PG lectures- Particle Physics Introduction
C.Lazzeroni
Outline
- Properties and classification of particles and forces- leptons and hadrons- mesons and baryons- forces and bosons
- Relativistic kinematics- particle production and decays- cross sections and lifetimes- centre-of-mass and lab frames- invariant mass
- Summary of Standard Model- Feynman diagrams- QED- QCD- Weak interactions
- Conservation laws- conserved quantum numbers and symmetries- allowed and forbidden reactions
Outline
- Angular momentum and Spin- basic quanyum mechanics- Clebsch-Gordon coefficients- conservation of angular momentum and spin
- Isospin- combination., conservation- pion-nucleon scattering
- C, P, CP, CPT- application in quark model- neutral kaon mixing- CP violation
- CKM matrix- weak decays of quarks- quark mixing- CPV in SM
essential books
- Particle physics, B.R. Martin and G.Shaw- Quarks and Leptons, F.Halzen and A.D.Martin- Introduction to High Energy Physics, 2nd ed.,D.H.Perkins- Review of Particle Physics, PDG
useful books
- Detectors for particle radiation, K.Kleinknecht- Large Hadron Collider Phenomenology, M.Kramer and F.J.P.Soler- Particle Detectors, C.Grupen and B.Shwartz- QCD and collider physics, R.K.Ellis and W.J.Stirling and B.R.Webber- A modern introduction to Quantum Field Theory, M.Maggiore- Prestigious discoveries at CERN, R.Cashmore, L.Maiani, J.P.Revol- An introduction to relativistic processes and the Standard Model ofElectroweak Interactions, C.M.Becchi and G.Ridolfi- Handbook of accelerator physics and engineering, A.Wu Chao and M.Tigner- Statistical methods in experimental physics, F.James
Section 1 - matter and forces
7
We now know that all matter is made of two types of elementaryparticles (spin ½ fermions):
LEPTONS: e.g. e-, νe
QUARKS: e.g. up quark (u) and down quark (d) proton (uud)
A consequence of relativity and quantum mechanics is that forevery particle there exists an antiparticle which has identicalmass, spin, energy, momentum, BUT has the opposite signinteraction.
ANTIPARTICLES: e.g. positron e+ , antiquarks (u, d), antiproton (uud)
8
LeptonsParticles which DO NO INTERACT via the STRONG interaction. Spin ½ fermions 6 distinct FLAVOURS 3 charged leptons: e−, µ−, τ−
µ and τ unstable 3 neutral leptons: νe, νµ, ντ
Neutrinos are stable andalmost massless
+antimatter partners, e+, νe
Charged leptons only experience the electromagnetic and weakforces
Neutrinos only experience the weak force Lepton number conserved (apart from neutrino oscillations)
< 18.20ντ
1777.0-1τ−
3rd
< 0.170νµ
105.7-1µ−
2nd
< 3 eV/c20νe
0.511-1e−1st
Approx. Mass(MeV/c2)
Charge (e)FlavourGen.
9
QuarksQuarks experience ALL the forces (electromagnetic, strong, weak) Spin ½ fermions Fractional charge 6 distinct flavours Quarks come in 3 colours Red, Green, Blue Quarks are confined within HADRONSe.g.
COLOUR is a label for the charge of the strong interaction. Unlike theelectric charge of an electron (-e), the strong charge comes in 3orthogonal colours RGB.
4.5-1/3b174+2/3t
3rd
0.5-1/3s1.5+2/3c
2nd
0.35-1/3d0.35+2/3u
1st
Approx. Mass(GeV/c2)
Charge (e)FlavourGen.
u u d u d
( )du!"+( )uudp !
+antiquarks u,d,…
10
Hadrons Single free quarks are NEVER observed, but are always CONFINED inbound states, called HADRONS. Macroscopically hadrons behave as point-like COMPOSITE particles.
Hadrons are of two types:
MESONS (qq)Bound states of a QUARK and an ANTIQUARKAll have INTEGER spin 0, 1, 2,… Bosonse.g.
antiparticle of a meson is a meson
BARYONS (qqq)Bound states of 3 QUARKSAll have HALF-INTEGER spin 1/2, 3/2,… Fermionse.g.
PLUS ANTIBARYONS e.g.
q q
q q q
( )du!"+
( )uudp ! ( )uddn !
( )du!"#
)( qqq ( )duup ! ( )ddun !
No other combination of quarks gives stable particle
Patterns follow from Group Theory: see Orlando’s lecturesSU(3) for 3 flavours: u,d,sSU(4) for 4 flavours: u,d,s,c
Quark Model
12
Forces
Quantum Mechanically: Forces arise due to exchange of VIRTUALFIELD QUANTA (Gauge Bosons)
Field strength at any point is uncertain
Number of quanta emitted and absorbed
1q 2qpv
Fv
crt~pr =h
21qq~
rr
dt
2
21 ˆ==!
vv Massless particle
e.g. photon
ħ =1 natural units
13
Gauge BosonsGAUGE BOSONS mediate the fundamental forces Spin 1 particles (i.e. Vector Bosons) No generations The manner in which the Gauge Bosons interact with the
leptons and quarks determines the nature of the fundamentalforces.
