1. Quarks, Colors, and Confinement
ChoeJo YeolLin
Dept. of Physics, KAIST
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2. Introduction
Baryons, Mesons, and Quarks
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3. Fall KAIST PH489 3

4. Historical Backgrounds
1961
The Eightfold Way
1935
Yukawa’s Meson
1949
Quantum Electrodynamics
1949
Quantum Electrodynamics
1947
Discovery of Strange Particle
1928
Dirac Equation
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5. Quantum Chromodynamics
1975-
The Standard Model
1974
Discovery of J/ψ Meson
1964
The Quark Model
Quantum Chromodynamics
Deep Inelastic Scattering Experiments
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6. Color Charge Fall 2017 KAIST PH489 6
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7. Color Confinement Fall 2017 KAIST PH489 7
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8. Review: QED Symmetry and Formalisms of Quantum Electrodynamics
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9. Quantum Electrodynamics
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10. 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
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11. The Feynman Rule Fall 2017 KAIST PH489 11
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12. 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
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13. 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
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14. 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 +…
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15. 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
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16. Color SU(3) and QCD Symmetry and Formalisms of Quantum Chromodynamics
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17. Quantum Chromodynamics
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18. 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 ′
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19. 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
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20. 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
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21. 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
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22. 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
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23. 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
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24. Running of Coupling Constant
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25. 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
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26. 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
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27. Running of Coupling Constant
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28. 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
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29. Running of Coupling Constant
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30. Running of Coupling Constant
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31. Asymptotic Freedom
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32. Asymptotic Freedom
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33. Color Confinement Fall 2017 KAIST PH489 33
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34. Color Confinement Fall 2017 KAIST PH489 34
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35. Color Confinement in Hadrons
Structures of Mesons, Baryons, Tertaquarks, and Pentaquarks
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36. 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
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37. 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)
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38. Tetraquarks 2 quarks, 2 antiquarks 𝑑 𝑢 𝑐 𝑐 - Bosonic particle
- Baryon number 𝐵=0 𝑢 𝑐 𝑐
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39. Pentaquarks 4 quarks, 1 antiquark 𝑐 𝑑 𝑐 𝑢 𝑢 - Fermionic particle
- Baryon number 𝐵=1 𝑐 𝑢 𝑢
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40. QCD Matter Brief Introduction to Non-Hadronic Quark Matter States
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41. QCD Matter Fall 2017 KAIST PH489 41
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42. QCD Matter Fall 2017 KAIST PH489 42
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43. Quark-Gluon Plasma Fall 2017 KAIST PH489 43
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44. Quark-Gluon Plasma Fall 2017 KAIST PH489 44
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45. 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
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46. 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),
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