Nuclear and Particle Physics Franz Muheim 1

Particle Physics Particle Physics --Measurements and TheoryMeasurements and Theory

Natural UnitsRelativistic KinematicsParticle Physics Measurements

LifetimesResonances and WidthsScattering Cross sectionCollider and Fixed Target Experiments

Conservation LawsCharge, Lepton and Baryon number, Parity, Quark flavours

Theoretical ConceptsQuantum Field TheoryKlein-Gordon EquationAnti-particlesYukawa PotentialScattering Amplitude - Fermi’s Golden RuleMatrix elements

OutlineOutline

Nuclear and Particle Physics Franz Muheim 2

Particle Physics UnitsParticle Physics Units

Particle Physicsis relativistic and quantum mechanical

c = 299 792 458 m/sħ = h/2π = 1.055·10-34 Js

Lengthsize of proton: 1 fm = 10-15 m

Lifetimesas short as 10-23 s

Charge1 e = -1.60·10-19 C

EnergyUnits: 1 GeV = 109 eV -- 1 eV = 1.60·10-19 Juse also MeV, keV

Massin GeV/c2, rest mass is E = mc2

Natural Units Set ħ = c = 1Mass [GeV/c2], energy [GeV]

and momentum [GeV/c] in GeVTime [(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

Natural UnitsNatural Units

Nuclear and Particle Physics Franz Muheim 3

Particle Physics Particle Physics MeasurementsMeasurements

How do we measure particle propertiesand interaction strengths?Static properties

Mass How do you weigh an electron?Magnetic moment couples to magnetic field Spin, Parity

Particle decaysLifetimesResonances & WidthsAllowed/forbiddenDecaysConservation laws

ScatteringElastic scattering e- p → e- pInelastic annihilation e+ e- → µ+ µ-Cross section

total σDifferential dσ/dΩ

Luminosity LParticle flux

Event rate N

Force Lifetimes

Strong 10-23 -- 10-20 s

El.mag. 10-20 -- 10-16 sWeak 10-13 -- 103 s

Force Cross sectionsStrong O(10 mb)

El.mag. O(10-1 mb)

Weak O(10-1 pb)

Nuclear and Particle Physics Franz Muheim 4

Relativistic KinematicsRelativistic Kinematics

Basics4-momentumInvariant massFour-vector notation

Useful Lorentz boosts relationsset 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 decaysP0 → P1 P2 work in P0 rest frame

Example: π+→µ+ νµwork in π+ rest frameuse mν

2 = 0

( ) 22222 /

,,,

cmpcEppp

pppcEp zyx

=−=⋅=

⎟⎠⎞

⎜⎝⎛=

rµ

µ

µ

( ) ( ) ( )( )

210

22

21

20

1

1021

20

22

121

221

22

2221110

2

2

2

,,0,

ppm

mmmE

Emmmm

ppppppp

pEppEpmp

rr

rrr

=−+

=

−+=

⋅−+=−=

===µµ

µµµ

( ) ( ) ( )

MeV/c 8.29

MeV 8.1092

,,0,

22

22

21

=−=

=+

=

===

µµ

π

µπµ

νµ

µµµ

πµ

µmEp

mmm

E

pEppEpmp v

r

rrr

Nuclear and Particle Physics Franz Muheim 5

LifetimesLifetimes

<L>

Decay time distributionMean lifetimeτ = <dΓ/dt>aka proper time, eigen-timeof a particle

Lifetime measurementsIn laboratory frameDecay Length L = γβcτ

Example: Bd → π+π-

in LHCb experiment<L> ≈ 7 mmAverage B meson energy<EB> ≈ 80 GeV

τ = 1.54 ps

Example: π+ discoveryDecay sequence

Emulsions exposed toCosmic rays

ττ1exp =Γ⎟

⎠⎞

⎜⎝⎛ −Γ=

Γ tdtd

µµ ννµνµπ ee++++ →→

µµ ννµνµπ ee++++ →→

Nuclear and Particle Physics Franz Muheim 6

ResonancesResonances and Widthsand Widths

Strong InteractionsProduction and decay of particlesLifetime τ ~ 10-23 s cτ ~ O(10-15 m) unmeasurable

Time and energy measurements are relatedNatural width

Energy width Γ and lifetime τ of a particle Γ = ħ/τ → Width Γ = O(100 MeV)

measurableExample - Delta(1232) Resonance

pp ++++ →∆→ ππProduction

Peak at EnergyE = 1.23 GeV(Centre-of-Mass)Width Γ = 120 MeVLifetimeτ = ħ/Γ ≈ 5·10-24 s

