The Static Quark Model Particle Physics

Particle Physics - Chapter 1 The static quark model

Paolo Bagnaia

last mod. 5-Mar-21 1 − The static quark model

1. Quantum numbers 2. Hadrons : elementary or composite ? 3. The eightfold way 4. The discovery of the Ω– 5. The static quark model 6. The mesons 7. Meson quantum numbers 8. Meson mixing 9. The baryons Caveats for this chapter: • arguments are presented in 10. SU(3) historical order; some of the results are incomplete, e.g. heavy flavors 11. Color are not mentioned here (wait a bit); 12. Symmetries and multiplets • large overlap with [FNSN, MQR, IE].

Paolo Bagnaia - PP - 01 2 short summary • in this chapter [see § 6 for QCD]:  no dynamics, only static classification, The roadmap: i.e. algebraic regularities of the states;  only hadrons, no photons / leptons; • operators associated with conserved quantum numbers; • modest program, but impressive results: • old attempts of classification;  all hadrons are (may be classified as) composites of the same elementary • first successes (multiplets, Ω−); objects, called quarks; • modern classification:  the quark dynamics, outside the scope  quarks; of this chapter, follow simple  group theory: flavor-SU(3); conservation rules;  color: color-SU(3);  QM and group theory are enough to  symmetries; produce the hadron classification; • "construction" of a a a a a a a a a  although quarks have not been mesons and baryons; observed, their static properties can be inferred from the particle spectra; → § 6, QCD. • does it mean that quarks are "real"? what really "real" means? [???]

Paolo Bagnaia - PP - 01 3 1/4 quantum numbers : the Mendeleev way • Many hadrons exist, with different ( ) quantum numbers (qn). * even if D.I.M. lived long before • Some qn show regularities (spin, parity, …). the advent of QM. • Other qn are more intriguing (mass, ...). an example from ( ) • A natural approach (à la D.I.M. * ): many years ago  investigate in detail the qn: [add antiparticles…]  the associated operators;  the qn conservation; the proliferation of hadrons started in  look for regularities; Dmitri Ivanovich Mendeleev the '50s – now they (Дми́трий Ива́нович Менделе́ев)  classify the states. are few hundreds …

Name π± π0 K± K0 η p n Λ Σ±,0 ∆ Mass (MeV) 140 135 494 498 548 938 940 1116 1190 1232 Charge ±1 0 ±1 0 0 1 0 0 ±1,0 2,±1,0 many other − − − − − Parity + + + + + hadrons Baryon n. 0 0 0 0 0 1 1 1 1 1 3 Spin 0 0 0 0 0 ½ ½ ½ ½ /2 other qn …

Paolo Bagnaia - PP - 01 4 2/5 quantum numbers : parity for complete definitions and discussion, [FNSN], [MQR], [BJ].

Definition* : |ψ(q,x,t)> = P|ψ(q,─x, t)> • Gauge theories → Pγ = Pg = −1. W± and Z do NOT conserveℙ parity in their • Particles atℙrest (= ⃗in their own ref⃗ .sys.) interactions, so their intrinsic parity is not defined. are parity eigenstates: |ψ(q,x=0,t)> = P|ψ(q,x=0,t)>. • For a many-body system, P is a multiplicative quantum number : • Eigenvalue P : intrinsic parity ℙ ⃗ ⃗ ψ(x1,x2…xn,t) = P1P2…Pnψ(-x1,-x2…-xn,t). 2 = , P real → P = (± 1). • Particlesℙ ⃗ ⃗ in⃗ a state of orbital⃗ ⃗ angular⃗ • Diracℙ 𝟙𝟙equation → for spin ½ fermions, momentum are parity eigenstates : P(antiparticle) = -P(particle) k Ykm(θ,φ) = (−1) Ykm(θ ,φ+π) → k • Convention: P(quarks/leptons) = +1 → |ψkm (θ,φ)> = (−1) |ψkm(θ,φ)> +1 = P = P = P = P = P = P = …; e− u d s • Therefore, for a two- or a three-particle −1 = P = P = P = P = P = P = … ℙ e+ µ+ τ+ ū d̄ s̄ system: L • Field theory: for spin-0 bosons → Psys(12) = P1P2 (−1) ; P(antiparticle) = +P(particle) : L1+L2 L1 Psys(123) = P1P2P3(−1) . Pπ+ = P = P , … 1 2

* here and in the following slides : = +1 or −1 ? L2 • q : charges + additive qn; … be patient … L 3 • x,a : polar / axial vectors; 1 2 • t : time. ⃗ Paolo Bagnaia - PP - 01 5 3/5 quantum numbers : charge conjugation

Definition : changes a particle p into its • Only particles (like π0, unlike K’s) which antiparticle p̄, leaving untouched the space are their own antiparticles,ℂ are and time variablesℂ : eigenstates of , with values C = (± 1) : |p,ψ(x,t)> = C |p̄, ψ(x,t)>. C = +1 for π0, η, η’; ℂ C = −1 for ρ0, ω, φ; • Therefore,ℂ ⃗ under : ⃗ C = −1 for γ. for Z, and are not defined charge q → −q; ℂ baryon n. B→ −B; • However, few particlesℂ areℙ an eigenstate flip lepton n. L → −L ; of ; e.g. |π+> = ─ |π−>. strangeness S → −S; ℂ position x → x ; • ℂWhy define ? E.g. use -conservation in e.-m. decays: momentum p → p; no-flip ⃗ ⃗ π0 → : +1ℂ→ (−1) (−1) ℂ ok; spin s → s. π0 → γ : +1→ (−1) (−1) (−1) no. • is hermitian; its eigenvalues are ±1; BR(π0 → γ ) measured to be ∼10−8. they are multiplicatively conserved in strongℂ and e.m. interactions. see later

