sign in sign up
<<
Home , Quark model

Introduction to the

Lecture 4

Noether’s Theorem (continued)

b.) For Dirac fields, in the fundamental representation of SU(N) with charge Q (of U(1)), the Lagrangian is given by

L = ψj i∂/ − m ψj and the Noether currents are 

a a Jµ = ψjγµTjlψl for SU(N)

Jµ = Qψjγµψj for U(1)

The Model

Hadrons are that feel the strong force; they are classified as:

1 • - 2 : ∼ qqq • spin-0 (or spin-1 vector mesons): ∼ qq¯

Prior to 1950’s these consisted of p, n , π±, π0

1 spin- /2 spin-0 |{z} | {z } mp ∼ mn ∼ 1 GeV

|pi Put p and n in SU(2) as an “” doublet (fundamental representation): .  |ni  [SU(2) generators often labeled by (I1, I2, I3)= ~σ/2]

Observations in the

Baryon number (B), number (L), charge are conserved; they are related to symmetry under global U(1) transformations:

′ ψ → ψ = eiΛB BeiΛLLeiΛQQψ

We also note

• Q is always measured in terms of charge

• anti-particles have opposite B, L, Q quantum numbers

1 B L Q e− 0 1 −1 For example: quantum numbers for β-decay γ 0 0 0 − νe 0 1 0 n → p + e +ν ¯e n 1 0 0 B 1=1+ 0+ 0 p 1 0 1 L 0=0+ 1+ −1 π0 0 0 0 Q 0 = 1+ −1+ 0 π± 0 0 ±1

Around 1950 new particles were observed K±, Σ±,... , typically produced in pairs →. An- other U(1) charge was postulated → conservation of total charge S = ‘’. Mesons and could be arranged in octet/decuplet structures:

S S

K0 K+ n ∼ (udd) p ∼ (uud)

π− π0 η η′ π+ Σ− Σ0 Λ Σ+ I3 I3 TT T T K− K¯ 0 Ξ− Ξ0

1 Octet (spin-0) Octet (spin-/2)

S ∆∗− ∆∗0 ∆∗+ ∆∗++

Σ∗− Σ∗0 Σ∗+

I3

Ξ∗− Ξ∗0

Ω

Patterns are naturally explained by introducing 3 underlying objects: ‘up’ quark ∼ |ui, ‘down’ quark ∼ |di, ‘strange’ quark ∼ |si; come in different ‘flavours’.

2 B I I3 S Q 1 1 1 2 u /3 /2 /2 0 /3 1 1 1 1 d /3 /2 − /2 0 − /3 1 1 s /3 0 0 −1 − /3 1 1 p ∼ uud 1 /2 /2 0 1 1 1 n ∼ udd 1 /2 − /2 0 0 1 Note: Q = I3 + 2 B + S where B + S = Y is called the ‘’.  Y Y d u s¯

I3 I3 s u¯ d¯

particles anti-particles

u We can write these in SU(3)Flavours, ψq ∼  d  s   As ms ≥ mu ≈ md, only approximate symmetry. Using representation theory:

• 3 ⊗ 3=1¯ ⊕ 8 for 2- states: SU(3) Meson octet + singlet

• 3 ⊗ 3 ⊗ 3=1 ⊕ 8 ⊕ 8 ⊕ 10 for 3-particle states: SU(3) Baryon octet + decouplet

This description shows how the quarks are fundamental and make-up all other representa- tions; the octet and decouplet structures above are thus explained.

Now, we can see

1 spin 0 spin 2 spin 1 0 p ∼ uud K ∼ sd¯ ∆++ ∼ uuu π− ∼ n ∼ udd +− ¯ Ξ ∼ dss . Σ0 ∼ uds¯ − . − Ω ∼ sss . Ξ ∼ dss¯

Meanwhile three more heavy quarks are discovered: ‘’ c 1974, with mc ∼ 1.2 GeV ‘bottom/beauty’ b 1977, mb ∼ 5 GeV ‘top/truth’ t 1995, mt ∼ 175 GeV

Also the τ-lepton in 1976 and ντ (τ ) in 2000.

3 p u isospin doublet → n d

The quark masses are not identical, mu ∼ md ≤ ms; no flavour symmetry ⇒ no SU(6)F since quark masses are very different; mass terms destroy the symmetry in the Lagrangian (check!); SU(3)F is approximate symmetry. md ∼ mu ∼ few MeV, ms ∼ 100 MeV, mb ∼ 5 GeV mc ∼ 1.2 GeV, mt ∼ 175 GeV

Note that md > mu ⇒ mn > mp, else no chemistry possible!

Parity Violation

Strong interactions and electromagnetism are invariant; + + however weak interactions (e.g. β-decay: u → d + e + νe; n → p + e + νe) violate parity symmetry. Is in agreement with having no right handed .

Under a parity operation, the parity operator reverses momentum: P (E, ~p)=(E, −~p); hence helicity (=projection of spin onto momentum) flips sign:

1 ψ L = 11 ∓ γ ψ such that P ψ L = ψ R /R 2 5 /R /L 

νe u νµ c ντ t

 e L dL  µ L s  τ L bL

uR cR tR eR µR τR dR sR bR

Left-handed doublets are in the fundamental representation of SU(2)L (weak isospin):

′ νe νe iΛaT a νe → ′ = e  e L  e L  e L

′ eR → eR = eR (singlet)

′ u u iΛaT a u → ′ = e dL d L dL

′ uR → uR = uR ′ dR → dR = dR Remark:

• Symmetry will be realised locally (see below). • We know meanwhile that also right-handed neutrinos must exist, else there would be no neutrino oszillations possible.

4