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# Chapter3 - P470 Chapter 3 Building Hadrons from Quarks...

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P470 Chapter 3 Building Hadrons from Quarks Mesons in SU(2) We are now ready to consider mesons and baryons constructed from quarks. As we recall, mesons are made of quark-antiquark pair and baryons are made of three quarks. Consider mesons made of u and d quarks first. ψ = ψ (space) ψ (spin) ψ (flavor) ψ (color) It is straight-forward to write down the spin wave function of a meson. Since both q and q have spin ½, the qq can form either a spin-triplet or a spin-singlet state: 1,1 =↑↑ () 1 1, 0 2 =↑ + spin 1 triplet 1, 1 −= 1 0,0 2 spin 0 singlet For the flavor wave function, we have an analogous situation. The u , d quarks are isospin doublet, and so are , du . However, a slight complication arises since ud transforms in the complex conjugate representation. Now and i uu I d ⎛⎞ = ⎜⎟ ⎝⎠ d k * ij i j k II i I ε ⎡⎤ = ⎣⎦ taking complex conjugate, we get ** i j ijk k I Ii =− I o r ( ) * , - i j kk I I

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P470 2 * i I transforms , ud into ′′ : () * i uu I d ⎛⎞ =− ⎜⎟ ⎝⎠ d (Eq. 1) write out explicitly: * i I * 1 01 1 10 2 I −= * 2 0 1 0 2 i I i * 3 1 2 I One can easily verify that is related to I * i I i by a unitary transformation * ii I I ⎛⎞⎛⎞ ⎜⎟⎜⎟ ⎝⎠⎝⎠ One can multiply (Eq. 1) by on the left 1 * 0101 1010 i I dd −− (Eq. 2) I i (Eq. 2) becomes i I = or i I ⎛⎞ ⎛ = ⎜⎟ ⎜ ⎝⎠ ⎝ Therefore, d u transforms just like and is the isospin doublet for u d .
P470 3 Combining the with the u d ⎛⎞ ⎜⎟ ⎝⎠ d u , one obtains isospin triplet and isospin singlet: ,1 , 1 z II u d == () 1 1, 0 2 uu dd =− I = 1 1, 1 ud −= 1 0,0 2 uu dd =+ I = 0 The total spin of the mesons can be S = 0 or 1. The lightest mesons have orbital angular momentum L = 0. Therefore, their total angular momentum J = 0 or 1. J = 0, I = 1 : π + , π o , π - (mass ~ 140 MeV) J = 1, I = 1 : ρ + , ρ o , ρ - (mass ~ 770 MeV) J = 1, I = 0 : ω o (mass ~ 780 MeV) (We will postpone the discussion of J = 0, I = 0 meson until later, since its wave function contains non- u , d components.) One can readily combine the flavor and spin parts of the wavefunction. For example: 11 1 0 2 22 o z Ju u d d u u u u d d d ρ = = ↑↓+↓↑ = ↓+ ↓ ↑− ↑ ↓− ↓ d 0 z d u d u d π + = ↑↓ −↓↑ Baryons in SU(2) Now, consider baryons made of u , d quarks. Let us start with the flavor part.

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P470 4 First, combining two quarks: 1,1 : uu () 1 1, 0 2 ud du + I = 1 triplet 1, 1 dd 1 0,0 2 ud du I = 0 singlet Note that I = 1 is symmetric with respect to the interchange of the two quarks, while I = 0 singlet is anti-symmetric. We therefore label the symmetry property of the representations as 223 1 S ⊗= ⊕ A where the dimension of the representation is 2I+1 (For example, I=1/2 2; I=1 3; I=0 1). Adding another u / d quark, we have ( ) ( ) ( ) 222 3 1 2 3 2 1 2 SA S A ⊗⊗= + ⊗= ⊗ ⊕ First consider 12 : A 1 2 u ud du d ⎛⎞ −⊗ ⎜⎟ ⎝⎠ 1 2 1 2 ud du u
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Chapter3 - P470 Chapter 3 Building Hadrons from Quarks...

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