Chapter 5.2 by Manu Verma

Chapter 5 Quark Model

Chapter 5 5.1 Introduction 5.2 Quark Model 5.3 Meson and Baryon wave function 5.4 Magnetic moment and masses of baryons 5.5 Interactive Exercise

5.2 Quark Model

Quark Model

Particle Physics

Quark Model An understanding of the Eightfold Way came in 1964, when Gell-Mann and Zweig independently proposed that all hadrons are composed of more elementary constituents which Gell-Mann called quarks. Quarks come in three types forming a triangular Eightfold way pattern. d u S=0

Q= 2/3 S=1

s Q = -1/3

Fig 1: The quarks

The up quark (u) carries a charge 2/3 and strangeness 0 , down quark (d) carries a charge of -1/3 and strangeness 0, strange quark (s) carries a charge of -1/3 and strangeness S=-1. To each quark (q) there corresponds an antiquark (q), with opposite charge and strangeness. s S = -1

S=0

Fig 2: The antiquarks Dayalbagh Educational Institute

d Q = -2/3 Q = 1/3 4

Quark Model

Particle Physics

And there are two composition rules for formation of baryon and meson: 1. Every baryon is composed of three quarks (and every antibaryon is composed of three antiquarks). 2. Every meson is composed of a quark and an antiquark. With this it is a matter of elementary arithmetic to construct baryon decuplet and meson octet. To construct baryon decuplet combination of three quarks is taken and we add up their charge and strangeness. Following is the table showing quark combinations and its corresponding charge and strangeness and baryon it forms. Quark combination

Q (charge)

S (strangeness)

Baryon

uuu

Δ++

uud

Δ+

udd

Δ0

ddd

-1

Δ-

uus

-1

Σ*+

uds

-1

Σ*0

dds

-1

Σ*-

uss

-2

Ξ*0

dss

-1

-2

Ξ*-

sss

-1

-3

Ω-

Table 1: Baryon decuplet Dayalbagh Educational Institute

Quark Model

Particle Physics

There are ten combinations of three quarks. Three u’s of Q = 2/3each, yield a total charge +2 and a strangeness of zero which constitute Δ++ particle. Continuing down the table, we find all the members of the decuplet ending with Ω-. Quark-antiquark combination yields the meson table: Quark content

Q (charge)

S (strangeness)

Meson

π0

π+

-1

π-

K+

-1

K-

-1

Table 2: Meson nonet

There are nine combinations and eight particles in the meson octet. Quark model required a third particle besides π0 and η with S = Q = 0. That particle is η‫ ׳‬was found experimentally. In Eightfold way it is classified as singlet. According to Quark Model η‫ ׳‬belongs with other eight mesons to form meson nonet. As uu, dd and ss all has Q = S = 0, it was not possible to say which is π0, η‫ ׳‬and η. So, final conclusion is we have three mesons with Q = S = 0. Dayalbagh Educational Institute

Quark Model

Particle Physics

For baryon octet if we take decuplet and knock off the three corners where quarks are identical uuu, sss, ddd and double the centre where all three quarks are different, we obtain eight states in baryon octet. Same set of quarks accounts for octet, it is just some combinations do not appear at all and one appears twice. The same combination of quarks can give a number of different particles delta-plus and proton composed of two u’s (two up quarks) and a d quark. Pi-plus and rho-plus both are composed of ud. Thus we can construct an infinite number of hadrons out of only three quarks. Some things were absolutely excluded in the quark model. No combination of quarks can produce a baryon with S = 1 or Q = -2. There is no meson with charge +2 and strangeness -3. There were major experimental searches for these socalled ‘exotic’ particles, but no evidence was found for these particles. Quark model suffered from one profound embarrassment that individual quark has not been seen. If proton is composed of three quarks, when we hit proton very hard quark should come out. It is not hard to recognize quark as they have fractional charge. An ordinary Millikan oil drop experiment would clinch the identification. At least one of the quarks should be absolutely stable, as there is no lighter particle present with fractional charge in which quark can decay. Dayalbagh Educational Institute

