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Particle Attributes – Quantum Numbers
Intro Lecture – Quantum numbers (Quantised Attributes –
subject to conservation laws and hence related to Symmetries)
listed NOT explained. Now we cover
Electric Charge
Baryon Number,
Lepton Number
Spin
Parity
Isospin
Strange, Charm,
Bottom & Top
Charge Conjugation
Colour etc….
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Particle Attributes – Quantum Numbers
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Electric Charge (Q) & Baryon Number (B) conserved in
all interactions– Strong, Weak & Electromagnetic !
Baryons:
B = +1
Anti-Baryons: B = -1
Everything else: B = 0
Baryons made up of 3 quarks: B = 1/3
For an anti-quark
B = -1/3
Nq − Nq
Strong Interaction: π- p → K0 Λ0
B: 0 1
0 1
Constant
Weak Interaction: n → p e- υ¯e
B: 1
1 0 0
Proton being the lightest Baryon cannot decay i.e p → e+ π0
B: 1
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0 0
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Lepton Number:
Leptons have L = +1
Anti-Leptons L = -1
All Else
L= 0
e ,ν e have Le = +1
µ − ,ν µ haveLµ = +1
τ − ,ν τ have Lτ = +1
−
X
X
Leptons of different type do NOT mix !
Unique Lepton Number for each Type
e ,ν e have Le = −1
+
µ + ,ν µ have Lµ = +1
τ + ,ν¯τ have Lτ = −1
Le , Lµ & Lτ
separately
conserved
µ± → e± γ (Both e & µ lepton numbers violated !!)
e+ e- → µ±τ (Both µ & τ lepton numbers violated !!)
Each Lepton Number is conserved in strong, weak & electromagnetic
interactions
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Particle Attributes – Quantum Numbers
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SPIN ( Angular Momentum)
Particles can have 2 types of Angular Momenta
Orbital – Classical Analogue – subject to Quantum Conditions.
A particle can be in any Orbital Angular Momentum State
Spin – Intrinsic Angular momentum – specific attribute of a
particle – discrete states (values) –i.e. a Quantum Mechanical effect
– for Spin S - (2S+1) states of different SZ
S = +1
z
S
Bosons
- 0 h , 1 h , 2 h ....
Fermions - 1 h , 3 h , 5 h....
2 2 2
z
For S = ½ - 2 states with Sz = +1/2 & -1/2
For S = 1 - 3 States with Sz = +1, 0, -1
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Sz = +½
Sz = 0
Sz = -½
Sz = -1
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Angular Momentum
Classically
r r r
L=r×p
But Quantum Mechanically
r
r
r and p are quantised by the Uncertainty Principle
r
L Can only take certain values and assume defined orientation
wrt to a given direction in Space
Particle Wave Function has both Orbital & Spin angular momentum
components besides the position & momentum components
r
2
2
ih ∇ Ψ p = p Ψ p
L Ψ
= l ( l + 1) h
Ψ
lm
L
z
Ψ
lm
= mh Ψ
lm
Jz = jz Ψ( j)
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lm
For a given l m = -l, -l+1,…-1,0,1…
l-1,l
With l = 0,1,2,3…
(2j+1) states for spin-j
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Parity
Parity is an entirely Quantum Mechanical Concept.
The operator (P) acts on the space part of the wave function
reversing the space coordinate r to -r
P Ψ (r ) → Ψ (−r ) = λ Ψ (r )
P
2
i.e.
2
But
2
Ψ (r ) = λ Ψ (r ) = Ψ (r ) , λ = 1
λ = ±1
Eigenvalues of P are +1 (even) or -1 (Odd)
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Parity (cont.)
Parity Operator P reverses r to -r
Equivalent to a Reflection in the x-y
plane
Followed by a Rotation about the z-axis
© 2004 S. Lloyd
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Parity (cont.)
For a given state of orbital angular momentum l, P = (-1)l
Consider the H-atom – bound by a central potential – wave function
a product of the spatial and angular functions
Ψ ( r , ϑ , ϕ ) = χ ( r )Y
Y
m
l
(θ , φ ) =
m
l
(θ , φ )
( 2 l + 1)( l − m )! m
im φ
P (cos θ ) e
l
4 π ( l + m )!
