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Quarks, Colors, and Confinement
ChoeJo YeolLin Dept. of Physics, KAIST Fall KAIST PH489 1
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Introduction Baryons, Mesons, and Quarks Fall KAIST PH489 2
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Fall KAIST PH489 3
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Historical Backgrounds
1961 The Eightfold Way 1935 Yukawa’s Meson 1949 Quantum Electrodynamics 1949 Quantum Electrodynamics 1947 Discovery of Strange Particle 1928 Dirac Equation Fall KAIST PH489 4
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Quantum Chromodynamics
1975- The Standard Model 1974 Discovery of J/ψ Meson 1964 The Quark Model Quantum Chromodynamics Deep Inelastic Scattering Experiments Fall KAIST PH489 5
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Color Charge Fall 2017 KAIST PH489 6
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Color Confinement Fall 2017 KAIST PH489 7
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Review: QED Symmetry and Formalisms of Quantum Electrodynamics
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Quantum Electrodynamics
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QED Lagrangian ℒ= 𝜓 𝛾 𝜇 𝑝 𝜇 𝜓−𝑚 𝜓 𝜓 Dirac Lagrangian
ℒ= 𝜓 𝛾 𝜇 𝑝 𝜇 𝜓−𝑚 𝜓 𝜓 Dirac Lagrangian ℒ=𝑖 𝜓 𝛾 𝜇 𝜕 𝜇 𝜓−𝑚 𝜓 𝜓− 𝑞 𝑒 𝜓 𝛾 𝜇 𝜓 𝐴 𝜇 − 1 4 𝐹 𝜇𝜈 𝐹 𝜇𝜈 ℒ= 𝜓 𝛾 𝜇 𝑝 𝜇 − 𝑞 𝑒 𝐴 𝜇 𝜓−𝑚 𝜓 𝜓 Free fermion Lagrangian (Dirac) Interaction term Free photon Lagrangian Fall KAIST PH489 10
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The Feynman Rule Fall 2017 KAIST PH489 11
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The Feynman Rule Interaction term − 𝑞 𝑒 𝜓 𝛾 𝜇 𝜓 𝐴 𝜇 Vertex factor
− 𝑞 𝑒 𝜓 𝛾 𝜇 𝜓 𝐴 𝜇 Vertex factor −𝑖 𝑞 𝑒 𝛾 𝜇 Fall KAIST PH489 12
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The Feynman Rule 𝑢 𝑐 𝑝 𝑐 −𝑖 𝑞 𝑒 𝛾 𝜇 𝑢 𝑎 𝑝 𝑎
𝑢 𝑐 𝑝 𝑐 −𝑖 𝑞 𝑒 𝛾 𝜇 𝑢 𝑎 𝑝 𝑎 ℳ LO = 𝑢 𝑐 𝑝 𝑐 −𝑖 𝑞 𝑒 𝛾 𝜇 𝑢 𝑎 𝑝 𝑎 − 𝑖 𝑔 𝜇𝜈 𝑞 𝑢 𝑑 𝑝 𝑑 −𝑖 𝑞 𝑒 𝛾 𝜈 𝑢 𝑏 𝑝 𝑏 − 𝑖 𝑔 𝜇𝜈 𝑞 2 𝑗 𝜇 𝑗 𝜈 𝑢 𝑑 𝑝 𝑑 −𝑖 𝑞 𝑒 𝛾 𝜈 𝑢 𝑏 𝑝 𝑏 Fall KAIST PH489 13
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The Feynman Rule Corrections to the QED matrix element
ℳ 𝑓𝑖 =𝛼 ℳ LO + 𝛼 2 𝑗 2 ℳ 2, 𝑗 𝛼 3 𝑗 3 ℳ 3, 𝑗 𝛼 4 𝑗 4 ℳ 4, 𝑗 4 +… ℳ 𝑓𝑖 2 = 𝛼 2 ℳ LO 2 +𝑂 𝛼 3 +𝑂 𝛼 4 +… Fall KAIST PH489 14