Quark Model

Particle Physics

So quarks seems to be easy to produce, easy to identify and easy to store. Then why individual quark has not been seen? Quark confinement is the answer of all these questions. Quark confinement means quarks are confined within baryons and mesons. E

absolutely

ven if all quarks are confined inside hadrons, this does not mean they are inaccessible to experimental study. One can explore the interior of proton in same way as Rutherford probed the inside of an atom. Such experiments were carried out in the late 1960s using high energy electrons at the Stanford Linear Accelerator Center (SLAC). They were repeated in early 1970s using neutrino beams at Cern. The results of these ‘deep inelastic scattering’ experiments were similar to that of Rutherford’s: most of the incident particles pass right through, whereas a small number bounce back sharply. This means that the charge of proton is concentrated in small lumps, just as Rutherford’s results indicated that the positive charge in an atom is concentrated at the nucleus. In case of proton the evidence suggests three lumps, instead of one. Lumps inside proton were called patrons. This was a strong support to quark model, but it was not conclusive support.

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Quark Model

Particle Physics

There was a theoretical objection to quark model that it violates the Pauli Exclusion Principle which states no two electrons or fermions can occupy the same state. Exclusion principle applies to quarks because they carry ½ spin. Delta plus plus Δ++ (uuu), delta minus Δ(ddd) and omega Ω (sss) constitutes three quarks in same state which is inconsistent with Pauli principle. In 1964, O.W. Greenberg proposed that quarks not only come in three flavors up, down and strange but each of these also comes in three colors red, green and blue. To make a baryon, we simply take one quark of each color thus three u’s in Δ++ , three d’s in Δ- and three s in Ω are no more identical. Since exclusion principle only applies to identical particles, the problem is solved. Color here has absolutely no connection with ordinary meaning of the word. Redness, greenness and blueness are some new property of quark in addition to strangeness and charge. A red quark carries one unit of redness, zero units of blueness and greenness. All naturally occurring particles are colorless. Colorless means either the total amount of each color is zero or all three colors are present in equal amounts. It mimics the optical fact that light beams of three primary colors combines to make white. This explains why we can’t have particle of two or four quarks and why individual quarks do not occur in nature. Dayalbagh Educational Institute

Quark Model

Particle Physics

November revolution and its aftermath (1974-1983 and 1995) 1964 to 1974 was barren time for elementary particle physics. Quark model was in an uncomfortable stage. Although it neatly explained eightfold way and correctly predicted the lumpy structure of proton, but it had two conspicuous defects. There was no experimental evidence of quarks and inconsistency with Pauli principle. What rescued quark model was not the discovery of free quark, or an explanation of quark confinement, or confirmation of color hypothesis, but something entirely different and unexpected that is discovery of psi meson (ѱ). The ѱ meson was discovered at Brookhaven by a group under C.C. Ting in summer in 1974. ѱ meson was discovered independently by Richter’s group at SLAC. The two teams then published simultaneously, Ting naming the particle j and Richter calling it ѱ.

j/ ѱ was an electrically neutral, extremely heavy meson whose weight is more than three times the weight of proton. This particle was unusual because of its extraordinarily long lifetime10-20 seconds which is 1000 times longer than any particle of comparable mass. Its long lifetime was beginning of a fundamental new physics, so the discovery of ѱ meson is known as November Revolution. Dayalbagh Educational Institute

Quark Model

Particle Physics

The nature of ѱ meson was explained by the quark model that ѱ is a bound state of a new fourth quark, charm quark (c) and its antiquark (c). ѱ (ѱ meson) = cc.

The idea of a fourth flavor and its name was introduced many years earlier by Bjorken and Glashow. One of the explanations given for the existence of fourth quark was a parallelism between leptons and quarks. Leptons: e, νe, μ, νμ Quarks: d, u, s As there are four leptons present then four quarks should also be there. Quark model explained the properties of ѱ meson but with some implications.