2
1
1
1
Spin
80, 9110-7W±, Z0W and ZWeak
Massless10-39?GravitonGravity
Massless10-2γPhotonElectromagnetic
Massless1gGluonStrong
Mass(GeV/c2)
StrengthBosonForce
(Mass Higgs H 0 to be found )
14
Range of ForcesThe range of a force is directly related to the mass of theexchanged bosons.
Due to quark confinement, nucleons start to experience thestrong interaction at ~ 2 fm
∞
10-15StrongStrong (Nuclear)
∞Gravity10-18Weak
∞Electromagnetic
Range (m)Force
Section II - Relativistic kinematics
16
UnitsCommon practise in particle and nuclear physics NOT to use SI units.
Energies are measured in units of eV:KeV (103 eV) MeV (106 eV) GeV (109 eV) TeV (1012 eV)
Nuclear Particle
Masses quoted in units of MeV/c2 or GeV/c2 (m = E/c2)
e.g. Electron mass
Atomic masses are often given in unified (or atomic) mass units
1 unified mass unit (u) ≡ Mass of an atom of 12 1 u = 1g/NA = 1.66 x 10-27 kg = 931.5 MeV/c2
Cross-sections are usually quoted in barns: 1 b = 10-28 m2
( )( )225
19283131
cMeV511.0ceV1011.5
10602.11031011.9kg1011.9
=!=
!!!=!= """
em
C12
6
17
Natural Units Choose energy as basic unit of measurement
Energy GeV Time (GeV/ ħ)-1
Momentum GeV/c Length (GeV/ ħc)-1
Mass GeV/c2 Cross-section (GeV/ ħc) -2
Simplify by choosing ħ=c=1
Energy GeV Time GeV-1
Momentum GeV Length GeV-1
Mass GeV Cross-section GeV-2
Convert back to SI units by reintroducing “missing” factors ofħ and c
ħc = 0.197 GeV fm ħ = 6.6 x 10-25 GeV s
18
Example:
Charge: Use “Heaviside-Lorentz” units:
Fine structure constant
becomes
mb39.0fm109.3
GeVfm)197.0(GeV1
cGeV1tion(S.I.)secCross-
22
22
222
=!=
!=
!=
"
"
"h
19
Relativistic Kinematics In Special Relativity, the total energy and momentum of a particle ofmass m are
and are related by
Note: At rest, E = m and for E >>m, E ~ p
The K.E. is the extra energy due to motion
In the non-relativistic limit
Particle physics T is of O (100 GeV) >> rest energies ⇒ relativisticformulas. Always treat β decay relativistically.
222mpE +=
20
In Special Relativity and transform between frames ofreference, BUT
are CONSTANT.
( )xt, v ( )pE, v
222
222
pEm
xtdv
v
!=
!= Invariant intervalINVARIANT MASS
21
Four-Vectors Define FOUR-VECTORS:
where
Scalar product of two four-vectors
Invariant:
or
22
Colliders and √ sConsider the collision of two particles:
The invariant quantity
is the energy in the zero momentum (centre-of-mass) frame
It is the amount of energy available to interaction e.g. themaximum energy/mass of a particle produced in matter-antimatterannihilation.
s
23
Fixed Target Collision
For
e.g. 100 GeV proton hitting a proton at rest:
Collider Experiment
For then and
e.g. 100 GeV proton colliding with a 100 GeV proton:
In a fixed target experiment most of the proton’s energy is used toprovide momentum to the C.O.M system rather than being availablefor the interaction.
GeV1411002 !""!= ppm2Es
GeV2001002s =!=
21
2
2
2
1m2Emms ++=
211m,mE >>
21m2Es =
21m2Es =!
( )!"++= cosppEE2mms 2121
2
2
2
1
vv
211m,mE >> Ep =
v
( ) 2Es4EcosEE2s222 =!="#=
Transform from a particle rest frame to a frame in which particle has velocity β=v/c
p0 = 0 , E0 =m p1, E1
Equivalent to boosting frame of reference in opposite direction -β:
p1 = γp0 + βγE0 = βγmE1 = γE0 + βγp0 = γm
1)
2) Calculate invariant mass M of AB system:
pA = (EA , pA ) pB = (EB , pB )
EA2 = | PA |2 + MA
2 EB2 = | PB |2 + MB
2
M2 = (EA + EB)2 - (pA + pB)2 = = EA
2 + 2 EA EB + EB2 - ( |pA|2 + 2 pA pB + | pB|2 ) =
= MA2 + MB
2 + 2 ( EA EB - pA pB )
3) Rapidity: useful in hadron-hadron collisions, related to longitudinal momentum distribution
|pL | = |p| cos θ
y = 1/2 ln ( E + pL ) / ( E - pL)
y = ln (E + pL ) / ( pT2 + m2)
Rapidity y is not Lorentz invariant but y distribution (shape)remains unchanged under Lorentz boostLorentz transform to frame moving with relative velocity βalong the beam direction causes a constant offset
y0 = y + 1/2 ln ( 1 - β ) / ( 1+ β)
√
Section III - decay rates
27
How do we study particles and forces ? Static Properties
Mass, spin and parity (JP), magnetic moments, bound states
Particle DecaysAllowed/forbidden decays → Conservation Laws
Particle ScatteringDirect production of new massive particles inmatter/antimatter ANNIHILATIONStudy of particle interaction cross-sections.