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The Feynman Rule Mandelstam Variables Value of 𝑞 2 𝑠= 𝑝 1 + 𝑝 2 2
𝑠= 𝑝 1 + 𝑝 2 2 𝑡= 𝑝 1 − 𝑝 3 2 𝑢= 𝑝 1 − 𝑝 4 2 Fall KAIST PH489 15
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Color SU(3) and QCD Symmetry and Formalisms of Quantum Chromodynamics
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Quantum Chromodynamics
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Color SU(3) 𝜓′= 𝐴 11 𝐴 12 𝐴 13 𝐴 21 𝐴 22 𝐴 23 𝐴 31 𝐴 32 𝐴 33
𝜓′= 𝐴 11 𝐴 12 𝐴 13 𝐴 21 𝐴 22 𝐴 23 𝐴 31 𝐴 32 𝐴 33 𝜓= 𝜓 𝑟 𝜓 𝑔 𝜓 𝑏 = 𝜓 𝑟 ′ 𝜓 𝑔 ′ 𝜓 𝑏 ′ Fall KAIST PH489 18
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Color SU(3) 𝐴 𝜇 = 1 2 𝛌∙ 𝐀 𝜇 = 1 2 𝑗=1 8 𝜆 𝑗 𝐴 𝜇 𝑗 𝐴 𝜇 →
𝐴 𝜇 = 1 2 𝛌∙ 𝐀 𝜇 = 1 2 𝑗=1 8 𝜆 𝑗 𝐴 𝜇 𝑗 𝐴 𝜇 → 𝐀 𝜇 = 𝐴 𝜇 1 𝐴 𝜇 2 ⋮ 𝐴 𝜇 8 𝐹 𝜇𝜈 → 𝐹 𝜇𝜈 = 𝜕 𝜇 𝐴 𝜈 − 𝜕 𝜈 𝐴 𝜇 +𝑖 𝑞 𝑐 𝐴 𝜇 , 𝐴 𝜈 𝛌= 𝜆 1 𝜆 2 ⋯ 𝜆 8 Fall KAIST PH489 19
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QCD Lagrangian ℒ=𝑖 𝜓 𝛾 𝜇 𝜕 𝜇 𝜓−𝑚 𝜓 𝜓− 𝑞 𝑐 𝜓 𝛾 𝜇 𝜓 𝐴 𝜇 − 1 4 𝐹 𝜇𝜈 𝐹 𝜇𝜈
ℒ=𝑖 𝜓 𝛾 𝜇 𝜕 𝜇 𝜓−𝑚 𝜓 𝜓− 𝑞 𝑐 𝜓 𝛾 𝜇 𝜓 𝐴 𝜇 − 𝐹 𝜇𝜈 𝐹 𝜇𝜈 ℒ= 𝜓 𝛾 𝜇 𝑝 𝜇 − 𝑞 𝑐 𝐴 𝜇 𝜓−𝑚 𝜓 𝜓 Free fermion Lagrangian Quark-gluon interaction term Gluon Lagrangian Fall KAIST PH489 20
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QCD Lagrangian 𝐹 𝜇𝜈 = 𝜕 𝜇 𝐴 𝜈 − 𝜕 𝜈 𝐴 𝜇 +𝑖 𝑞 𝑐 𝐴 𝜇 𝐴 𝜈 − 𝐴 𝜈 𝐴 𝜇
𝐹 𝜇𝜈 = 𝜕 𝜇 𝐴 𝜈 − 𝜕 𝜈 𝐴 𝜇 +𝑖 𝑞 𝑐 𝐴 𝜇 𝐴 𝜈 − 𝐴 𝜈 𝐴 𝜇 ⇒ 𝐹 𝜇𝜈 𝐹 𝜇𝜈 =𝑂 𝐴 𝜇 𝐴 𝜈 +𝑂 𝐴 𝜇 𝐴 𝜈 𝐴 𝜉 +𝑂 𝐴 𝜇 𝐴 𝜈 𝐴 𝜉 𝐴 𝜊 Fall KAIST PH489 21
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QCD Lagrangian Interaction term − 𝑞 𝑐 𝜓 𝛾 𝜇 𝜓 𝐴 𝜇
− 𝑞 𝑐 𝜓 𝛾 𝜇 𝜓 𝐴 𝜇 =− 1 2 𝑞 𝑐 𝜓 𝛾 𝜇 𝜆 𝑗 𝜓 𝐴 𝜇 𝑗 Vertex factor − 1 2 𝑖 𝑞 𝑐 𝜆 𝑗 𝛾 𝜇 Fall KAIST PH489 22
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QCD Lagrangian QED fermion current 𝑗 𝜇 = 𝑢 3 𝑝 3 −𝑖 𝑞 𝑒 𝛾 𝜇 𝑢 1 𝑝 1
𝑗 𝜇 = 𝑢 3 𝑝 3 −𝑖 𝑞 𝑒 𝛾 𝜇 𝑢 1 𝑝 1 QCD fermion current 𝑗 𝜇 = 𝑢 3 𝑝 3 𝑐 𝑛 † − 1 2 𝑖 𝑞 𝑐 𝜆 𝑗 𝛾 𝜇 𝑐 𝑚 𝑢 1 𝑝 1 Fall KAIST PH489 23
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Running of Coupling Constant
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Running of Coupling Constant
Effective photon propagator 𝑃 0 = 𝑒 0 𝑞 2 ⇒𝑃= 𝑃 0 + 𝑃 0 𝜋 𝑞 2 𝑃 0 + 𝑃 0 𝜋 𝑞 2 𝑃 0 𝜋 𝑞 2 𝑃 0 +⋯= 𝑃 −𝜋 𝑞 2 ≡ 𝑃 − 𝑒 0 2 Π 𝑞 2 ≡ 𝑒 2 𝑞 2 𝑞 2 Fall KAIST PH489 25
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Running of Coupling Constant
