Chapter 2.1 by Manu Verma

Chapter 2 Elementary particles

2.1 Classification of Particles 2.2 Leptons 2.3 Quarks 2.4 Hadrons 2.5 Interactive Exercise

2.1 Classification Of Particles

Elementary particles

Particle Physics

Classification Of Particles Elementary Particle Physics is concerned with the basic forces of nature treats the smallest objects in the Universe. Fundamental Particles: Leptons and Quarks Composite Particles: Hadrons composed of quarks. In the real world, a general classification of elementary particles proton, electron, neutrino and photon is fermions and bosons. We further subdivide these groups according to the types of interaction in which they participate. All electrically charged particles by virtue of their charge can interact electromagnetically. Some particles called leptons respond only to the weak force. They are the familiar electron e-, neutrino νe , muon μ- , νμ , τ(tau). All these leptons have intrinsic spin ½ and are therefore fermions. Particles which can participate in the strong interactions are called hadrons. The hadron family contains both fermions and bosons. Hadrons with half integer spin are called baryons (n and p are most familiar). The mesons (named so because they had masses intermediate between the light or zero mass leptons and the heavier baryons called bosons).

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Table: 1

Note: All electrically charged particles can interact electromagnetically. Some hadrons decay via the weak interaction.

The Particle World Stable particles are particles which are stable against decay via strong interactions. The particles which decay, decay via weak interactions with relatively long life time ~10-10s; or via electromagnetic interaction with much shorter lifetime of 10-16s. In the table, we have the broad categories of leptons, mesons and baryons. The particles are arranged in order of increasing mass. The properties of charge, rest mass, intrinsic spin, parity, for them. In addition, to every particle, their corresponds an antiparticle. Dayalbagh Educational Institute

Elementary particles

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Particle Properties What you should know, or be able to work out, is: the quantum numbers, and if they are conserved are or not the force responsible for a decay, from the lifetime the main decay modes, using a Feynman diagram that the +, − and 0 superscripts correspond to the electric charge of the particles the relationship between a particle and its antiparticle. Anti-particles are not named in the tables. Anti-particles have the same mass, same lifetime, opposite quantum numbers from the particle. Anti-particles decay into the anti-particles of the shown modes. For example, an antimuon, µ+, has a mass of 105.7 MeV/c2 and a lifetime of 2.197×10−6 s. Its quantum numbers are Le = 0, Lµ = −1, Lτ = 0 and Q = +1. Its main decay mode is µ+ → e+ νe νµ. Quantum Numbers The following quantum numbers are conserved in all reactions: Total quark number, Nq = N(q) − N(q). Nq = +1 for all quarks Nq = −1 for all anti-quarks Nq = 0 for all leptons and anti-leptons Nq = 3 for baryons and Nq = 0 for mesons. Dayalbagh Educational Institute

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The three lepton flavour quantum numbers: Electron Number: Le = N(e- ) − N(e+ ) + N(νe) − N(νe) Muon Number: Lµ = N(µ- ) − N(µ+) + N(νµ) − N( νµ) Tau Number: Lτ = N(τ-) − N(τ+) + N(ντ) − N(ντ) Electric charge, Q. There are six quantum numbers are used to describe quark flavour, which hadrons also carry: Up quark number Nu ≡ N(u) − N(u) Down quark number, Nd ≡ N(d) − N(d) Strange quark number Ns ≡ N(s) − N(s) Charm quark number, Nc ≡ N(c) − N(c) Bottom quark number, Nb ≡ N(b) − N( b ) Top quark number, Nt ≡ N(t) − N( t ) These quark flavour quantum numbers are conserved in strong and electromagnetic interactions, but not in the weak interactions. For historical reasons, quark quantum numbers are often re-formulated into similar quantum numbers called: strangeness S = −Ns, charmness C = Nc, bottomness B = −Nb topness, T = Nt, strong isospin |I, IZ〉 where IZ = ½ (Nu − Nd) and baryon number, B = 1/3Nq. The physics described by both sets of quantum numbers is identical.

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Elementary particles

Particle Physics Mass (MeV/c2)

Lµ

Lτ

Q (e)

Lifetime (s)

Main Decay Modes

Lepton

Symbol

Antiparticle

electron

0.511

−1

Stable

muon

µ-

µ+

105.7

−1

2.197 × 10−6

e- νeνµ

tau

τ-

τ+

1777

−1

2.91 × 10−13

e- νeντ , µ νµντ , hadrons + ντ

electron neutrino

νe

∼0

muon neutrino

νµ

∼0

tau neutrino

ντ

∼0

Table 2: The leptons of the Standard Model. The masses of the neutrinos are so small, that we can ignore them in most reactions. The concepts of lifetime and decay mode don’t really make sense for the neutrinos.

