Chapter 5.3 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

5.3 Meson and Baryon wave function

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Meson and Baryon wave function Mesons Every meson is composed of a quark and an antiquark. They have integral spin so they are bosons. Lighest mesons are composed of up, down or strange quarks. Lighest mesons observed experimentally are a family of nine particles with spin-parity 0- is called pseudoscalar meson nonet and family of nine particles with spin parity 1- called vector meson nonet. The pseudoscalar nonet includes the familar pions and kaons, together with Îˇ meson and a heavier meson Îˇ'. Vector mesons are all resonances particles and include the k* mesons. Same set of internal quantum number values occurs for both nonets. The spin of mesons is just the sum of the quark and antiquark spins, and can be 0 or 1. Since there are nine possible combinations of a quark and antiquark, so two nonets of mesons with spin-parities 0 - and 1- . The hypercharge and isospin values for each quark combination ab, are obtained by adding values for a and b.

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Pseudoscalar and vector meson nonet K+

π0 π

π-

η '

K-

Fig 1: Pseudoscalar nonet of light quark [2]

K*+

K*0

ρ0 ρ+

K*-

ρ-

K*0

Fig 2: Vector nonet of light quark [2]

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The hypercharge and isospin values for each quark combination ab, is obtained by adding hypercharge and isospin values for a and b. Light mesons are made of up, down and strange quarks. We confine to the ground state with angular momentum, l = 0. Quarks spin can be antiparallel or parallel. Antiparallel spin state with spin, S = 0 is singlet state and parallel spin state with spin, S = 1 is triplet state. S = 0 state gives pseudoscalar nonet and S = 1 gives vector nonet. We obtained nine mesons by combining a quark and an antiquark in all possible combinations with three neutral states with strangeness zero (uu, dd, ss). It is not clear which of these uu, dd, ss was π0, η, η' in pseudoscalar nonet and ρ0, ω, ϕ in vector nonet. Now this doubt can be resolved, up and down quarks constitute an isospin doublet: u=

and

Antiquarks too have isospin doublet as: and Anti down quark (d) has isospin value I3 = +1/2 and anti up quark (u) has isospin value I3 = -1/2, when we combine two particles with I = ½ we obtain an isotriplet: = - ud = (uu - dd)/ √2 = du = (uu + dd)/ √2 (isosinglet) Dayalbagh Educational Institute

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In case of pseudoscalar mesons the triplet is pion and for vector mesons it is ρ. π0 and ρ0 is neither uu nor dd but linear combination of these two, π0 and ρ0 = (uu - dd)/ √2 η and η' or (ω and ϕ) is represented as linear combination of singlet state. These particles carry identical quantum numbers and they tend to mix. Thus in case of pseudoscalars the physical state appears to be: η = 1/√6 (uu + dd - 2ss) η' = 1/√3(uu + dd + ss) Whereas for vector mesons ω = 1/√2 (uu + dd) ϕ = ss The pseudoscalar combinations are more natural, since η' treats u, d and s symmetrically and is unaffected by SU (3) transformations. It is singlet under SU (3), in exactly the same manner the π0 is a singlet under SU (2) isospin. The η transforms as part of SU (3) octet, three pions and four kaons are whose other members. In contrast neither ω nor ϕ is an isosinglet. Strange mesons are constructed by combining an s quark with u or d quark: K+ = us, K0 = ds, K0 = sd, K- = su Here is table showing properties and quark content of mesons in l = 0 state.

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Mesons table: The states of light summmarised in a table:-

L=0

meson

nonets

Particle

I(isospin)

Y=S hypercharge

Quark Content

π+

π0

1/√2 (uu-dd)

π-

-1

K+

1/2

+1/2

1/2

-1/2

1/2

+1/2

-1

K-

1/2

-1/2

-1

1/√6(uu+dd-2ss)

η'

1/√3(uu+dd+ss)

are

Table 1: Properties and Quark Content of Pseudoscalar Mesons

Particle

I(isospin)

Y=S hypercharge

Quark Content

ρ+

ρ0

1/√2 (uu-dd)

ρ-

-1

K*+

1/2

+1/2

K*0

1/2

-1/2

K*0

1/2

+1/2

-1

K*-

1/2

-1/2

-1

1/√2(uu+dd)

Table 2: Properties and Quark Content of Vector Mesons Dayalbagh Educational Institute