Paolo Bagnaia - PP - 01 6 4/5 quantum numbers : G-parity • charge conjugation is defined as proposed by Lee and Yang, 1956. |q, B, L, S > = ± |-q, -B, -L, -S >; • -parity is multiplicative : ℂ 𝔾𝔾 • therefore, only states q = B = L = S = 0 |nπ+ mπ− kπ0> = ℂ 𝔾𝔾 may be -eigenstates (e.g. π0, η, γ, = (−)n+m+k |nπ+ mπ− kπ0>; [π+π−]). 𝔾𝔾 |qq̄> = (−)L+S+I |qq̄> → G = (−)L+S+I; ℂ ≡ • is useful: Generalization [G-parity]: 2, 𝔾𝔾 where = rotation in the isospin space:  -parity is conserved only in strong 2 𝔾𝔾 ℂ ℝ 𝔾𝔾 ≡ exp (-i ); interactions ( and isospin are valid); 2ℝ 2 𝔾𝔾 I-I3  it produces selection rules (e.g. a 2|I, I3> = (−) |I, −I3> ℝ ℂ π |q, x, t, I, I > = (−)I-I3|-q, x, t, I, -I >; decay in odd/even number of 's is ℝ2 3 3 allowed/forbidden). • hasℝ more⃗ eigenstates than ⃗ ; e.g.: • e.g. ω(782) is IG(JPC) = 0−(1− −) : π± − π | > = | >; BR (ω → π+π−π0) = (89.2±0.7)% 𝔾𝔾 0 0∓ ℂ |π > = + |π >; + − ℂ BR (ω → π π ) = ( 1.5±0.1)% π± π 2| > = + | >; opposite to the obvious phase-space ℂ 0 ∓0 2|π > = − |π >. predictions (more room for 2π than 3π ℝ • therefore: decay). ℝ ±,0 ±,0 ±,0 |π > = 2 |π > = − |π >. • [see also J/ψ decay].

Paolo𝔾𝔾 Bagnaia - PP -ℂ01 ℝ 7 5/5 quantum numbers : some proofs

• See [FNSN, §7]. • For a qq̄ (or particle-antiparticle) state: |q,x,s -q,-x,-s> = • Be |q, x, a> an eigenstate of a generic = C |-q,x,s, q,-x,-s> = 𝕊𝕊 ℙ ℂ ⃗ ⃗ ⃗ ⃗ "flip" operator ( = , , , [spin- = C P |-q,-x,s, q,x,-s> = ⃗ 𝕊𝕊 ℙ ⃗ ⃗ ⃗ ⃗ flip], [time-reversal]). = C P |-q,-x,-s,q,x,s> = 𝕂𝕂 𝕂𝕂 ℂ 𝔾𝔾 ℙ 𝕊𝕊 𝕊𝕊 ⃗ ⃗ ⃗ ⃗ the spin s is an = C P |q,x,s, -q,-x,-s>. axial vector (a) 𝕋𝕋 𝒮𝒮 ⃗ ⃗ ⃗ ⃗ • From their definition: → = CP = ± 1; ⃗ 2 → −1 𝒮𝒮 ⃗ ⃗ ⃗ ⃗ = = . → = ± -1 -1 = ± . 𝕊𝕊 ℙ ℂ 𝒮𝒮 𝕂𝕂 𝟙𝟙 𝕂𝕂 𝕂𝕂 • an eigenstate of has eigenvalue K: • for suchℂ a state𝕊𝕊 ℙ[BJ, 335𝕊𝕊 ℙ]: |q, x, a> = K |±q, ±x, ±a>; L+1 𝕂𝕂 P = (−1) (see before); 2 |q, x, a> = K2 |q, x, a> = |q, x, a>; 𝕂𝕂 ⃗ ⃗ = (−1)S+1 (Pauli principle); K2 = 1 → K = real = ±1. 𝕂𝕂 ⃗ ⃗ ⃗ → C = P × = (−1)L+S; 𝒮𝒮 → G = (-1)L+S+I (see before). • P(γ) = −1 (vector potential [FNSN, 7.2.2]) 𝒮𝒮 : eigenvalue of ; S : value of the spin. 𝒮𝒮 𝕊𝕊 Paolo Bagnaia - PP - 01 8 1/3 hadrons : "elementary" or composite ? Over time the very notion of "elementary (???) particle" 1890 1900 1910 1920 entered a deep crisis. e− p The existence of (too) many 1920 1930 1940 1950 hadrons was seen as a contradiction with the n e+ µ± π± K± elementary nature of the 1950 1960 fundamental component of

matter. 0 0 ± 0 0 π Λ Σ p̄νe Σ Λ ρ νµ α2 and many more K0 ∆ Ξ n̄ Ξ0 ω φ η' It was natural to interpret the − � η f Ω hadrons as consecutive K* − resonances of elementary components. too many hadronic states: resonances ? The main problem was then to the figure shows the particle discoveries from 1898 to measure the properties of the the '60s; their abundance and regularity, as a function components and possibly to of quantum numbers like charge and strangeness, were observe them. suggesting a possible sequence, similar to the Mendeleev table [FNSN]. [and the leptons ? ... see later]

Paolo Bagnaia - PP - 01 9 2/3 hadrons : "elementary" or composite ?

1949 : E.Fermi and C.N. Yang proposed that ALL the resonances were bound state p-n.

Chen-Ning Yang Shoiki Sakata Enrico Fermi (杨振宁 - 楊振寧, (坂田昌一, Yáng Zhènníng) Sakata Shōichi) 1956 : Sakata extended the Fermi- Yang model including the Λ, to account for strangeness : all hadronic states were then composed by (p, n, Λ) and their antiparticles.

Paolo Bagnaia - PP - 01 10 3/3 hadrons : "elementary" or composite ? 1961 : M. Gell-Mann and Y. Ne’eman mention an internal structure. The (independently) proposed a new name was invented by Gell-Mann classification, the Eightfold Way, and comes from the "eight based on the symmetry group SU(3). commandments" of the Buddhism. The classification did NOT explicitly

The Quark Idea bottom (up, down, strange) charm 1960 1970 1980 1990

Murray Yuval Ne'eman ± (יובל נאמן) Gell-Mann J/ψτ D ϒ Λcηc B W Z top ψ'χc ϒ' Σc Ds ψ" ϒ" Ξc 1990 2000

Bs t Many more hadrons Λb have been discovered. Warning : "t" is a quark, not a hadron (in modern language).