Changing r to –r implies θ → π - θ & φ → π + φ
e
imφ
m imφ
→ ( −1) e
l+m m
P (cos θ )
and P (cos θ ) → ( −1)
l
l
m
m
l m
Y (θ, φ) = (−1) Y (θ, φ) P Ψ( r , θ , φ = ( −1)l Ψ( r , θ , φ
l
l
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Parity (cont.)
For Fermions, P (anti-particle) = -1 x P (particle)
For Bosons,
P (anti-particle) = P (particle)
Arbitrarily define P = 1 for nucleons and then determine P for other
particles from experiment (angular distributions in interactions)
Parity for π+, π0, π- P = -1
N.B. – P is conserved in Strong & electromagnetic interactions
but not in Weak
Particles (& Physical attributes) are labelled by their total spin (J)
and Parity (P) – JP
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JP
Name of Object
Example
0-
PseudoScalar
Pressure
0+
Scalar
M, T, λ
-
Vector
1
1
+
2+
Vector P = -1
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r r
p, x
Axial Vector
r
S, L
Tensor
Spin-spin coupling
Axial Vector P = +1
©2004 S.Lloyd
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Isospin (I)
1932 Heisenberg suggested n & p sub-states of nucleon ascribed
new quantum number I = 1/2 – isospin analogous to spin with
Cartesian coordinates I1, I2, I3, in an imaginary Isospin space
Related to charge & Baryon Number Q/e = B/2 + I3 this gives the
Proton I3 = +1/2 and the neutron I3 = -1/2
I, I3 are conserved in Strong interactions BUT not in
electromagnetic interactions due to coupling to Q.
Quark Model p = (uud) & n = (udd) => u-quark I3 = +1/2 while dquark has I3 = -1/2. So knowing the quark-content of a particle
we can determine the isospin
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Particle Attributes – Quantum Numbers
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Isospin (I) – Cont.
π
+
π
= ud , I 3 = +1
0
=
1
2
( dd − uu ) , I 3 = 0
-
π = du , I 3 = −1
Isospin represented by a vector with component I3
designating ‘upness’ and ‘downness’
I3 :: Up (u)
Isospin predicts:
σ(p+p→d+
π+)
~ 2σ (p + n → d +
π0 )
I3 :: Down (d)
As observed !
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Strangeness
Associated Production of long-lived neutral particles π- p → K0 Λ0
K0 & Λ0, decaying weakly, were labelled ‘Strange’
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Strangeness (cont.)
Associated production explained by introduction of a 3rd quark
(strange – s)
s-quark is assigned Strangeness S = -1, anti-quark
¯ (s) has S = +1
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Strangeness (cont.)
Quark Content of some Strange Particles
Λ0 (uds), Λ0 (u d s ), K 0 (ds ), K 0 (d s ), K + (us ) & K − (u s )
Q, B, I & S – What’s the Connection ?
Gellman – Nishijima Relation:
Q = I3 +
B+S
2
Strong & EM forces Conserve S – Weak does not.
Charm (C), Bottom (B) & Top (T) – analogous to Strange.
u,d,s,c,b & t are the 6 Quarks
Weak force changes quark flavour
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Charge Conjugation (C)
Charge Conjugation Operator (C) changes particle to anti-particle
e.g. e- → e+ Only if no Conservation Law is violated.
C has definite Eigenvalues for a neutral boson which is its own
anti-particle
Cπ
0
=η π
0
And
C
2
2
2
0
0
0
π
=η π
= π
∴ η = ±1
EM fields are produced by moving charges. But Q → -Q under C
so Cγ = -1 And C is a multiplicative Quantum Number, so Cγγ= 1.
With π0 → γγ,
C π0
= π
0
i.e. η = 1
If EM forces are C-invariant π0 → γγγ
is forbidden
π 0 → 3γ
−8
<
3
.
0
×
10
π 0 → 2γ
C is conserved in Strong & EM Interactions but not in Weak
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Particle Attributes – Quantum Numbers
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Quark Colour
The Interaction π+ p → p π+ π0 the final state π+ p form a resonance
of mass 1232 MeV
∆++ has I3 = 3/2, J= 3/2, P = +1, Wave function is totally symmetric