Quark

Symbol

Antiquark

Q(e)

down

−1/3

+2/3

strange

−1/3

charm

+2/3

bottom

−1/3

top

+2/3

Table 3: The quarks of the Standard Model. Quarks are always found in bound states, therefore it doesn’t always make much sense to talk about the masses and lifetimes of the individual quarks.

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Meson

Symbol

Anti-particle

Mass (MeV/c2)

Lifetime (s)

Charged Pion

π+

π-

139.6

2.60 × 10−8

Neutral Pion

π0

self

135.0

0.83 × 10−16

Charged Kaon

K+

K-

493.7

1.24 × 10−8

Neutral Kaon

K-short

K0S

497.7

0.89 × 10−10

K-long

K 0L

497.7

5.2 × 10−8

Eta

η0

self

547.5

< 10−18

Eta-Prime

η'0

self

957.8

< 10−20

Charged Rho

ρ+

ρ-

770

0.4 × 10−23

Neutral Rho

ρ0

self

770

0.4 × 10−23

Omega

ω0

self

782

0.8 × 10−22

Phi

self

1020

20 × 10−23

D+-meson

D+

D-

1869

10.6 × 10−13

D0 -meson

1864.6

4.2 × 10−13

DS-meson

D+S

D-S

1969

4.7 × 10−13

J/Psi

J/ψ

self

3097

0.8 × 10−20

B+-meson

B+

B-

5279

1.7 × 10−12

B0 -meson

B0d

5279

1.5 × 10−12

BS-meson

B0S

5370

1.5 × 10−12

Upsilon

self

9460

1.3 × 10−20

Table 4

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Meson

quark composition

Main Decay Modes

Charged Pion

-1

µ+ν µ

Neutral Pion

(dd - uu)/√2

γγ

Charged Kaon

-1

µ+ νµ, π+π0

Neutral Kaon

-1

K-short

(K0 + K0)/√2

π+π−, 2π0

K-long

(K0 + K0)/√2

π+e−νe

Eta

(dd + uu-2ss)/√6

γγ, 3π0

Eta-Prime

(dd + uu-2ss)/√6

π+π−η, ρ0 γ, π0π0η

Charged Rho

-1

π +π 0

Neutral Rho

uu, dd

π+π−

Omega

uu, dd

π+π−π0

Phi

K+ K −, K0 K0

D+-meson

-1

+ 1

D0 -meson

-1

+ 1

DS-meson

-1

+ 1

J/Psi

e+e−, µ+µ−...

B+-meson

-1

K++ something

B0 -meson

-1

BS-meson

-1

D−S + something

Upsilon

e+e−, µ+µ−, B0d ,B0d

Table 5: Selected mesons. Notes: The neutral kaons mix with each other and appear physically as K0L and K0S. Decay modes are only shown for some the mesons. Dayalbagh Educational Institute

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Lifetime (s)

Main Decay Modes

Baryon

Symbol

quark composition

Proton

uud

Stable

Neutron

ddu

920

pe−νe

Lambda

Λ0

uds

2.6 × 10−10

pπ−, nπ0

Sigma Plus

Σ+

uus

0.8×10−10

pπ0, nπ+

Sigma Zero

Σ0

uds

6 × 10−20

Λ0γ

Sigma Minus

Σ−

dds

1.5 × 10−10

nπ−

Delta

∆++

uuu

0.6 × 10−23

pπ+

Delta

∆+

uud

0.6 × 10−23

pπ0

Delta

∆0

udd

0.6 × 10−23

nπ0

Delta

∆−

ddd

0.6 × 10−23

nπ−

Cascade Zero

Ξ0

uss

2.9 × 10−10

Λ0π0

Cascade Minus

Ξ−

dss

1.64 ×10−10

Λ0π−

Omega Minus

Ω−

sss

0.82 ×10−10

Ξ0π−, Λ0K−

Lambda-C

Λ+c

udc

2 × 10−13

Table 6: Selected baryons. Anti-baryons are symbolised by an overline, e.g. Σ− = uus is the antiparticle of Σ+.