Heisenberg’s Uncertainty PrincipleHeisenberg’s Uncertainty Principle

h≈∆∆ tE

Nuclear and Particle Physics Franz Muheim 7

ScatteringScattering

Fixed Target Experimentsa + b → c + d + …

na # of beam particlesva velocity of beam particlesnb # of target particles

per unit areaIncident flux F = nava

effective area of any scattering happeningnormalised per unit of incident fluxdepends on underlying physics

What you want to studydN # of scattered particles in solid angle dΩdσ/dΩ differential cross section in solid angle dΩσ total cross section

L LuminosityN Event rate

Incident flux times number of targetsDepends on your experimental setup

Event Rate = Luminosity times Cross Section

LNd

dd

ddN

Ldd

=⇒ΩΩ

=

Ω=

Ω⇒

∫ σσ

σ

σ 1

Cross SectionCross Section

NLN

ddN

Ldd

RateEvent

1

σ

σ

=Ω

=Ω

LuminosityLuminosity

σσσ LddFndnvndN bbaa ===

1234...30224 scm10][ Luminositycm 10b 1 barn 1 −−− === L

Nuclear and Particle Physics Franz Muheim 8

ScatteringScattering

Centre-of-Mass Energya + b → c + d + … Collision of two particles s is invariant quantity Mandelstam

variable

centre-of-mass energyTotal available energy in centre-of-mass frameECoM is invariant in any frame, e.g. laboratory

Energy Thresholdfor particle production

Fixed Target Experiments

Example:100 GeV proton onto proton at restECoM = √s = √(2Epmp) = 14 GeVMost of beam energy goes into CoM momentumand is not available for interactions

imEmEEmEmmsE

mppEp

>>≅⇒++==

==

lab2labCoM2lab22

21CoM

221lab1

if 22

)0,(),(rr µµ

( ) ( ) ( )

( )θ

µµ

cos2

2

212122

21

2122

21

221

221

221

ppEEmm

pppp

ppEEpps

rr

rr

−++=

⋅++=

+−+=+=

sE =CoM

∑=

≥=,...,

CoMdcj

jmsE

Nuclear and Particle Physics Franz Muheim 9

ScatteringScattering

Collider Experiments

Head-on collisionsof two particles

θ = 1800

All of beam energy available forparticle production

ExampleLEP - Large Electron Positron Collider at CERN100 GeV e- onto 100 GeV e+Centre-of-mass energyECoM = √s = 2E = 200 GeV

Cross section σ(e+ e- → µ+ µ-) = 2.2 pbLuminosity ∫Ldt = 400 pb-1

Number of recorded events N = σ ∫Ldt = 870

( ) ii mEEEEppEEmmE >>≅⇒+++= if 42 21CoM212122

21CoM

rr

( )θcos2 212122

21 ppEEmms rr

−++=

Nuclear and Particle Physics Franz Muheim 10

Conservation LawsConservation Laws

Noether’s TheoremEvery symmetry has associated with it a conservation law and vice-versa

Energy and Momentum, Angular Momentumconserved in all interactionsSymmetries – translations in space and time,rotations in space

Charge conservationWell established|qp + qe| < 1.60·10-21 eValid for all processesSymmetry – gauge transformation

Lepton and Baryon number (L and B)|L+B| conservation = matter conservationProton decay not observed (B violation)Lepton family numbers Le, Lµ, Lτ conservedSymmetry – mystery

Quark Flavours, Isospin, Parityconserved in strong and electromagn processes Violated in weak interactionsSymmetry – unknown

Nuclear and Particle Physics Franz Muheim 11

Theoretical ConceptsTheoretical Concepts

Standard Model of Particle Physics

Quantum Field Theory (QFT)Describes fundamental interactions of Elementary particlesCombines quantum mechanics and special relativity

Natural explanation for antiparticlesand for Pauli exclusion principleFull QFT is beyond scope of this course

Introduction to Major QFT conceptsTransition RateMatrix elementsFeynman DiagramsForce mediated by exchange of bosons

Quantum field theory

Special relativity

Very fastv → c

Quantum mechanics

Classical Physics

Very small∆x ∆p ≈ ħc

Standard Model of Particle PhysicsStandard Model of Particle Physics

Nuclear and Particle Physics Franz Muheim 12

KleinKlein--Gordon EquationGordon Equation

Schroedinger EquationFor free particlenon-relativistic1st order in time derivative2nd order in space derivativesnot Lorentz-invariant