If fourth quark exists, there should be all kind of new baryons and mesons carrying various amounts of charms. ѱ meson itself carries no net charm, for if c is assigned a charm of +1, then cc will have a charm of -1 then the net charm of ѱ meson is zero. To confirm charm hypothesis, it was important to produce a particle with naked charm. The first evidence for charmed baryons, = udc and

= uuc appeared in 1975, followed by

Ξc = usc and Ωc = ssc. The first charmed mesons D0 = cu and D+ =cd was discovered in 1976. After this quark model was put back on its feet.

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Quark Model

Particle Physics

In 1975 a new lepton was discovered which spoiled Glashow symmetry of leptons and quarks. This new particle τ has its own neutrino, so there were six leptons and only four quarks. Two years later a new heavy meson (the upsilon) was discovered and was recognized as carrier of fifth quark b (beauty or bottom quark), upsilon meson is represented as ϒ and thus: ϒ (upsilon meson) = b b Immediately the search began for hadrons exhibiting ‘naked beauty’ or ‘bare bottom’. The first bottom baryon, = udb, was discovered in the 1980’s and the second = uub in 2006, in 2007 the first baryon with a quark from all three generations was discovered = dsb. The first bottom mesons B0 = bd and B- = bu were found in 1983. Early search for ‘toponium’ a tt meson analogous to the ѱ and ϒ was unsuccessful, both because the electron-positron colliders did not reach enough high energy and top quark is too short lived to form bound states. Until 1995 the existence of top quark was not established. In 1995 Tevatron finally accumulated enough data to sustain strong indications that in reaction: u + u (or d + d ) → t + t The top and anti-top particle is formed and it immediately decays, and by analyzing the decay products one is able to infer their fleeting appearance. This was a breif description of the discovery and stablization of quark model. Dayalbagh Educational Institute

Quark Model

Particle Physics

Isospin Heisenberg suggested in 1932 that proton and neutron could be thought of as different states of same particle, (spin up) and (spin down). Isospin is an abstract property of e`lementary particles. Particles are assigned a total isospin quantum number I and an isospin projection in one direction I 3 Strong force is invariant under any rotation in isospin space. Strong force treats all particles with same total isospin I.

SU(2) Symmetry SU(2) is a special Unitary group of matrices which follow the rule: a b If A ϵ SU(2), A= -b* a* Then, determinant of A=aa*+bb*=1 SU(2) has three dimension or three degrees of freedom because it requires four real numbers,the real and imaginary components of a and b. If τ1, τ2 and τ3 are the representation of isospin -1/2 matrix operators, isospin wavefunctions can be expressed as a two component vector with p and n representing the spin-up and spin-down basis elements. To verify that τ3 admits ±1/2 as its eigen values,when we compute τ3n=-1/2 n. Dayalbagh Educational Institute

Quark Model

Particle Physics

There are three isospin 1/2 matrices and 3 pauli spin matrices which generates the group SU(2). SU(2) can be considered to be acting on two dimensional space spanned by (p,n). This is called group representation. Representation of a group is a map which preserves multiplication and sends a member of a group to linear operator on a vector space. Internal isospin symmetry of actual particles is not exactly SU(2), its due to mass splitting between quarks. Mass splitting between lightest quarks (up and down) is at most 2 to 3%, thats why SU(2) is relatively good symmetry of lighest baryons.

SU (3) symmetry Just as in 1932 the proton and neutron were seen to form a pair, in 1950’s the Λ, the Σ’s and the Ξ’s together were observed to constitute a natural grouping within baryon family. All these particles carry spin ½ and have similar masses. They all were thought to be different states of one particle as Heisenberg suggested for proton and neutron. Eight baryons were thought to be as supermultiplets, means they belong in same representation of some enlarged symmetry group, in which SU(2) of isospin would be incorporated as a subgroup. The symmetry group is SU(3), the octets constitute eight dimensional representation of SU(3) and the decuplet a ten dimensional representation. Dayalbagh Educational Institute