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Baryons Baryons are composed of three quarks and antibaryons are composed of three antiquarks. Baryons have half integral spin, thus it is fermions. Most familiar baryons are protons and neutrons. Baryons participate in strong interactions. Baryons are triquarks and baryon number of each quark is 1/3, thus baryon's baryon number is 1. Baryons wave functions have 1, 8 or 10 members and are called singlets, octets and decuplets respectively. Lighest baryons supermultiplets are octet J p=1/2+ and decuplet, Jp=3/2+. Baryon spin is equal to sum of quark spins. Baryons which are fermions with half integral spin have antisymmetric wave functions. If we take baryon decuplet and knock off the three corners and double the centre, we obtain eight states of baryon octet. Same combination of quark can go to a number of different particles. For example ∆+ and proton are both composed of two up and one down quark. +

The 1/2 octet includes the nucleons, the Λ (lambda) and Σ (sigma) particles, together with cascade particles Ξ (cascade), which decay by weak -10

interactions with life time of 10 s. The states are all resonances except for Ω which decay by weak -10

interaction with life time of 10 s. Dayalbagh Educational Institute

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The following table shows various baryons and their quark composition. The corresponding value of isospin (I), third component of isospin (I3) ans strangeness(s) is also shown. Quark composition

Observed state

uud

1/2

udd

-1/2

1/2

uds

-1

uus

Σ+

-1

uds

Σ0

-1

dds

Σ-

-1

uss

Ξ0

1/2

-2

dss

Ξ-

-1/2

1/2

-2

Table 3: The states of the L=1/2+ octet of light baryons Quark composition

Observed state

uuu

Δ++

3/2

uud

Δ+

1/2

3/2

udd

Δ0

-1/2

3/2

ddd

Δ-

-3/2

3/2

uus

Σ+

-1

uds

Σ0

-1

dds

Σ-

-1

uss

Ξ0

1/2

-2

dss

Ξ-

-1/2

1/2

-2

sss

Ω-

-3

Table 4:The states of the L=3/2+ decuplet of light baryons

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Baryon Wave Functions Baryon is a three body system, there are two orbital angular momentum to be considered. We will consider the ground state for l=l'=0. Angular momentum of baryon comes entirely from combined spins of three quarks.each quark can have '(↑) spin up' or '(↓) spin down'. The combinations for spins are (↑↑↑), (↑↑↓), (↑↓↑), (↑↓↓), (↓↑↑), (↓↑↓), (↓↓↑), (↓↓↓). The quarks spin can can combine to give either a total 1/2 or 3/2. Spin -3/2 combinations are completely symmetric in the sense that interchanging any two particles leaves the state untouched. |3/2 3/2> = (↑↑↑) |3/2 1/2> = (↑↑↓ + ↑↓↑+ ↓↑↑)/ √3 |3/2 -1/2> = (↓↓↑ + ↓↑↓+ ↑↓↓)/ √3

Spin 3/2(Ψs)

|3/2- 3/2> = (↓↓↓)

The spin-1/2 combinations are partially antisymmetric, means interchange of any two particles reverses the sign. The below set is antisymmetric in particles 1 and 2 hence the subscript is 12. |1/2 1/2>12 = (↑↓-↓↑)↑/√2 |1/2 -1/2>12 = (↑↓-↓↑)↑/√2 Dayalbagh Educational Institute

Spin 1/2(Ψ12)

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The below set is antisymmetric in particles 2 and 3 hence the subscript is 23. |1/2 1/2>23 = ↑(↑↓-↓↑)/√2 |1/2 -1/2>23 = ↑(↑↓-↓↑)/√2

Spin 1/2(Ψ23)

The below set is antisymmetric in particles 1 and 3 hence the subscript is 13. |1/2 1/2>13 = ↑(↑↓-↓↑)/√2 |1/2 -1/2>13 = ↑(↑↓-↓↑)/√2

Spin 1/2(Ψ13)

These states are not independent of each other as | >13 = | >12 +| >23 By Quantum field theory, the connection between spin and statistics is Boson (Integral spin) have symmetric wave function: ψ (1,2) = ψ (2,1) Fermions (1/2-integral spin) have antisymmetric wave function: ψ (1,2) = -ψ (2,1) If we have two particles , one in state ψα and other in state ψβ and particles are distinct. Then Wave Function for system is ψ(1,2) = ψα(1) ψβ(2), if particle 1 is in ψα and other is in state ψβ. Or ψ(1,2) = ψβ(1) ψα(2) , if particle 1 is in ψβ and other is in state ψα . Dayalbagh Educational Institute

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If the particles are indistinguishable, we cannot state which one is in which state. But if the particles are identical bosons , the wave function is the symmetric combination Ψ(1,2) = 1/√2 [ψα (1) ψβ (2) + ψβ (1) ψα (2)] For identical fermions, antisymmetric combination

the

wave

function

Ψ(1,2) = 1/√2 [ψα (1) ψβ (2) - ψβ (1) ψα (2)] If we put two fermions into same state we get wavefunction zero which is the Pauli exclusion principle. Pauli principle does not apply to bosons thus we can put as many pions into the same state as we like. But in baryons we are putting three quarks together thus antisymmetrization requirement should be taken into account. The wavefunction of a baryon consists of several factors, spatial part, describing the location of three quarks, spin part, representing their spins, flavor component, which indicates what combination of u, d and s quark is involved and a color term which specifies colors of the quarks. Ψ = Ψ(Space) Ψ(Spin) Ψ(Flavor) Ψ(Color) Spatial function is symmetric as l=l'=0. Spin state is completely symmetric (j = 3/2) or of mixed symmetry (j = 1/2).