Klein-Gordon (K-G) EquationStart with relativistic equationE2 = p2 + m2 (ħ = c = 1) Apply quantum mechanical operators

2nd order in space and time derivativesLorentz invariantPlane wave solutions of K-G equation

negative energies (E < 0) also negative probability densities (|ψ|2 < 0)

Negative Energy solutionsDirac Equation, but –ve energies remainAntimatter

∇−→∂∂

→r

hr

h ipt

iE

ψψ

ψψ

ti

m

Emp

∂∂

=∇−

=

hh 2

2

2

2

ˆ2ˆ

0or 222

222

2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+∇−

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∇+

∂∂

− ψψψ mt

mt

rr

( ) ( ) 22exp mpExipNx +±=⇒−= νν

µψ

Nuclear and Particle Physics Franz Muheim 13

KleinKlein--Gordon EquationGordon Equation

InterpretationK-G Equation is for spinless particlesSolutions are wave-functions for bosons

Time-Independent SolutionConsider static case, i.e. no time derivative

Solution is spherically symmetric

Interpretation - Potentialanalogous to Coulomb potentialForce is mediated by exchange of massive bosons

Yukawa PotentialIntroduced to explain nuclear force

g strength of force – “strong nuclear charge”m mass of bosonR Range of force see also nuclear physicsFor m = 0 and g = e → Coulomb Potential

mcR

Rr

rgrV h

=⎟⎠⎞

⎜⎝⎛ −−= exp

4)(

2

π

ψψ m=∇ 2

( )mrr

gr −−= exp4

)(2

πψ

Yukawa PotentialYukawa Potential

Nuclear and Particle Physics Franz Muheim 14

AntiparticlesAntiparticles

Klein-Gordon & Dirac Equationspredict negative energy solutions

Interpretation - DiracVacuum filled with E < 0 electrons 2 electrons with opposite spinsper energy state - “Dirac Sea”Hole of E < 0, -ve chargein Dirac sea -> antiparticleE > 0, +ve charge-> positron, e+ discovery (1931)Predicts e+e- pair production and annihilation

Modern Interpretation – Feynman-StueckelbergE < 0 solutions: Negative energy particle moving backwards in space and time correspond to

AntiparticlesPositive energy,opposite charge moving forward in space and time

( )[ ]( )[ ]xpEti

xptEirr

rr

⋅−−=−⋅−−−−−

(exp)()())((exp

Nuclear and Particle Physics Franz Muheim 15

Scattering AmplitudeScattering Amplitude

Transition Rate WScattering reaction a + b → c + d

W = σ FInteraction rate per target particlerelated to physics of reaction

Fermi’s Golden Rule

non-relativistic1st order time-dependent perturbation theorysee e.g. Halzen&Martin, p. 80, Quantum Physics

Contains all physics of the interaction

Hamiltonian H is perturbation – 1st orderIncoming and outgoing plane wavesworks if perturbation is small Born

Approximation

ffiMW ρπ 22h

=

Matrix Element Mfiscattering amplitude

Density ρf# of possible final states“phase space”

Fermi’s Golden RuleFermi’s Golden Rule

Matrix Element Matrix Element

iffi HM ψψ)

=

Nuclear and Particle Physics Franz Muheim 16

Matrix ElementMatrix Element

Scattering in PotentialExample: e- p → e- pIncoming and outgoing plane wavesMatrix elementMomentum transfer

Mfi (q) is Fourier transform of Potential V(r)

Scattering in Yukawa Potential

Cross section

Result still holds relativistically4-momentum transfer

( ) ( )

( ) fi

if

iffi

ppqrdrVrqiN

rdrpirVrpiN

rdrVM

rrrrrrr

rrrrrr

rr

−=⋅=

⋅⋅−=

=

∫

∫

∫

32

32

3*

)(exp1

exp)(exp1

)( ψψ

( )mrr

grV −−= exp4

)(2

π

( ) ( )

( ) ( )( ) ( )

( )22

2

0

2

2

0 0

2

0

2

expexpexp2

sinexpcosexp4

qmg

drmrrqirqiqi

g

dddrrr

mrrqigM fi

r

rrr

r

+−=

−−−−=

−−=

∫

∫ ∫ ∫∞

∞φθθθ

ππ π

( )22

2

qmgM fi r+

−=

∫= rdrVM iffirr 3* )( ψψ

Propagatorterm in Mfi 1/(m2 +q2)

( ) 0114222

2 =∝Ω

⇒+

∝∝Ω

mqd

d

qmM

dd σσ

r

( )fifi ppEEq rr−−= ,µ

fi ppq rrr−=