Quark Model

Particle Physics

The fact which made this case more difficult than Heisenberg’s isospin symmetry was that no naturally occurring particles fall into the fundamental three dimensional representation of SU (3), as nucleons and later kaons do for SU (2). Up, down and strange quark together form a threedimensional representation of SU (3), which breaks down into an isodoublet (u and d quark) and an isosinglet (s quark) under SU (2). When the charmed quark came into existence, the flavor symmetry group of strong interaction expanded once again to SU (4). After the discovery of bottom quark, group representation expanded to SU (5) and finally to SU (6) after arrival of top quark. Isospin SU (2) is a very good symmetry as the members of an isospin multiplet differs in mass by at most 2 to 3%. But the Eightfold Way SU (3) is a badly broken symmetry because mass splitting within the baryon octet is around 40%. Symmetry breaking is even worse when we include charm because (udc) weight more than twice the Λ (uds) although they are in same SU (4) supermultiplets. It is worse still with bottom an absolutely terrible with top quark which doesn’t form bound states at all. Isospin is a good symmetry because the effective u and d quark masses are so nearly equal. The Eightfold Way is a fair symmetry because the effective mass of strange quark is not too far from up and down quark. Dayalbagh Educational Institute

Quark Model

Particle Physics

Classification of hadrons in terms of valence quarks [M. Gell-Mann, G. Zweig, 1964] Basic quark multiplet: (u, d, s) ⇔ fundamental representation of SU(3) Assign S(u) = 0, S(d) = 0, S(s) = -1 Hypercharge: Y ≡ B +S

⇨ centers multiplets in origin

Charge: Q = I3 + Y/2 Baryon number conservation ⇨ single quarks cannot be made or destroyed, only creation and annihilation of qq pair possible

Quark Multiplets

s 2/3

2/3 d

u 1/2

-1/2

-1/2 I3

1/2 d -2/3

s -2/3

Antiquark

Quark

Fig. 3: SU(3) quark and antiquark multiplets: Y ≡ B +S

Quantum numbers of quarks: Y ≡ B +S

Q = I3 + Y/2

Quark

Spin

1/2

1/3

2/3

1/2

1/3

1/2

1/3

-1/3

-1/2

1/3

1/2

1/3

-1/3

-1

-2/3

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Quark Model

Particle Physics

Mesons Made of q and q' For two flavors: u, d ⇨ isotriplet and isosinglet of qq' combinations (as for spin): 2 ⊗ 2 = 3 ⊕ 1 |I = 1, I3 = 1〉 = -ud isospin triplet

|I = 1, I3 = 0〉 = 1/√2 (uu -dd) |I = 1, I3 = -1〉 = du

isospin singlet

|I = 1, I3 = 0〉 = 1/√2 (uu + dd)

For three flavors: u, d, s ⇨ nine possible combinations (nonet) 3⊗3=8⊕1 SUf(3) decomposition of 9 possible qq' combinations (a)

(b)

us A

s su 3⊗3

⊕

sd 8

Fig. 4: The quark content of the meson nonet, showing the SU(3) decomposition in the I3 , Y plane.

Rule: superimpose center of gravity of antiquark multiplet on top of every site of the quark multiplet Dayalbagh Educational Institute

Quark Model

Particle Physics

A,B,C: I3 = Y = 0 ⇨ linear combinations of states uu, dd, ss states C (singlet) contains each flavor on equal footing:

C = √ 1/3 (uu + dd + ss ) A is taken to be a member of isospin triplet:

A = √ 1/2 (uu - dd ) B (isospin singlet) orthogonal to A and C:

B = √ 1/6 (uu + dd - 2ss ) Generally: singlet and octet states with I=0 (B,C) mix to form physical states These multiplets represent lowest energy bound states for given spin values expect them to have orbital angular momentum L=0 Excitation spectrum corresponding to qq' vibrations, rotations, etc. observed meson spectrum Quantum numbers of mesons in quark model (Intrinsic) spin: S = 0, 1 (coupling of two spin-½ quarks) Total angular momentum (“spin”): J= L+S L: relative orb. Ang. mom. of quarks