𝑃= 𝑒 2 𝑞 2 𝑞 2 = 𝑒 𝑞 − 𝑒 0 2 Π 𝑞 2 𝛼 𝑞 2 = 𝑒 2 𝑞 2 4𝜋 = 𝛼 𝜇 2 1−𝛼 𝜇 𝜋 ln 𝑞 2 𝜇 2 ⇒ 𝑒 2 𝑞 2 = 𝑒 − 𝑒 0 2 Π 𝑞 2 , 𝑒 2 𝜇 2 = 𝑒 − 𝑒 0 2 Π 𝜇 2 ⇒ 𝑒 2 𝑞 2 = 𝑒 2 𝜇 2 1− 𝑒 2 𝜇 2 Π 𝑞 2 −Π( 𝜇 2 ) = 𝑒 2 𝜇 2 1− 𝑒 2 𝜇 𝜋 2 ln 𝑞 2 𝜇 2 Fall KAIST PH489 26
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Running of Coupling Constant
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Running of Coupling Constant
QED coupling constant QCD coupling constant (𝑛=3 in our QCD) 𝛼 𝑠 𝑞 2 = 𝛼 𝑠 𝜇 𝐵 𝛼 𝑠 𝜇 2 ln 𝑞 2 𝜇 2 𝛼 𝑞 2 = 𝛼 𝜇 2 1− 1 3𝜋 𝛼 𝜇 2 ln 𝑞 2 𝜇 2 𝐵= 1 4𝜋 11∙ 1 3 𝑛−2 𝑁 𝐹 Gluon-gluon coupling Fermion Fall KAIST PH489 28
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Running of Coupling Constant
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Running of Coupling Constant
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Asymptotic Freedom Fall KAIST PH489 31
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Asymptotic Freedom Fall KAIST PH489 32
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Color Confinement Fall 2017 KAIST PH489 33
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Color Confinement Fall 2017 KAIST PH489 34
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Color Confinement in Hadrons
Structures of Mesons, Baryons, Tertaquarks, and Pentaquarks Fall KAIST PH489 35
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Mesons 1 quark, 1 antiquark 𝜓 𝑐 = 1 3 𝑟 𝑟 +𝑔 𝑔 +𝑏 𝑏
- Bosonic particle with spin 𝑠=0 or 1 - Baryon number 𝐵=0 𝜓 𝑐 = 𝑟 𝑟 +𝑔 𝑔 +𝑏 𝑏 - Quark and antiquark carry opposite color Fall KAIST PH489 36
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Baryons 3 quarks 𝜓 𝑐 = 1 6 (𝑟𝑔𝑏−𝑟𝑏𝑔 +𝑔𝑏𝑟−𝑔𝑟𝑏 +𝑏𝑟𝑔−𝑏𝑔𝑟)
- Fermionic particle with spin 𝑠=1/2 or 3/2 𝜓 𝑐 = (𝑟𝑔𝑏−𝑟𝑏𝑔 - Baryon number 𝐵=1 - 3 quarks carry 3 different colors +𝑔𝑏𝑟−𝑔𝑟𝑏 +𝑏𝑟𝑔−𝑏𝑔𝑟) Fall KAIST PH489 37
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Tetraquarks 2 quarks, 2 antiquarks 𝑑 𝑢 𝑐 𝑐 - Bosonic particle
- Baryon number 𝐵=0 𝑢 𝑐 𝑐 Fall KAIST PH489 38
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Pentaquarks 4 quarks, 1 antiquark 𝑐 𝑑 𝑐 𝑢 𝑢 - Fermionic particle
- Baryon number 𝐵=1 𝑐 𝑢 𝑢 Fall KAIST PH489 39
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QCD Matter Brief Introduction to Non-Hadronic Quark Matter States
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QCD Matter Fall 2017 KAIST PH489 41
