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

qq

dt

pdF

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