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The following tables list the stable particles with some properties: Particle

Spin-parity J p

Mass/MeV

Principal decay modes

Mean lifetime/s

Leptons υe

J=½

< 7.3 x 10-3

Stable

J=½

0.511

Stable

υμ

J=½

< 0.27

Stable

J=½

105.66

eυυ

2.20 x 10-6

υτ

J=½

< 35

Stable

J=½

1784.1

μ υ υ, e υ υ, hadrons

3.1 x 10-13

Table 7: Stable Particle Table: Leptons Particle

Spin-parity Jp

Mass/MeV

Principal decay modes

Mean lifetime/s

Non-strange mesons π±

139.57

μυ

2.6 x 10-8

π0

134.97

γγ

0.83 x 10-16

547.5

γ γ, 3π0, π+, π-, π0

Γ=1.19 ± 0.11 keV 1.24 x 10-8

Strange mesons K±

493.65

μ υ, π± π0 ,3π

K0 K0

497.67

50% KS0, 50% KL0

KS0

π+π-, π0π0

0.89 x 10-10

KL0

3π0, π+ π- π0, π± μ+υ,

5.17 x 10-8

π± e + υ Table 8: Stable Particle Table: Mesons

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Elementary particles Particle

Spinparity J p

Particle Physics Mass/Me V

Principal decay modes

Mean lifetime/s

Charmed non-strange meson D±

1869.3

eX, KX, K0X, K0 X

10.7 x 10-13

D0 D0

1864.5

eX, μX, K X K0 X, K0X

4.2 x 10-13

Charmed strange meson F±

KX, K0X, K0X

1971

4.5 x 10-13

non-KKX, eX

(now Ds±)

Bottom meson B±

5279 DX, D0/D0 X, D*X, F X F D, F* D, F D*, F* D*

B0 B0

(12.9 ± 0.5) x 10-13

5279

Table 9: Stable Particle Table: Mesons

The charmed baryon X stands for any particles consistent with the appropriate conservation laws. Particle

Spin-parity J p

Mass/MeV

Principal decay modes

Mean lifetime/s

Baryons Non-strange baryons p

½+

938.3

Stable (>1032a)

½+

939.6

p e- υ

889.1± 2.1

Strangeness – 1 baryons Λ

½+

1115.6

p π-, n π0

2.6 x 10-10

Σ+

½+

1189.4

p π0 , n π+

0.8 x 10-10

Σ0

½+

1192.6

Λγ

7.4 x 10-20

Σ-

½+

1197.4

nπ-

1.5 x 10-10

Strangeness – 2 baryons Ξ0

½+

1314.9

Λ π0

2.9 x 10-10

Ξ-

½+

1321.3

Λ π-

1.6 x 10-10

Table 10: Stable Particle Table: Baryons Dayalbagh Educational Institute

Elementary particles Particle

Particle Physics

Spin-parity Jp

Mass/MeV

Principal decay modes

Mean lifetime/s

1672.4

ΛK-, Ξ0π-, Ξ-π0

0.82 x 10-10

Baryons Strangeness – 3 Baryons Ω-

3/2 +

Charmed Baryons Λ c+

½+

2282.0

ΛX, pK-π+, pK0

1.9 x 10-13

Σc(2455)

½+

2453

Λ c+ π

Ξc+

½+

2466

ΛK-π+π+, Σ+K-π+

≈3 x 10-13

Σ0K-π+π+, Ξ-π+π+ Ξc0

2473

½+

Ξ-π+, Ξ-π+π+π-,

≈0.8 x 10-13

pK-K*(892)0 Bottom Baryons Λb0

½+

≈5641

J/ψ (1S)Λ, pD0π-,

Λc+π+π-πTable 11: Stable Particle Table: Baryons

The spin-parity assignments for the charmed and bottom baryons are quark model predictions. Particle

Spinparity J p

Mass/MeV

Principal decay modes

Mean lifetime/s

Gauge bosons γ

Stable

W±

80.22± 0.26 GeV

e υ, μ υ, τ υ

Γ = 2.12±0.11 GeV

91.173±0.020 GeV

e+ e-, μ+ μ-, τ+ τ-

Γ = 2.487±0.01 GeV

υ υ, hadrons g

Stable

(gluon) Table 12: Stable Particle Table: Bosons

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Resonances

Apart from this list, many hadrons have been discovered in high energy accelerators, with lifetimes 10-23s. This short lifetime as compared to other stable particles, indicates that these states decay via strong interactions. These particles are called resonances, e.g. N(1520), where the bracket indicates mass in MeV (Mega-electron Volts) and the particle symbol indicates that the basic properties are similar to the nucleon.

Fig: 1 The nucleon N(939) and its first existed state N (1520).

