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−=