Paolo Bagnaia - PP - 01 11 1/5 the Eightfold Way: 1961-64

All hadrons (known in the '60s) are classified in the plane (I3 − Y), (Y = strong hypercharge): Q=0 Q=+1 mesons Y JP = 0− K0 K+ I3 = Iz = third component of isospin; +1 Y = B + S [baryon number + strangeness]. Q=−1

π0,η0, η’ The strangeness S, which contributes to Y, had 0 π– π+ the effect to enlarge the isospin symmetry group SU(2) to the larger SU(3): Special Unitarity group, with dimension=3. -1 K– K0 � -1 0 +1 I3 The Gell-Mann − Nishijima formula (1956) was :

Q = I3 + ½(B+S) including heavy flavors [B:baryon, B:bottom] :

Q = I3 + ½(B+S+C+B+T)

This symmetry is now called "flavor SU(3) [SU(3)F]", to distinguish it from the "color SU(3) [SU(3)C]", which is the exact symmetry of the strong interactions in QCD. see later

Paolo Bagnaia - PP - 01 12 2/5 the Eightfold Way: SU(3) The particles form the multiplets of SU(3) . F mesons Each multiplet contains particles that have Q=0 Q=+1 Y JP = 0− 0 + the same spin and intrinsic parity. The basic +1 K K multiplicity for mesons is nine (3×3), which Q=−1 splits in two SU(3) multiplets: (octet + singlet). For baryons there are octects� + π0,η0, η’ 0 π– π+ decuplets. The gestation of SU(3) was long and difficult. It both explained the multiplets of -1 K– K0 known particles/resonances, and (more � exciting) predicted new states, before they -1 0 +1 I3 were actually discovered (really a triumph). In principle, in a similar way, the discovery ± However, the mass difference p − n (or π of heavier flavors could be interpreted with − π0) is < few MeV, while the π − K (or p − higher groups (e.g. SU(4)F to incorporate Λ) is much larger. Therefore, while the the charm quark, and so on). However, isospin symmetry SU(2) is almost exact, the these higher symmetries are broken even symmetry SU(3)F, grouping together more, as demonstrated by the mass strange and non-strange particles, is values. Therefore, SU(6)F for all known substantially violated. mesons JP = 0− is (almost) never used.

Paolo Bagnaia - PP - 01 13 3/5 the Eightfold Way: mesons JP=1−

Another example of a multiplet: the octet of Q=0 Q=+1 vector mesons : Y +1 K*0 K*+ m ≈ 890 MeV Q=−1 Y Q=0 Q=+1 0 K+ +1 K m ≈ 495 MeV ρ0,ω,φ 770 (ρ) Q=−1 0 ρ– ρ+ 780 (ω) π0,η0, η’ 1020 (φ) 0 π– π+ ≈ 140 MeV

*0 -1 K– K0 ≈ 495 MeV -1 K*– K 890 MeV -1 0 +1 � I3 � -1 0 +1 I3

meson resonances JP = 1− P − [mesons J = 0 ] (all discovered by 1961).

Paolo Bagnaia - PP - 01 14 4/5 the Eightfold Way: baryons JP=½+

Y Q=0 Q=+1 P + + Octet of baryons J = ½ . 0 K +1 K m ≈ 495 MeV Q=−1 Y Q=0 Q=+1 π0,η0, η’ – + 0 π π ≈ 140 MeV +1 n p m ≈ 939 MeV Q=−1

-1 K– K0 ≈ 495 MeV 0 -1 0 +1 Σ ,Λ 1116 (Λ) � I3 0 Σ– Σ+ 1193 (Σ)

mesons: Y = S -1 Ξ– Ξ0 1318 MeV baryons: Y = S + B -1 0 +1 I3 notice the masses: for mesons, because of (K ↔ K) the masses of an octet are symmetric wrt (S=0, I3=0), while for baryonsℂℙ𝕋𝕋 the mass� increases as –S [because the s-quark (S= –1) is heavier than u/d, but they did not know it]

Paolo Bagnaia - PP - 01 15 5/5 P 3 + the Eightfold Way: baryons J = /2

P 3 + The next multiplet of baryons is a Decuplet of baryon resonances J = /2 Y decuplet JP = ³∕₂+. ∆− ∆0 ∆+ ∆++ +1 m ≈ 1232 MeV When the E.W. was proposed, they knew only 9 members of the 153 MeV multiplet, but can predict the last Σ*– Σ*0 Σ*+ member: 0 1385 MeV • it is a decuplet, because of E.W.; 148 MeV • the state Y = –2, I3 = 0 (→ Q = – Ξ*– Ξ*0 Q=+1 B -1 1533 MeV 1, S = –3, =1) must exists; • call it Ω–; 150 MeV ??? Q=0 • look the mass differences vs Y: – -2 Ω ≈1680 MeV • mass linear in Y → mΩ– ≈ 1680 Q=−1 MeV (NOT an E.W. requirement, but a reasonable assumption); -1 0 +1 I3 • the conservation laws set the dynamics of production and when the Eightfold Way was first decay of the Ω–. proposed, this particle (now called Ω−) was not known → see next slide.

Paolo Bagnaia - PP - 01 16 1/2 the discovery of the Ω– The particle Ω−, predicted () in 1962, was discovered in 1964 by N.Samios et al., using () From a 1962 report: the 80-inch hydrogen bubble chamber at Ξ ∼ Brookhaven (next slide). Discovery of * resonance with mass 1530 MeV is announced […]. Ω− The can only decay weakly to an S = –2 [As a consequence,] Gell-Mann and (1) final state : Ne’eman […] predicted a new particle and Ω− → Ξ0 π− ; → Ξ− π0 ; → Λ0 K− ; all its properties: − [a posteriori confirmed by the measurement • Name = Ω (Omega because this particle -10 is the last in the decuplet); τΩ- ≅ 0.82 × 10 s] • Mass ≈ 1680 MeV (the masses of ∆, Σ* ______and Ξ* are about equidistant ∼150 MeV); (1) Since the electromagnetic and strong • Charge = −1; interactions conserve the strangeness, the lightest 3 (non-weak) S- and B- conserving decay is : • Spin = /2; Ω− → Ξ0 K− [S : −3 → −2 −1, B : +1 → +1 +0] • Strangeness = −3, Y = −2; which is impossible, because • Isospin = 0 (no charge-partners); -10 m(Ω) ≈ 1700 MeV < m(Ξ) + m(K) ≈ 1800 MeV. • Lifetime ∼10 s, because of its weak decay, since strong decay is forbidden(1); Therefore the Ω− must decay via strangeness- − 0 − − − 0 violating weak interactions : the Ω− lifetime • Decay modes: Ω → Ξ π or Ω → Ξ π . reflects its weak (NOT strong NOR e.m.) decay.

