecture 2 Introduction to Measurements in

Measuring properties of and interactions Particle quantum numbers and conservation laws Review of relativistic dynamics Natural Units Fermi’s Golden Rule Measurements in scattering and cross sections

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Introduction: Measurements in Particle Physics

Properties of the particles Static Particle Properties ! Static properties • , m, Charge, Q ! Dynamic properties • Magnetic moment • Spin and Parity, J! Properties of interactions • Colour charge, weak hypercharge ! Decay properties • More quantum numbers... ! Scattering properties Dynamic Particle Properties • Energy, E • Momentum, $$p

Particle Scattering Particle Decays • Total cross section, !. Particle lifetime, ", and decay width, # • • Differential cross section, d!/d" Allowed and forbidden decays → • conservation laws • Collision Luminosity, L • Event Rate, N

2 and Flavour Quantum Numbers

• Lepton Number, L: Total number of ! total number of anti-leptons ! number, Le + Le = N(e−) N(e ) + N(νe) N(ν¯e) ! number, Lµ − + − ! L = N(µ−) N(µ ) + N(ν ) N(ν¯ ) number, L" µ − µ − µ + • L = Le + Lµ + L" Lτ = N(τ −) N(τ ) + N(ντ ) N(ν¯τ ) − −

• Quark Number, Nq: Total number of ! total number of anti-quarks

! number, Nu: e.g. Nu = N(u) ! N(u)̅ ! number, Nc

! number, Nd ! number, Nb

! number, Ns ! number, Nt • Nq = Nu + Nd + Ns + Nc + Nb + Nt

• The lepton flavour quantum numbers (L, Le, Lµ, L") are conserved in all interactions: strong, electromagnetic and weak. • Quark number (Nq) is also conserved in all interactions. • [Individual quark flavours (Nu, Nd, Ns, Nc, Nb, Nt) are conserved in strong and electro- magnetic interactions. They are not (necessarily) conserved in weak interactions.]

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Conservation Laws

Noether’s Theorem: Every symmetry of nature has a conservation law associated with it, and vice-versa.

• Energy, Momentum and Angular Momentum ! Conserved in all interactions ! Symmetry: translations in time and space; rotations in space • e.g. electric charge Q, colour charge ! Conserved in all interactions ! Symmetry: gauge transformation - underlying symmetry in QM description of electromagnetism / strong force

• Lepton Flavour Le, Lµ, L" and total quark number Nq Emmy Noether ! Conserved in all interactions 1882-1935 ! Symmetry: mystery!

• Quark Flavour Nu, Nd, Ns , Nc, Nb, Nt, Parity, ! ! Conserved in strong and electromagnetic interactions ! Violated in weak interactions ! Symmetry: unknown!

4 Review: Relativistic Dynamics Please review JH D&R §14 • Two important quantities for Lorentz transformations: β = v/c γ(v) = 1/ 1 β2 − • Four-momentum of a particle: p = (E/c, p￿x, py, pz) = (E/c, p￿ ) • Energy of a particle E2 = p￿ 2c2 + m2c4 E = γmc2 Scalar product of 4-momentum: (p)2 = (E/c)2 p￿ 2 = m2c2 • − • Particles with m=0 travel at the

Natural Units Lorentz boosts: γ = E/m γβ = p￿ /m β = p￿ /E | | | |

Four momentum: p = (E, px, py, pz) = (E, p￿ )

Invariant mass (p)2 = E2 p￿ 2 = m2 −

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Natural Units I

kg m s GeV c ℏ SI units: [M] [L] [T] Natural units: [Energy] [velocity] [action]

• For everyday physics SI units are a natural choice: M(SH student)~75kg. #27 • Not so good for particle physics: Mproton~10 kg

• PP chooses a different basis - Natural Units, based on: " quantum mechanics (ℏ); " relativity (c); " appropriate unit of energy 1 GeV = 109 eV = 1.60 $ 10#10 J

Energy GeV Time (GeV/ℏ)!1 Momentum GeV/c Length (GeV/ℏc)!1 Mass GeV/c2 Area (GeV/ℏc)!2

6 Natural Units II

Simplify even further by ★ measuring speeds relative to c ★ measuring action/angular momentum/spin relative to ℏ Equivalent to setting c = ℏ = 1 ! All quantities are expressed in powers of GeV