10-8
10-20
10-23
Typical Lifetime (s)
10-13Weak
10-2Electromagnetic10Strong
Typical Cross-section (mb)Force
28
Particle DecaysMost particles are transient states – only the privileged few liveforever (e−, u, d, γ, …)
A decay is the transition from one quantum state (initial state) toanother (final or daughter).
The transition rate is given by FERMI’S GOLDEN RULE:
where λ is the number of transitions per unit timeMfi is the matrix elementρ (Ef) is the density of final states.
⇒ λ dt is the probability a particle will decay in time dt.
29
1 particle decayLet p(t) be the probability that a particle will survive until at leasttime t, if it is known to exist at t=0.
Prob. particle decays in the next time dt = p(t)λ dtProb. particle survives in the next time dt = p(t)(1-λ dt)= p(t+dt)
EXPONENTIAL DECAY LAW
30
Probability that a particle lives until time t and then decays in timedt is
The AVERAGE LIFETIME of the particle
Finite lifetime ⇒ UNCERTAIN energy ΔE
Decaying states do not correspond to a single energy – they havea width ΔE
The width, ΔE, of a particle state is– Inversely proportional to the lifetime τ– Equal to the transition rate λ using natural units.
ħ =1 natural units
31
Many particles
NUMBER of particles at time t,
where N(0) is the number at time t=0.
RATE OF DECAYS
ACTIVITY
HALF-LIFE i.e. the time over which 50% of the particlesdecay
32
Decaying States → ResonancesQM description of decaying statesConsider a state with energy E0 and lifetime τ
i.e. the probability density decays exponentially (as required).
The frequencies present in the wavefunction are given by theFourier transform of
Probability of finding state withenergy E = f(E)*f(E)
33
Probability for producing the decaying state has this energydependence, i.e. RESONANT when E=E0
Consider full-width at half-maximum Γ
TOTAL WIDTH (using natural units)
1
0.5
E0-Γ/2 E0 E0+Γ/2 E
P(E)
Γ
Breit-Wigner shape
(ΔEΔt~1 h=1)!=
"=#1
34
Partial Decay WidthsParticles can often decay with more than one decay mode, eachwith its own transition rate
The TOTAL DECAY RATE is given by
This determines the AVERAGE LIFETIME
The TOTAL WIDTH of a particle state is
DEFINE the PARTIAL WIDTHS
The proportion of decays to a particular decay mode is calledthe BRANCHING FRACTION
Section III - reactions
36
Reactions and Cross-sectionsThe strength of a particular reaction between two particles is specifiedby the interaction CROSS-SECTION.
A cross-section is an effective target area presented to the incomingparticle for it to cause the reaction.
UNITS: σ 1 barn (b) = 10-28 m2 Area
The cross-section, σ, is defined as the reaction rate per target particle,Γ, per unit incident flux, Φ
where the flux, Φ, is the number of beam particles passing through unitarea per second.
Γ is given by Fermi’s Golden Rule
!"=#
37
Consider a beam of particles incident upon a target:
Number of target particles in area A = n A dxEffective area for absorption = σ n A dxRate at which particles are = −dN = N σ n A dxremoved from beam A −dN = σ n dx N
n nuclei/unit volume
A
N particles/sec
TargetBeam
dx
σ = No scattered particles /sec N n dx
38
Beam attenuation in a target (thickness L)
thick target (σ n L>>1)
thin target (σ n L<<1, e -σ n L ≈ 1-σ n L)
MEAN FREE PATH between interactions = 1 /nσ
For nuclear interactions: nuclear interaction lengthFor electromagnetic interactions: radiation length
39
Rewrite cross-section in terms of the incident flux,
Hence,
σ = No scattered particles /sec N n dx = No scattered particles /sec (ΦA) (NT/A dx) dx
σ = No scattered particles /sec Flux ∗ Number target particles
Number of target particlesin cylindrical volume NT = n ∗ Volume = n A dx
dxA
40
Differential Cross-sectionThe angular distribution of scattered particles is not necessarilyuniform
Number of particles scattered into dΩ = ΔNΩ
The DIFFERENTIAL CROSS-SECTION is the number of particlesscattered per unit time and solid angle divided by the incident fluxand by the number of target nuclei defined by the beam area.
Beam
Targetsolid angle, dΩ
DIFFERENTIALCROSS-SECTIONUnits: area/steradian
41
Most experiments do not cover 4π and in general we use dσ/dΩ .
Angular distributions provide more information about themechanism of the interaction.
Different types of interaction can occur between particles.
where the σi are called PARTIAL CROSS-SECTIONS.
Types of interaction:
Elastic scattering:
a + b → a + b only momenta of a and b changeInelastic scattering:
a + b → c + d + …… final state not the same as initial state
σXY ! PQ = exclusive σXY ! anything = inclusive