Paolo Bagnaia - PP - 01 17 2/2 the discovery of the Ω–: the event

the Ω– observation required both genius and luck (e.g. K− + p → Ω− + K+ + K0 compute the probability of the two γ conversions in H2): Ξ0 + π− (∆S = 1 weak decay)

π0 + Λ (∆S = 1 w.d.)

π− + p(∆S = 1 w.d.)

γ + γ (e.m. decay)

e+e−

Brookhaven National Laboratory 80-inch hydrogen bubble chamber - 1964

Paolo Bagnaia - PP - 01 18 1/2 the static quark model In 1964 M. Gell-Mann and G. Zweig proposed independently that all the hadrons are composed of three constituents, that Gell-Mann called(1) quarks. This model, enriched by both extensions (other quarks) and dynamics (electroweak interactions and QCD) is still the basis of our understanding of the elementary particles, the Standard Model(2).

In this chapter we consider only the static 1969 : Gell-Mann is awarded Nobel Prize "for properties of the three original quarks. his contributions and discoveries concerning Sometimes, in the literature, it is referred as the classification of elementary particles and their interactions" the naïve quark model. . ______(1) The name so whimsical was taken from the quark hypothesis was a mathematical (now) famous quote "Three quarks for Muster convenience or reality. Today, as shown in the Mark !", from James Joyce's novel "Finnegans following, our understanding is clearer, but Wake" (book 2, chapt. 4). complicated: the quarks are real (to the extent

(2) that all QM particles are), but they cannot be At that time it was not clear whether the seen as isolated single objects.

Paolo Bagnaia - PP - 01 19 2/2 the static quark model: u d s The hypothesis: u d s c b t • three quarks u, d, and s (up, down, strange); B baryon ⅓ ⅓ ⅓ • quarks (q): standard Dirac fermions with spin ½ and fractional charge (±⅓e ±⅔e); J spin ½ ½ ½ • antiquarks (q̄): according to Dirac theory, I isospin ½ ½ 0 the q-antiparticles; rd I3 3 i-spin ½ ½ 0 • baryons: combinations qqq (e.g. uds, uud); S strang. 0 0 1 − • antibaryons: three antiquarks (e.g ūūd̄); Y B+S ⅓ ⅓ ⅔ − • mesons: pairs qq̄ (e.g uū, ud̄, sū); Q I3 + ½Y ⅔ ⅓ ⅓ − • "antimesons": a q̄q pair: the mesons are their own antiparticles, i.e. "anti-mesons" = − − c, b, t not yet discovered in mesons. the '60 !!! see § 3

The quarks form a triplet, which is a basic d Y u s̄ Y representation of the group SU(3). Quarks I3

may be represented in a vector shape in the I3

plane I3 − Y; their combinations (= hadrons) s ū d̄ are the sums of such vectors.

Paolo Bagnaia - PP - 01 20 1/3 The mesons

"Build" the mesons qq̄ with these rules : • in the space I - Y, sum "vectors" (i.e. quarks and d u s̄ 3 , antiquarks) to produce qq̄ pairs, i.e. mesons; s ū d̄ • all the combinations are allowed:

– 8 1 3 ⊗ 3 = = ⊕

• the pseudoscalar mesons (JP=0-) are qq̄ states in s-wave with opposite spins ( ⇑ ⇓ ).

Paolo Bagnaia - PP - 01 21 2/3 The mesons: JPC=0–+

More specifically, with s-wave (JPC=0–+), Notice that π0, η, η’ are combinations we get the "pseudoscalar" nonet : (mixing) of the three possible qq̄ states see later d u s̄ (for the mixing parameters ) : , s ū d̄

Y Y ds̄ us̄ K0 K+ +1 +1

d u

dū uū + dd̄ + ss̄ ud̄ 0 3 0 π– π+ π0, η0, η’

s -1 -1 – sū sd̄ K– K0

-1 0 +1 -1 0 +1 I3 I3

Paolo Bagnaia - PP - 01 22 3/3 The mesons: JPC=1– –

If JPC = 1– – (i.e. spin ⇑ ⇑ ), the "vector" Notice that ρ0, ω, ϕ are combinations nonet : (mixing) of the three possible qq̄ states :

d u Y Y ds̄ us̄ K*0 K*+ s +1 +1

dū uū + dd̄ + ss̄ ud̄ – + 0 3 0 ρ ρ ρ0, ω, φ

-1 -1 – sū sd̄ K*– K*0

-1 0 +1 -1 0 +1 I3 I3

Paolo Bagnaia - PP - 01 23 1/5 Meson quantum numbers: JPC • Parity : the quarks and the antiquarks • Therefore we can apply (see before): have opposite P : P = (−1)L+1; L L L+1 Pqq̄ = P1P2 (−1) = −1 (−1) = (−1) . C = P × = (−1)L+S; G = (−1)L+S+I (only for the eigenstates Charge conjugation : for mesons, which are 𝒮𝒮 of , e.g. π±0/η, NOT K0). also eigenstates, = , parity followed by spin swap (see before). 𝔾𝔾 ℂ ℂ ℙ𝕊𝕊 • Conclusion: the first multiplets are (+ many others):

q̄ q L S J=L⊕S P C I G 0 + ℂ 0 0 − + 1 − q q̄ 0 q 0 − 1 1 − − 1 + 0 − 0 1 + − q̄ 1 + (swap spins of 1 ℙ 𝕊𝕊 0 + q and q)̄ 1 0,1,2 + + 1 − the "intrinsic spin" s of a meson (seen as an elementary object) becomes the "full PC −+ − − +− ++ ++ ++ angular momentum"⃗ J of a qq̄ composite. J = 0 , 1 , 1 , 0 , 1 , 2 , ... ⃗ Paolo Bagnaia - PP - 01 24 2/5 Meson quantum numbers : multiplets • For the lowest state nonets, these are • method (mainly bubble chambers) : the quantum numbers :  measure (zillions of) events; e.g. :