Fig: 2 The spectrum of the strange and non-strange baryons. Only those states which are well established experimentally are shown

Similar cases of resonances are known for other stable baryons, as shown in fig (2) and mesons. Dayalbagh Educational Institute

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Resonances are highly unstable particles which decay by the strong interaction (lifetimes about 10-23 s)

ground state

resonance

Fig 3: Example of a qq system in ground and first excited states

If a ground state is a member of an isospin multiplet, then resonant states will form a corresponding multiplet too Since resonances have very short lifetimes, they can only be detected by registering their decay products:

Invariant mass of the particle is measured via masses of its decay products: Resonance peak shapes are approximated by the BreitWigner formula (fig. 4):

Mean value of the Breit-Wigner shape is the mass of a resonance: M=W0 Γ is the width of a resonance, and is inverse mean lifetime of a particle at rest: Γ ≡ 1/τ Dayalbagh Educational Institute

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Fig 4

Fig 5: A typical resonance peak in K+Kinvariant mass Distribution (Source: )

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Peaks in the observed total cross-section of the Ď&#x20AC;Âąpreaction correspond to resonances formation Fig 6: Formation of a resonance R and its subsequent inclusive decay into a nucleon N

Fig 7(a): Scattering of p+ and p- on proton (Source: )

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Introduction: Measurements in Particle Physics We know that particles generally do two things: Scatter e.g. collisions of protons on protons at LHC collision of cosmic rays in atmosphere Decay e.g. decay of particles produced at LHC decay of cosmic ray muons Measurements of scatterings and decays are used to infer properties of the particles and interactions. Can also measure directly some properties of the longer-lived particles. Static Particle Properties Mass, m, Charge, Q Magnetic moment Spin and Parity, Jπ Particle Scattering Total cross section, σ. Differential cross section, dσ/dΩ Collision Luminosity, ℒ Event Rate, N

Particle Decays Particle lifetime, τ, and decay width, Γ Allowed and forbidden decays → conservation laws

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Natural Units Lorentz boosts: Four momentum: Invariant mass

Particle Decay Introduction Most fundamental particles and hadrons decay. We can measure: particle lifetime, τ: average time taken particle to decay decay width, Γ≡ ħ/τ, measured in units of energy decay length, L: average distance travelled before decaying “Tracks” branching ratio, decay left by charged modes: particles in particles final state, how often a given final state occurs decay kinematics ∑ pinitial = ∑ pfinal Look at the following examples: decay of π+ meson into muon: π+ → μ+νμ decay of KS mesons into two pions:

Ks → π+π−,Ks → π0π0 Dayalbagh Educational Institute

Fig 8

Here: one invisible (neutral) particle has decayed into two particles 20

Elementary particles

Particle Physics

Particle Lifetime Particle lifetime, τ, the time taken for the sample to reduce to 1/e of original sample. Different forces have different typical lifetimes. Also define total decay width, Γ≡ ħ/τ.

In its own rest frame particle travels υτ = βcτ before decaying. In the lab, time is dilated by γ. If τ is large enough, energetic particles travel a measurable distance L = γβcτ in lab. Force

Typical Lifetime

Strong

10-20 - 10-23 s

Electromag

10-20 - 10-16 s

Weak

10-13 - 103 s

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Decay Modes Particles can have more than one decay mode. e.g. The Ks meson decays 99.9% of the time in one of two ways: Ks → π+π−,Ks → π0π0 Each decay mode has its own matrix element, M. Fermi’s Golden Rule gives us the partial decay width for each decay mode: Γ(Ks → π+π−) ∝ |M (Ks → π+π−)|2 Γ(Ks → π0π0) ∝ |M (Ks → π0π0)|2 The total decay width is equal to the sum of the decay widths for all the allowed decays. Γ(Ks) = Γ(Ks → π0π0) + Γ(Ks → π+π−) The branching ratio, BR, is the fraction of time a particle decays to a particular final state:

Particle Decay Kinematics Most particles decay. e.g. Ks meson can decay as: Ks → π+π− Reconstruct the mass of a particle from the momenta of the decay products: ∑ pinitial = ∑ pfinal p (Ks) = p (π+) + p (π−)

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squaring each side …

M(Ks) is reconstructed invariant mass of KS K0 → π+π16000 14000 12000 10000 8000 6000 4000 2000 0

Fig 10

0.49 0.51 0.48 0.5 0.52 0.505 0.475 0.495 0.515 0.485 m(π+π-) GeV

Decay Kinematics II Decay of an unstable particle at rest: A → b d

Fig 11

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Four-momentum conservation:

For moving particles, apply appropriate Lorentz boost. Example: π+ → μ+ υμ

work in rest frame of pion. mυ ≈ 0

Scattering 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+eInelastic collision: new particles are created a b → c d ... e.g. e+e- → μ+μTwo main types of particle physics experiment: Collider experiments beams of a and b are brought into collision. Often