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QCD Matter Fall 2017 KAIST PH489 42
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Quark-Gluon Plasma Fall 2017 KAIST PH489 43
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Quark-Gluon Plasma Fall 2017 KAIST PH489 44
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References [1] Thomson, M. (2013). Modern Particle Physics. Cambridge University Press. [2] Griffiths, D. (2008). Introduction to Elementary Particles. John Wiley & Sons. [3] Greiner, W., Schramm, S., & Stein, E. (2007). Quantum Chromodynamics. Springer Science & Business Media. [4] Halzen, F., Martin, A. D. (1984). Quarks and Leptons. [5] Ryder, L. H. (1996). Quantum Field Theory. Cambridge university press. [6] Greiner, W., & Reinhardt, J. (2012). Quantum Electrodynamics. Springer Science & Business Media. [7] Yoo, J. H. (2017). KAIST 2017 Spring Undergraduate Physics Course Lecture Series PH450 Particle Physics [8] Riordan, M. (1992). The Discovery of Quarks. Science, 256(5061), [9] Ollitrault, J. Y. (2015). The Littlest Liquid. Physics, 8, 61. [10] Instituto de Física Corpuscular, Lattice QCD, the numerical approach to the strong force [11] Walsh, K. M. (2016). Tracking the transition of early-universe quark soup to matter-as-we-know-it. Brookhaven National Laboratory's Relativistic Heavy Ion Collider News, Web, 4 Fall KAIST PH489 45
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References (For Further Readings)
[12] Aaij, R., Adeva, B., Adinolfi, M., Affolder, A., Ajaltouni, Z., Albrecht, J., ... & Cartelle, P. A. (2014). Observation of the resonant character of the 𝑍 − state. Physical review letters, 112(22), [13] Aaij, R., Adeva, B., Adinolfi, M., Affolder, A., Ajaltouni, Z., Akar, S., ... & Alkhazov, G. (2015). Observation of 𝐽/𝜓𝑝 Resonances Consistent with Pentaquark States in 𝛬 𝑏 0 →𝐽/𝜓 𝐾 − 𝑝 Decays. Physical review letters, 115(7), [14] Park, W., & Lee, S. H. (2014). Color spin wave functions of heavy tetraquark states. Nuclear Physics A, 925, [15] Ruester, S. B., Werth, V., Buballa, M., Shovkovy, I. A., & Rischke, D. H. (2005). Phase diagram of neutral quark matter: Self-consistent treatment of quark masses. Physical Review D, 72(3), Fall KAIST PH489 46
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