PC 2s+1 + + − − 0 L S J LJ I=1 state p̄p → π π π π π ; − + 1 0 0 S0 π(140)  look for "peaks" in final state 0 − − 3 combined mass, e.g. m(π+ π− π0); 1 1 S1 ρ(770) + − 1 0 1 P1 b1(1235)  the peaks are associated with high + + 3 mass resonances, decaying via strong 0 P0 a0(1450) 1 interactions (width → Γ → strength); + + 3 1 1 P1 a1(1260)  2 + + 3P a (1320) the scattering properties (e.g. the 2 2 angular distribution) and decay modes identify the other quantum numbers; • all these multiplets have main qn n = 1; • result : an overall consistent picture; • as of today ~20 meson multiplets have been (partially) discovered [PDG]. • Great success !!! • important activity from the ’50 to the ’70; still some addict;

Paolo Bagnaia - PP - 01 25 3/5 Meson quantum numbers : example

read it !!! only 3 pages !!! Three examples in π+p → X c) |pπ+| = 2.3-2.9 GeV b) a) m(π+π0) for X = π+π0p b) m(π+π−) for X = π+ π+ π−p c) m(π+ π−π0) for X = π+π+π−π0p

0 Q: which resonances ? why not the ρ ? a) ρ+(770) → π+π0; b) ρ0(770) → π+π−; c) η(548) → π+π− π0 ω(782) → π+π−π0.

Paolo Bagnaia - PP - 01 26 4/5 Meson quantum numbers : ρ0 →/ π0π0 Problem: ρ0 → π0π0 is allowed ? NO, because of : a) -parity b) Clebsch-Gordan coeff. in c) Spin-statistics

0 0 isospin space C(ρℂ) = −1; C(π ) = +1 [Povh, problem 15-1] |ρ0〉 = |I=1, I =0〉; therefore, since the initial 3 • S(ρ0) = 1, S(π0) = 0 0 state is a C-eigenstate, |π 〉 = |1, 0〉; → L(π0π0) = 1; −1 = (+1) × (+1) → NO therefore the decay is • ρ0 is a boson → wave 〈π0π0 ρ0〉 〈 〉 NB. A general rule : "a vector | = j1j2m1m2|J M = function symmetric; = 〈1 1 0 0 | 1 0 〉 = 0; cannot decay into two equal • the π0's are two equal (pseudo-)scalars". → NO. bosons → space wave But (a) and (b) do not hold function symmetric; [PDG, § 44 : for weak decays. Instead (c) • L=1 makes the wave is due to statistics + angular … 1 function anti-symmetric 1⊗1 momentum conservation, … 0 and is valid for all → NO. interactions. … … … … 0 0 … 0 [(c) also forbids Z → HH]

Paolo Bagnaia - PP - 01 27 5/5 Meson quantum numbers : decays General comments on particle decays ( D ): • all these D actually exist and contribute Γ Σ Γ τ Γ • D exist if no selection rule S forbids them; to the particle T = j j and = 1/ T; for a particle the partial Γ 's may vary a • only S valid for ALL interactions count; • j lot, mainly because of their couplings, e.g. • [e.g. strangeness conservation, which is Γ(ρ0 → π+π−) >> Γ(ρ0 → e+e−); violated in weak interactions is NOT included;] • in practice, when strong D exist, they are • [be careful: some S are "non-obvious", but still dominant; valid, e.g. NO ρ0 → π0π0 ;] • the other D have small, but non-zero • the S include conservation of "numbers" probability, e.g. BR(ρ0→e+e−) ≈ 4.7×10-5. (charge, baryon n., lepton n.), of 4- momentum, ang. momentum, …; NB the decay mode(s) and the ΓT, Γj are specific to the process and not necessarily similar for 0 + − 0 + − 0 0 • e.g. ρ π , ρ →e e , π γ, pπ ; ± 0 similar particles; e.g. ΓT(π ) << ΓT(π ) are very − τ ≈ • some strong D (e.g. ρ0 → π+π ), some different, as p (stable !) and n ( n 879 s). e.m. (ρ0 → e+e−), some weak (Λ → pπ0);

• for each D there is a matrix element Mj Γ and a partial width j; ? • in Γj the phase-space factor may be important (e.g. φ → K+K−);

Paolo Bagnaia - PP - 01 28 1/3 Meson mixing

Light JPC Q mass qq̄ of I =0 qq̄ I I S 3 mesons (1) 3 (1) (MeV) (2) π+, π0, π– ud̄, qq̄(2), dū 0–+ 1 1, 0, -1 0 1, 0, -1 140 ∼(uū−dd̄)/√2 η qq̄(2) 0–+ 0 0 0 0 550 ∼(uū+dd̄−2ss̄)/√6 η’ qq̄(2) 0–+ 0 0 0 0 960 ∼(uū+dd̄+ss̄)/√6 K+, K0 (3) us̄, ds̄ 0– ½ ½,-½ +1 1, 0 495 K0, K− (3) sd̄, sū 0– ½ ½, -½ -1 0, -1 495 + 0 – (2) – – �ρ , ρ , ρ ud̄, qq̄ , dū 1 1 1, 0, -1 0 1, 0, -1 770 ∼(uū−dd̄)/√2 ω qq̄(2) 1– – 0 0 0 0 780 ∼(uū+dd̄)/√2 φ qq̄(2) 1– – 0 0 0 0 1020 ∼ss̄ K*+, K*0 (3) us̄, ds̄ 1– ½ ½,-½ +1 1, 0 890 K*0, K*− (3) sd̄, sū 1– ½ ½, -½ -1 0, -1 890 �Notes : L+1 L+S S (1) (L=0, B=0) → P = (−) = −; C = (−) = (−) ; Q = I3 + ½Y = I3 + ½S; (2) The mesons π0, η, η', ρ0, ω, φ are mixing of uū ⊕ dd̄ ⊕ ss̄ (see next);

(3) States with strangeness ≠ 0 are NOT eigenstates of C; since they have I=½, no I3=0 exists.