LHC

p-p collider

Fig 12

Fixed Target Experiments: A beam of a are accelerated into a target at rest. a scatters off b in the target. Dayalbagh Educational Institute

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Fig 13

NA48 Fixed Target: p+Be→K

Fig 14

Measuring Scattering The cross section, σ, measures the how often a scattering process occurs. σ is characteristic of a given process (force) from Fermi’s Golden Rule σ ∝ |M|2 and energy of the colliding particles. σ measured in units of area. Normally use barn, 1b = 10-28m2. ℒ, Luminosity, is characteristic of the Force Typical Cross beam. Measured in units sections of inverse area per unit Strong 10 mb time. Electromag 10-2 mb Weak Dayalbagh Educational Institute

10-13mb 25

Elementary particles

Particle Physics

Integrated luminosity, ∫ ℒ dt is luminosity delivered over a given period. Measured in units of inverse area, usually b-1. What, and how often, particles are created in the final state.

Cross Section

Event rate: ω =ℒσ Total number of events: N = σ ∫ ℒ dt

We have a beam of particles incident on a target (or another beam).

Fig 15

Flux of incident beam, f : number of particles per unit area per unit time. Beam illuminates N particles in target. We measure the scattering rate, dw/dΩ, number of particles scattered in given direction, per unit time per unit solid angle, dΩ.

Integrate over the solid angle, rate of scattering: w = f Nσ Define luminosity, ℒ = f N Scattering rate w = ℒ σ

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Collision Centre of Mass Energy,√s For a collision define Lorentz-invariant quantity, s : square of sum of four-momentum of incident particles:

√s = ECM is the energy in centre of momentum frame, energy available to create new particles! Fixed Target Collision, b is at rest. Ea >> ma , mb

Fig 16

Collider Experiment, with E = Ea = Eb >> ma, mb, θ = π s = 4E2

ECM = 2E

Collision Examples

Fig 17

Fig 18 Dayalbagh Educational Institute

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The previous collider at CERN collided electrons and positrons head-on with E(e-) = E(e+) = 45.1 GeV.

σ(e+e- → μ+μ-) =1.9 nb at ECM = 91.2 GeV Total integrated luminosity ∫Ldt = 400 pb-1 Nevts(e+e- → μ+μ-) = 400,000 & 1.9 = 760,000 To make hadrons, a 45.1 GeV electron beam was fired into a Beryllium target. Electrons collide Beryllium.

with

protons

and

neutrons

In fixed target electron energy is wasted providing momentum to the CM system rather than to make new particles. Dayalbagh Educational Institute

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Exercise 1 Write down typical life times for particles that decay by: a) the strong force: b) the electromagnetic force: c) the weak force:

Solution 1 a) The strong force: 10-23 to 10-20 sec. b) The Electromagnetic force: 10-20 to 10-16 sec. c) The weak force: 10-13 to 103 sec.

Exercise 2 By looking at the lifetimes on Particle data sheets, which force is responsible for the decay of Ď&#x20AC;0, B+, w0?

Solution 2 a) Ď&#x20AC;0 lifetime is 0.83x10-16 sec, the electromagnetic decay. b) B+ life time is 1.5x10-12 sec, weak decay. c) w0 lifetime is 0.8x 10-22 sec, strong decay. Dayalbagh Educational Institute

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Exercise 3 The lifetime of

η'0

has not been measured directly. 0

The total width of n10 has been measured to be Γ(η' ) = 0.203 + 0.016 MeV. What is the lifetime of force is responsible for its decay?

η'0. What

Solution 3 The total width of any particle is gives by Γ = ħ/τ ⇒

τ (η'0) = ħ/Γ (η'0).

Using ħ = 6.58 x 10-22 MeV sec, gives

τ (η'0) = 3.24 x 10-21 sec. Seeing the lifetime, we see that strong force is responsible for the decay of

η'0.

Exercise 4 If a resonance has I=3/2, B=1 and S=0. What are its change states?

Solution 4 There are 2I+1 = 2. (3/2) + 1 = 4 components of I with I3 = 3/2, ½,-1/2 and -3/2. Using Q = I3+

, these correspond to charge

states Dayalbagh Educational Institute

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Exercise 5 The ∑0 hyperon decays to +γ with a mean life time of 7.4 x 10-20 sec. Estimate its width.

Solution 5 ΔE Δt ≈ ћ

Hence width ΔE ≈ ħ=6.6 x 10-22 MeV. sec We have ΔE ≈ 9 keV Which is typical of an electromagnetic decay

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