J = L+S...|L-S|

Parity: P = -1. (-1)L opposite intrinsic parity of q and qbar (Dirac eq.) Dayalbagh Educational Institute

from space inversion properties of spherical harmonics YLM 18

Quark Model

Particle Physics

Particle - antiparticle conjugation”)

conjugation

(“charge

Cq = q, Cq = q neutral qq' state consisting of a quark and its own antiquark is an eigenstate of C Eigenvalue of C: Cqq = qq = ± qq c

q exchange q, qbar (-1)

=± q

interchange spins (-1)S+1

interchange positions (-1)L

C = (-1)· (-1)L ·(-1)S+1 = (-1)L+S Extension to charged particles made of ud and du G Parity = Charge conjugation + rotation in isospin space about y-axis G = CeiπI2 = C (-1)I = (-1)L+S+I System of n pions: G = (-1)n approximately conserved in strong interactions Mixing of singlet and octet states with I = 0 ⇨ physical states JP = 0 - :

η = B · cos θP- C · sin θP

B: octet state

η' = B · sin θP + C · cos θP C: singlet state pseudoscalar mixing angle: θP = -10°...-20° Dayalbagh Educational Institute

Quark Model

Particle Physics

Φ = B · cos θv - C · sin θv

JP = 1 - :

ω = B · sin θv + C · cos θv vector mixing angle: θv = 35°

(≈ ideal mixing)

⇨ Φ ≈ ss ω ≈ 1/√2 (uu + dd)

Baryons Quark model: qqq

1. Combine two of the quarks: 3⊗3 = 6⊕3 sym. antisym. under interchange of 2 quarks Combination of 2 isospin doublets (as for 1/√2 (ud + du) spin) Y 1/√2 (ud - du)

2/3 0 -2/3 -4/3

u d

s ss

Fig. 5: The qq SU(3) multiplets; 3⊗3 = 6⊕3

2. Add 3rd quark triplet: 3⊗3⊗3 = (6⊗3) ⊕ (3⊗3) Y 1

=10 ⊕ 8 ⊕ 8 ⊕ 1 ddd

uuu

0 -1 2

sss

Fig. 6: The qqq SU(3) multiplets; 3⊗3⊗3 = 10 ⊕ 8 ⊕ 8 ⊕ 1 Dayalbagh Educational Institute

Quark Model

Particle Physics

Quark model: qqq Example: uud combinations combine non-strange member of 3 with u of 3

pA = √1/2 (ud - du)u Decuplet states: tot. sym. under interchange of quarks

Δ = √1/3 [uud + (ud + du)u] Orthogonality to pA and ∆

ps = √1/6 [ (ud + du)u - 2uud]

Color

Naïve quark model unsatisfactory: Consider baryons at corner of decuplet: uuu, ddd, sss three identical quarks orbital angular momentum zero identical spin configuration: ↑↑↑ ⇨ total spin 3/2 ⇨ violation of Pauli principle (completely antisymmetric wavefunction) Ψ=

each factor symmetric under interchange ψspin ·ψspace ·ψflavor of two quarks for ∆++, ∆−, Ω−

What about qq, qq configurations or single quarks? ⇨ not observed! Solution: assign new property (quantum number) to quarks (not leptons!): color ⇨ label the three (otherwise identical) quarks: R, G, B only mnemonic labels, no relation to ordinary colors! Dayalbagh Educational Institute

Quark Model

Particle Physics

∆++ = uR, uG, uB, i.e. quarks are distinguishable ⇨ Antiquarks: opposite (complementary) color R, G, B ⇨ Quarks form a fundamental triplet of SU(3)color group, which is believed to be exact (in contrast to SU(3)flavor) Yellow B Red R Green G

Cyan R

Magenta G Fig. 7

Blue B require all observed meson and baryon states to be “colorless”, i.e. singlet representations of SU(3)color group equal mixture of R, G, B equal mixture of R, G, B equal mixtures of RR, GG, BB ⇨ Total wavefunction Ψ = ψspin ·ψspace ·ψflavor ·ψcolor completely antisymmetric under interchange of two quarks

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