Paolo Bagnaia - PP - 01 29 2/3 Meson mixing: JP = 0−, 1− Mesons are bound states qq̄. Consider only P − − ψ=−8 ,1 (u u dd) / 2  uds quarks (+ ūd̄s̄) in the nonets (J = 0 1 ,  ψ pseudo-scalar vector multi,I the and nonets) : ψ8,0 =(u u +− dd 2s s ) / 6   ideal case π+ π− + 0 • the states ( =ud̄, =dū, K =us̄, K =ds̄, ψ1 =(u u ++ dd s s ) / 3  K−=sū, K0=sd̄) have no quark ambiguity;

• but (uū dd̄ ss̄) have the same quantum 0 ps  � π(140) ≈ψ =(u u − dd) / 2 − numbers and the three states (ψ ψ 8 ,1  JP = 0, 8,0 8,1 η =ψps θ −ψps θ  ψ ) mix together (→ 2 angles per nonet); (550)8,0 cos ps 1 sin ps θ ≈− ° 1  ps 25 ; ps ps − η =ψ θ +ψ θ  • the physical particles (π0, η, η’ for 0 , ρ0, '(960)8,0 sin ps 1 cos ps ω, φ for 1−) are linear combinations qq̄; • (ψ ) decuples (π0 ρ0) (→ 1 angle only); ρ0v(770) ≈ψ =(uu − dd)/ 2  8,1 8 ,1  P − • θ and θ are computed from the mass φ(1020) =ψvv cos θ −ψ sin θ ≈ ss  J= 1, ps v 8,0 v 1 v  v v matrices* [PDG, §15.2]; θvect ≈36 ° . ω=ψ(780) 8,0 sinθv1 +ψ cos θ v ≈  θ ≈ ≈ -1  • notice: the vector mixing v 36° tan ≈+(u u dd) / 2  (1/√2), i.e. the φ meson is almost ss̄ only [i.e. φ → KK̄, see KLOE exp.]; * in principle, both the mass spectra and the mixing angles can be computed from QCD lagrangian ℒQCD ... waiting (... continue) for substantial improvements in computation methods.

Paolo Bagnaia - PP - 01 30 ? 3/3 Meson mixing: JP = 1− The decay amplitudes in the e.m. channels may be computed, up to a common factor, 1  and compared to the experiment; ρ=0 (770) (uu − dd); 2   0 +− q γ + e 1  Mfi(ρ ωφ → ee) ω(780)= (uu+ dd);  → j e− ∝α Q ; q̄ Qq √α 2  ∑ j q φ(1020)= ss;   Few problems :  2 0 + −  • the values are small*, e.g. BR(ρ →e e ) ≈ 0 +− 12− 1 1 -5 Γρ( → ee) ∝ − = ; 4.7×10 ;  2 33 2  2 • the phase-space factor is important,  +− 12− 1 1 → Γ( ω→ ee) ∝ + = ; especially for φ, which is very close to the  2 33 18 ss̄ threshold (mφ − 2 mK = few MeV).  2 +− 1 1  Γ( φ → ee) ∝ = ; However, the overall picture is clear: the   39 theory explains the data very well. ______9 : 1 : 2 (theo) ρ0ωφ →Γρωφ:: Γ Γ= * warning: the dominant decay modes are ±± strong; however, the e.m. decays ρ0ωφ → e+e−, with 8.8 2.6 : 1 : 1.7 0.4 (exp). a much smaller BR, are detectable → Γe.m. measurable → quark charges compared.

Paolo Bagnaia - PP - 01 31 1/5 The baryons The construction looks complicated, but in fact is quite simple : • add the three quarks one after the Y other; ddd udd uud uuu +1 1 3 3 1

• count the resultant multiplicity. 1 2 1

In group’s theory language : 1 1

3 6 3 3 ⊗ 3 ⊗ 3 = 10 ⊕ 8 ⊕ 8' ⊕ 1 0 dds uds uus i.e. a decuplet, two octects and a 2 2 singlet. 1

3 3 [proof. : -1 dss uss (3 ⊗ 3) ⊗ 3 = (6 ⊕ 3) ⊗ 3; 1 d u 6 ⊗ 3 = 10 ⊕ 8; � -2 1 s 3 ⊗ 3 = 8 ⊕ 1. q.e.d.] sss -3/2 -1/2 +1/2 +3/2 Both� for 10, 8, 8' and 1 the three I3 quarks have L = 0.

Paolo Bagnaia - PP - 01 32 2/5 The baryons: quantum numbers

mass Baryons qqq JP I I S Q(1) 3 (MeV) p, n uud, udd ½+ ½ ½, -½ 0 1, 0 940 Λ uds ½+ 0 0 -1 0 1115 Σ+, Σ0, Σ− uus, uds, dds ½+ 1 1, 0, -1 -1 1, 0, -1 1190 Ξ0, Ξ− uss, dss ½+ ½ ½,-½ -2 1, 0 1320 ++ + 0 − 3 + 3 3 1 1 3 ∆ , ∆ , ∆ , ∆ uuu, uud, udd, ddd /2 /2 /2, /2, - /2, - /2 0 2, 1, 0, -1 1230 + 0 − 3 + Σ* , Σ* , Σ* uus, uds, dds /2 1 1, 0, -1 -1 1, 0, -1 1385 *0 *− 3 + Ξ , Ξ uss, dss /2 ½ ½,-½ -2 1, 0 1530 − 3 + Ω sss /2 0 0 -3 -1 1670

Notes :

(1) Q = I3 + ½Y = I3 + ½(B + S); B = 1. 10 8 ⇑ ⇑ ⇑ ⇑ ⇑ ⇓

Paolo Bagnaia - PP - 01 33 3/5 The baryons: the octet JP = ½+ The lowest mass multiplet is an octet, which contains the familiar p and n, a triplet of S=-1 (the Σ’s) a Y singlet S=-1 (the Λ) and a doublet n p of S=-2 (the Ξ’s, sometimes called +1 “cascade baryons”). ddu duu

− + The three quarks have ℓ = 0 and Σ Σ0 Λ Σ ⇓ 0 spin ( ), i.e. a total spin of ½. dds uds uus

The masses are : Ξ− Ξ0 -1 dss uss • ~ 940 MeV for p and n; • ~1115 MeV for the Λ; d u Σ • ~1190 MeV for the ’s; -2 s • ~1320 MeV for the Ξ’s;

(difference of < few MeV in the -3/2 -1/2 +1/2 +3/2 I3 isospin multiplet, due to e-m interactions.)