BUT It is a Fermion
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Quark Colour (Cont.)
qqq = space
l, m s
flavour
All components are symmetric l=0, s=3/2 and flavour (uuu)
Need extra anti-symmetric component
Each Quark is assigned a new quantum number Colour – can have
3 values Red, Green & Blue and the particle wave function has an
extra factor |Ψcolor> Which is anti-symmetric
All particles – baryons & mesons are colourless – so each of the
3 quarks in a baryon has a different colour (r, g, b) !
In a meson the quark and the anti-quark carry opposite colour
to be colourless !
Strong force between quarks carried by Gluons – carry colour !
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Particle Attributes – Quantum Numbers
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Quark-Parton Model
Hadronic ( Baryons & mesons) states classified into Groups by spin
& parity e.g. 10 excited baryons with JP = 3/2+ and a ground state 8
with JP = 1/2+
d→u
d
Decuplet
→
s
missing
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QPM – Ground State Baryons
JP = 1/2+ each with 3 quarks with spins 1/2, -1/2,1/2 (
)
939 MeV
1193 ∑
1116 Λ
1318 MeV
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Baryons
With 3 different flavour quarks get 27 possible combinations
SU(3) symmetry arguments: 3⊗3 ⊗3 = 27
27 = 1(Singlet) + 8(Octet) + 8(Octet) + 10(Decuplet)
uud Combinations
Flavour symmetric (uud + udu + duu)
Spin symmetric
(↑↑↑ + ↑↑↑ + ↑↑↑)
or anti-symmetric (uud)
spin anti-
J = 3/2
(↑↓↑)
J =1/2
A (uud) baryon in both the decuplet & the octet !
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Particle Attributes – Quantum Numbers
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Baryons
Note: No (uuu), (ddd) or (sss) states in the ½+ Baryons !!
(uuu/ddd/sss) Combinations
(uuu) Flavour symmetric
And aligned spins (↑↑↑) with J = 3/2 Spin Symmetric
giving a |flavour>|spin> product symmetric.
Colour component has to be anti-symmetric.
(↑↑↓) with J= ½ makes the flavour-spin anti-symmetric BUT colour
component must be anti-symmetric. Overall product symmetric !!
So J=1/2 not possible
(uuu), (ddd) and (sss) states only in the Decuplet NOT in an Octet
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QPM - Mesons
With 3 different flavour quarks get 9 possible combinations SU(3)
symmetry arguments:
3⊗ 3 = 1+ 8
Pseudo-Scalar (JP = 0+) Mesons – (qq)
 pairs with opposite spins (↑↓)
S K + ( us )
K 0 ( ds )
π − (ud)
π
0
(dd − uu )
2
K − ( us )
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π + ( ud )
η
I3
 uu + dd − 2 ss 


2


K 0 ( ds )
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QPM – Mesons (Cont.)
Vector Mesons – quark anti-quark pairs with aligned spins (↑↑)
S
K *0 ( ds )
K *+ ( us )
ρ
ρ (u d )
−
0
( uu − dd )
2
K *− ( us )
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φ (ss )
ω ( uu + dd )
ρ + ( ud )
I3
2
K *0 ( ds )
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QPM – Higher Mass states
Hadron states seen so far are all with 0 orbital angular momentum
between the component quarks.
When the partons are excited into higher orbital angular momenta
heavier excited states are produced.
J = 2,3… for mesons & J = 5/2, 7/2… for Baryons
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QPM – Evidence
1st evidence from di-muon production in pion- Carbon scattering
C
12 Is an ‘Isoscalar’ target – i.e. N = N = 18
u
d
6
If QPM then q q annihilate &
the photon materialises into a
d-muon pair.

σ α (charge of q)(charge of q)
α (charge of q)
 2

With π-(ud)
 & π+(ud)
π−
C→µ
+
µ ......
π+
C→µ
+
µ ......
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-
-
≅4
In good agreement
with data

Confirming fractional quark charges
Particle Attributes – Quantum Numbers
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