Energy GeV Time GeV!1 Momentum GeV Length GeV!1 Mass GeV Area GeV!2

Convert to SI units by reintroducing missing factors of ℏ and c • Example: Area = 1 GeV!2 2 2 n m 2 n n m m [L] = [E]− [￿] [c] = [E]− [E] [T ] [L] [T ]− n = 2, m = 2 Area (in SI units) = 1 GeV!2 " ℏ2 c2 = 3.89 " 10!32 m2 = 0.389 mb

Other common units: !15 • and energies measured in MeV • lengths in fm = 10 m • cross section measured in barn, b # 10!28 m2 • electric charge in units of e 22 Two useful relations: ￿c = 197 MeV fm ￿ = 6.582 10− MeV s × 7

Quantum Mechanical Description of Interactions

• Each particle can be described as a quantum state, |%! • The electromagnetic, weak and strong forces acting on these states can be represented by (three different) quantum operators, Ô • Rates of interactions such as particle lifetimes and scattering cross sections are given by Fermi’s Golden rule: • Transition between an initial state |%i! and a final state |%f! are related to the matrix element M = Vfi = "%f | Ô | %i !: 2π Transition probability,T = 2ρ ￿ |M| • T is related to the cross section of scattering, &. + ' + ' + ' + ' 2 e.g. &(e e (µ µ ) # |M (e e (µ µ )| . We will see • T is related to the inverse lifetime of a decay, ". how to ' ' ' ' 2 e.g. "(µ (e )*e )µ) 1/| (µ (e )*e )µ)| . # M calculate M in Measuring T (e.g. & or ") yields information future lectures about the particles (|%!) and/or the forces (Ô) 8 Particle Decay Introduction Most fundamental particles and decay. “Tracks” We can measure: left by charged particles • particle lifetime, !: average time taken particle to decay • decay width, " # ℏ/!, measured in units of energy. " (or !) is characteristic of force causing decay through Fermi’s GR: " ! |M|2 • decay length, L: average distance travelled before decaying • decay modes, branching ratio: particles in final state, how often a given final state occurs Here: one p = p invisible (neutral) • decay kinematics initial final particle has decayed into ￿ ￿ two particles Use the following examples (next lecture): + + • decay of !+ meson into muon: π µ νµ → + 0 0 decay of KS mesons into two : KS π π−,KS π π • → → 9

Scattering Introduction Consider a collision between two particles: a and b. • Elastic collision: a and b scatter off each other a b $ a b. e.g. e+e'(e+e' • Inelastic collision: new particles are created a b $ c d ... e.g. e+e'(µ+µ'

Two main types of particle physics experiment: • Collider experiments beams of a and b are brought into collision. Often p￿a = p￿b − LHC a b p-p collider (Ea, p￿a) (Eb, p￿b)

• Fixed Target Experiments: A beam of a are accelerated into a target at rest. a scatters off b in the target. a b (E, p￿ ) NA48 a Fixed Target: p+Be→K

10 Measuring Scattering

• The cross section, &, measures the how often a scattering process occurs. Typical Cross Force • & is characteristic of a given process (force) Sections 2 from Fermi’s Golden Rule & # |M| and Strong 10 mb energy of the colliding particles. !2 • & measured in units of area. Normally use Electromag 10 mb #28 2 barn, 1 b = 10 m . Weak 10!13 mb • Luminosity, L, is characteristic of the beam. Measured in units of inverse area per unit time. • Integrated luminosity, +Ldt is luminosity • Event rate: delivered over a given period. Measured in w = σ #1 L units of inverse area, usually b . • Total number of events: • What, and how often, particles are created N = σ dt in the final state. L ￿ 11

Summary of Lecture 2 • Measurements in particle physics Particle Scattering are used to understand and test • Two types of scattering experiment: the underlying theory. collider and fixed target. • Measure properties of particles and • Cross section, &, measure of how their interactions. often a process happens, & # |M|2 • Conserved quantum numbers point #28 2 • Measured in barn, b: 1 b = 10 m . to underlying symmetries. • Number of events observed is cross • Rates of decays and scatterings section times integrated luminosity yield information about the forces (+ dt) of experiment: N = & ! dt and particles involved through the L L matrix element. M. Particle Decay 2π • Lifetime, ", time taken for sample T = 2ρ to decrease by 1/e. ￿ |M| • Decay width, % = ℏ/! # |M|2 Use relativistic kinematics and Lepton flavour quantum numbers: • natural units ! L, Le, Lµ, L" p =(E/c, px,py,pz)=(E/c , p￿ ) • Quark flavour quantum numbers: 2 2 E 2 2 2 ! Nu, Nd, Ns, Nc, Nb, Nt p = p￿ = m c c2 − ￿ ￿ 12