Paolo Bagnaia - PP - 01 34 4/5 P 3 + The baryons: the decuplet J = /2 The decuplet is rather simple (but there is a spin/statistics problem, Y see later). The spins are aligned (⇑ ⇑ ⇑), to produce an overall ∆− ∆0 ∆+ ∆++ 3 +1 J= /2. ddd ddu duu uuu

The masses, at percent level, are : Σ*− Σ*0 Σ*+ 0 dds uds uus ~ 1230 MeV for the ∆’s; ~ 1385 MeV for the Σ*’s, − Ξ* Ξ* Ξ*0 ~ 1530 MeV for the ’s -1 ~ 1670 MeV for the Ω-. dss uss d u Notice that the mass split among Ω− -2 s multiplets is very similar, ~150 sss MeV (important for the Ω− -3/2 -1/2 +1/2 +3/2 discovery, lot of speculations, no I3 real explanation).

Paolo Bagnaia - PP - 01 35 5/5 The baryons: example

0 0* +− ++ + − + + A - Σcc , Σ →Λ c π B - Ξcc →Λ c K π π

++ Check conservation of: quark Ξcc :ucc;  a. baryon n. Recently, the LHCb Collaboration at content: ΣΣ0 0* cc, : ddc;  b. charge LHC has realized a nice search for +  c. charm Λc :udc; baryons made with heavy quarks. −  d. strangeness K− : us; [these two examples should stay in § 3, Write the Feynman because they contain the c quark] π± : ud, ud. diagrams of the decays → after § 6

Paolo Bagnaia - PP - 01 36 1/5 SU(3)

• For the SU(2) symmetry, the generators are 01 +−11 1 1  σ= ψ= ψ= the Pauli matrices. The third one is 1 ;; 11    ; 10 22 1 −1  associated to the conserved quantum 01− +−11  1  1  number I . σ2 = i; ψ 22 =  ; ψ =   ; 3 10 22  i −i 

10 +−  1  0 σ=3  ; ψ= 33 ;; ψ=  • For SU(3), the Gell-Mann matrices Tj (j=1-8) 01−   0  1 are defined (next page). 222 σ=σ=σ=−σσσ=123i 123 I; σσ = εσ Pauli matrices • The two diagonal ones are associated to i, j 2i∑ ijk k. k and eigenvectors the operators of the third component of

isospin (T3) and hypercharge (T8).

• The eigenvectors |u〉 |d〉 |s〉 are associated with the quarks (u, d, s).

Paolo Bagnaia - PP - 01 37 2/5 SU(3) : Gell-Mann matrices

010 0− 10 1 0 0 λ=λ= λ= − 121 0 0; i1 0 0; 3 0 1 0;  000 000 000 Gell-Mann matrices λi 001  00− 1   000     λ = 000;λ = i00 0;λ = 001; 1 45   6  Tii= λ     2 100  10 0   010 00 0  10 0 1 λ= − λ=  78i0 0 1;  0 1 0 . 3  01 0  00− 2

100 8 16 λ=2  ∑ j 0 1 0 diagonal. j1= 3  001 i 8 U= 1 +∑ ελjj unitary matrix, det U= 1. 2 j1=

Paolo Bagnaia - PP - 01 38 3/5 SU(3) : eigenvectors

Definition of I3, Y, quark eigenvectors

and related relations : Y

100 100 d +1/3 u 111 1  Tˆ = λ= 0 − 1 0; Y = λ= 0 1 0 ; 33223 83   − 000 002 -1/2 +1/2 I3 10  0    XXXu= 0; d= 1; s= 0;    -2/3 s    00  1 11 Y Tˆˆˆ u=+= u; T d− d; T s = 0; +2/3 s̄ 322 33 11 2 Yuˆˆˆ=+= u; Yd+ d; Ys =− s; 33 3 +1/2 I ˆˆˆ11 -1/2 3 T3 u=−= u ; T 33 d+ d ; T s = 0; 22 ū -1/3 d̄ 1 12 Yuˆ = − u; Ydˆˆ=−= d; Ys+ s . 33 3

Paolo Bagnaia - PP - 01 39 4/5 SU(3) : operators

The ladder operators T±, U±, V± : T±±±=±=±=± T12 iT ; U T 67 iT ; V T 45 iT ; As an example, take V+ :

001 00− i 001V+ u= − s; d +1/3 u 1i  =+= + = = V++ T45 iT 000 000 000;Vd 0; 22 100 100 000 V+ s= 0; -1/2 +1/2 I3 

001  1     -2/3 s = = Vu+  0000   0; Y    Y 0000   +2/3 s̄ V+ U+ 001  0  =    = Vd+  0001   0;    0000   T- T+ I3 -1/2 +1/2 I3 001  0 1 ū -1/3   d̄ V U- = = = - Vs+  0000  0 u.   0001  0

Paolo Bagnaia - PP - 01 40 5/5 SU(3) : ladder operators

T+|d> = |u> T |u> = |d> – The ladder operators T±, U±, V±. d u

V+ U+

U+|s> = |d> V+|s> = |u> V |u> = |s> U–|d> = |s> s –

T- T+ I3

s̄ U V- -

V+|ū> = -|s̄> U+|d̄> = -|s̄>

V–|s̄> = -|ū> U–|s̄> = -|d̄>

ū d̄

T+|ū> = -|d̄> § QCD T–|d̄> = -|ū>

Paolo Bagnaia - PP - 01 41 1/2 Color : a new quantum number Consider the ∆++ resonance: • JP=3/2+ (measured); → quark/spin content [no ambiguity]: |∆++> = | u⇑ u⇑ u⇑ > • wave function : ++ ψ(∆ ) = ψspace × ψflavor × ψspin NO !!! Oskar W. Greenberg Moo-Young Han Yoichiro Nambu (한무영) (南部陽一郎, Nambu Yōichirō) But from symmetry considerations: ∆++ 3 the ∆++ is lightest uuu state Anomaly : the is a spin /2 fermion and its → L = 0; function MUST be antisymmetric for the exchange → ψ symmetric; of two quarks (Pauli principle). However, this space function is the product of three symmetric → ψ ψ flavor and spin symmetric; functions, and therefore is symmetric → ???. → ψ(∆++) = sym.×sym.×sym. = sym. The solution was suggested in 1964 by Greenberg, ++ → the ∆ is a fermion, i.e. asym. later also by Han and Nambu. They introduced a → NO !!! new quantum number for strongly interacting particles, composed by quarks : the COLOR.

