Mathematical Principles of Theoretical Physics

5.2 Quark Model Table 5.4: Baryons Baryons J m Qe Qw Qs B I I 3 S Y τ p 1 / 2 938.3 1 9 3 1 1 / 2 1 / 2 0 1 ∞ n 1 / 2 939.6 0 9 ...

268 CHAPTER 5. ELEMENTARY PARTICLES Table 5.5: Mesons Mesons J m Qe Qw Qs B I I 3 S Y τ π^00135062010008. 4 × 10 −^17 π+ 0 139.6 ...

5.2. QUARK MODEL 269 S= 0 n p S=− 1 S=− 2 Σ − Σ 0 Σ+ Λ I 3 Ξ − Ξ 0 Q=− 1 Q= 0 Q= 1 Y = 1 Y= 0 Y =− 1 -1 −^1 2 0 1 2 1 I 3 Figure ...

270 CHAPTER 5. ELEMENTARY PARTICLES K^0 K+ S= 1 S= 0 S=− 1 π − π 0 π+ η I 3 K− K ̄^0 Q=− 1 Q= 0 Q= 1 Y = 1 Y= 0 Y =− 1 -1 −^1 2 ...

5.2. QUARK MODEL 271 ∆− ∆ 0 S= 0 S=− 1 S=− 2 Σ∗− Σ∗^0 ∆+ ∆++ Σ∗+ Ξ∗^0 Q=− 1 Q= 0 Q= 1 Y= 1 Y = 0 Y=− 1 − 1 −^120121 I 3 S=− 3 Ξ∗ ...

272 CHAPTER 5. ELEMENTARY PARTICLES K∗^0 K∗+ S= 1 S= 0 S=− 1 ρ− ρ^0 ρ+ ω I 3 K ∗− K ̄ ∗ 0 Q=− 1 Q= 0 Q= 1 Y = 1 Y= 0 Y =− 1 -1 − ...

5.2. QUARK MODEL 273 Thus, each matrixU∈SU(N)is a linear transformation: (5.2.14) U:CN→CN, and eachU∈SU(N)gives (5.2.15) U∗:C N ...

274 CHAPTER 5. ELEMENTARY PARTICLES To understand the implications of (5.2.18) and (5.2.19), we have to know the irreducible rep ...

5.2. QUARK MODEL 275 and their complex linear combination constituteCN: CN= { N ∑ j= 1 zkψk|zk∈C, 1 ≤k≤N } , which contains all ...

276 CHAPTER 5. ELEMENTARY PARTICLES contains 9 new particles ψij=ψiψj, 1 ≤i,j≤ 3 , composed of a fundamental particleψiand an an ...

5.2. QUARK MODEL 277 can also be decomposed in the form (5.2.36) H(U) =H 1 (U)⊕ ··· ⊕HK(U) ∀U∈SU(N), andHj(U) ( 1 ≤j≤K)are as in ...

278 CHAPTER 5. ELEMENTARY PARTICLES then under the new basisE ̃={Ψ ̃ 1 ,···,Ψ ̃Nk}, the linear mapping H(U):X→X (X as in (5.2.22 ...

5.2. QUARK MODEL 279 The second step provides the rule and method to compute the dimensionalmjfrom thej-th square diagrams j on ...

280 CHAPTER 5. ELEMENTARY PARTICLES (c)Case k 1 = 3 and k 2 = 0 :We define the right-hand side of (5.2.42) as shown (5.2.47) N⊗N ...

5.2. QUARK MODEL 281 represents a particle system, then its conjugate (5.2.50) N︸⊗ ··· ⊗︷︷ N︸ k 1 ⊗N︸⊗ ··· ⊗︷︷ N︸ k 2 represents ...

282 CHAPTER 5. ELEMENTARY PARTICLES Example 5.6.ForN= 3 ,consider the two cases given by (5.2.57) 3 ⊗3 and 3⊗ 3 ⊗ 3 , which are ...

5.2. QUARK MODEL 283 Then we can compute that m= 1 for m= 15 for m= 4 for m= 20 for and Hence we deduce that (5.2.61) 4 ⊗ 4 = 1 ...

284 CHAPTER 5. ELEMENTARY PARTICLES Y K+(pΛ) ̄ π+( ̄np) I 3 K−( ̄pΛ) K ̄^0 ( ̄nΛ) π−(np ̄) η π^0 K^0 (n ̄Λ) Figure 5.6 Table 5.6 ...

5.2. QUARK MODEL 285 Table 5.7 mesons I I 3 Q S Jπ m(MeV) pn π+ 1 1 1 0 0 − ∼ 140 −pn π− 1 -1 -1 0 0 − ∼ 140 nn√−pp 2 π 0 1 0 0 ...

286 CHAPTER 5. ELEMENTARY PARTICLES 1) The spins of quarks have to beJ=^12 , as required by (5.2.69) spins of mesons: J= 0 (↑↓), ...