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Flavor have 33=27 possibilities: uuu, udu, uud, udd, .........sss, which we reshuffle into symmetric, antisymmetric and mixed combinations. Every baryon must one quark of each color,color state for baryons is Ψ (color) = (rgb – rbg + gbr – grb + brg – bgr)/√6 Color wave function is same for all baryons, we do not include it. Ψ (color) is antisymmetric, means rest of the wave function must be symmetric. In ground state Ψ(space) is symmetric, thus the product of Ψ(spin) and Ψ(flavor) has to be completely symmetric. Thus the wavefunction of baryon becomes antisymmetric as baryon is a fermion. For symmetric spin configuration, we should have symmmetric flavor state, thus we obtain spin-3/2 baryon decuplet. Ψ( baryon decuplet) = Ψs (Spin) Ψs (Flavor) For baryon octet we must put together states of mixed symmetry to make a completely symmetric combination. The product of two antisymmetric functions is itself symmetric. Thus Ψ12 (Spin) x Ψ12 (Flavor) is symmetric in 1 and 2. Likewise, Ψ23 (Spin) x Ψ23 (Flavor) is symmetric in 2 and 3, and Ψ13 (Spin) x Ψ13 (Flavor) is symmetric in 1 and 3. If we now add these, we get symmetric wave funtion. Ψ( baryon octet) = √2/3{Ψ12 (Spin) Ψ12 (Flavor) + Ψ23 (Spin) Ψ23 (Flavor) + Ψ13 (Spin) Ψ13 (Flavor)} Dayalbagh Educational Institute

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Wave function of baryon octet (uud) p

(udd) n (uds) Σ0 Λ (uds)

Σ(uus)

Fig 3: Baryon octet showing quark contents

Ξ(dss)

Σ+ (dds)

Ξ0 (uss)

By using formula for wavefunction of octet: Ψ( baryon octet) = √2/3{Ψ12 (Spin) Ψ12 (Flavor) + Ψ23 (Spin) Ψ23 (Flavor) + Ψ13 (Spin) Ψ13 (Flavor)} We can calculate wavefunction of all baryon octet particles as shown in the following tables.

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Wave function of proton

Table 5: wavefunction of proton

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Wave function of neutron

Table 6: Wavefunction of neutron

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Wave function of sigma minus

Table 7: Wavefunction of sigma minus

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Wave funtion of sigma zero

Table 8: Wavefunction of sigma zero

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Wave function of sigma plus

Table 9: Wavefunction of sigma plus

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Wave function of cascade minus

Table 10: Wavefunction of cascade minus

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Wave function of cascade zero

Table 11: wavefunction of cascade zero

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Wave function of lambda

Table 12: Wavefunction of lambda particle

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Wavefunction of Baryon decuplets. Y=B+S (ddd) Δ1-

(ddu) Δ0

Σ*(dds)

(uud) Δ+

Σ*+ (uus)

Σ*0 (uds) Ξ*(dss)

-1-

(uuu) Δ++

-2-

Ξ*0 (uss)

Ω(sss) | -3/2

| -1

| -1/2

| 0

| 1/2

| 1

| 3/2

Fig 4: Baryon Decuplet showing quark contents

By using Baryon decuplet wave function formula: Ψ( baryon decuplet) = Ψs (Spin) Ψs (Flavor) we can calculate wavefunction of decuplet.

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Wave function of delta plus and delta zero particle

Table 13: Wavefunction of delta plus and lambda zero particle

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Wave function of delta plus and delta zero particle

Table 14: Wavefunction of delta plus and delta plus plus particle

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Wave function ofsigma star plus and sigma star minus particle

Table 15: Wavefunction of sigma star plus and sigma star minus particle

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Wave function of cascade star minus and cascade star zero particle

Table 16: Wavefunction of cascade star minus and cascade star zero particle

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Wave function of sigma star zero and omega minus particle

Table 17: Wavefunction of sigma star zero particle

Table 18 : Wavefunction of omega minus particle

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