Paolo Bagnaia - PP - 01 42 2/2 Color : why's and how's

The idea [see §6, the following is quite (where ur, ug, ub are the color functions for naïve] : u quarks of red, green, blue type).

1. quarks exist in three colors (say Red, Then ψcolor is antisymmetric for the Green and Blue, like the TV screen(*); exchange of two quarks and so is the global 2. they sum like in a TV-screen : e.g. when wave function. RGB are all present, the screen is iwhite ; The introduction of the color has many 3. the "anticolor" is such that, color + other experimental evidences and anticolor = white (e.g. R = G + B); theoretical implications, which we will discuss in the following. 4. anti-quarks bring ANTI� -colors (see ______previous point); (*) however, these colors are in no way similar to 5. Mesons and Baryons, which are made of the real colors; therefore the names "red-green- quarks, are white and have no color: blue" are totally irrelevant. they are a "color singlet". Therefore, we have to include the color in the complete wave function; e.g. for ∆++ : R C ++ ψ(∆ ) = ψspace × ψflavor × ψspin × ψcolor 1 2 3 1 2 3 1 2 3 ψcolor = (1/√6) (ur ugub + ugubur + ubur ug G B M Y 1 2 3 1 2 3 1 2 3 − ugur ub − ur ubug − ubugur )

Paolo Bagnaia - PP - 01 43 1/3 Summary: Symmetries and Multiplets for a complete discussion, [BJ 10]. isospin multiplet must have the same 1. Since the strong interactions conserve spin and the same parity. isotopic spin (" "), hadrons gather in I- 5. s is also invariant with respect to multiplets. Within each multiplet, the unitary representations of SU(2). The 𝕀𝕀 states are identified by the value of I3. quantumℍ numbers which identify the 2. If no effect breaks the symmetry, the components of the multiplets are as members of each multiplet would be many as the number of generators, mass-degenerate. The electromagnetic which can be diagonalized interactions, which do not respect the I- simultaneously, because are mutually symmetry, split the mass degeneration commuting. This number is the rank of (at few %) in I-multiplets. the Group. In the case of SU(2) the rank is 1 and the operator is 3. 3. Since the strong interactions conserve I, ε -operators must commute with the 6. Since [ j , k] = i jkm 𝕀𝕀m, each of the generators commutes with 2 : strong interactions Hamiltonian (" s") 2 2 𝕀𝕀 2 𝕀𝕀 2 𝕀𝕀 and𝕀𝕀 with all the operators which in turn = 1 + 2 + 3 . ℍ 𝕀𝕀 commute with s. Therefore 2, obviously hermitian, can 𝕀𝕀 𝕀𝕀 𝕀𝕀 𝕀𝕀 4. Among these operators, consider the be diagonalized together with 3. ℍ 𝕀𝕀 angular momentum and the parity . (continue𝕀𝕀 …) As a result, all the members of an 𝕁𝕁 ℙ Paolo Bagnaia - PP - 01 44 2/3 Summary: Symmetries and Multiplets

7. The eigenvalues of and 3, can "tag" 11. In reality, since the differences in mass the eigenvectors and the particles. between the members of the same 𝕀𝕀 𝕀𝕀 multiplet are ~20%, the symmetry is 8. This fact gives the possibility to regroup "broken" (i.e. approximated). the states into multiplets with a given value of I. 12. Since the octets are characterized by two quantum numbers (I and Y), the 9. We can generalize this mechanism from 3 symmetry group has rank = 2, i.e. two the isospin case to any operator : if we of the generators commute between can prove that is invariant for a given them. kind of transformations, then: a. look for anℍappropriate symmetry 13. We are interested in the "irreducible group; representations" of the group, such b. identify its irreducible representations that we get any member of a multiplet and derive the possible multiplets, from everyone else, using the transformations. c. verify that they describe physical states which actually exist. (… continue …) 10. This approach suggested the idea that Baryons and Mesons are grouped in two octets, composed of multiplets of isotopic spin.

Paolo Bagnaia - PP - 01 45 3/3 Summary: Symmetries and Multiplets 14. The non-trivial representation (non-trivial = other than the Singlet) of lower dimension is called "Fundamental Y representation". d +1/3 u 15. In SU(3) there are eight symmetry generators. Two of

them are diagonal and associated to I3 and Y. -1/2 +1/2 I3 16. The fundamental representations are triplets (→ quarks), from which higher multiplets (→ hadrons) are derived : -2/3 s mesons: 3 ⊗ 3 = 1 ⊕ 8 ; Y baryons: 3 ⊗ 3 ⊗ 3 = 1 ⊕ 8 ⊕ 8 ⊕ 10. s̄ � +2/3 17. This purely mathematical scheme has two relevant applications: a. "flavour SU(3)", SU(3) with Y and I for the quarks F F 3F -1/2 +1/2 I3 uds – this symmetry is approximate (i.e. "broken"); ū -1/3 d̄ b. "color SU(3)", SU(3)C with YC and I3C for the colors rgb; this symmetry is exact.

Paolo Bagnaia - PP - 01 46 References

1. e.g. [BJ, 8]; 2. large overlap with [FNSN, 7] 3. isospin and SU(3) : [IE, 2]; 4. group theory : [IE, app C]; 5. color + eightfold way : [IE, 7-8] 6. G.Salmè – appunti.

Paolo Bagnaia - PP - 01 47 End

End of chapter 1

Paolo Bagnaia - PP - 01 48