1. Non-Abelian Gauge Theories Or: Finally! References: [HM 14.1-4, - PowerPoint PPT Presentation

PHYS 6610: Graduate Nuclear and Particle Physics I H. W. Grießhammer INS Institute for Nuclear Studies The George Washington University Institute for Nuclear Studies Spring 2018 III. Descriptions 1. Non-Abelian Gauge Theories Or: Finally! References: [HM 14.1-4, 2.15; HG 12.3; CL 8.1] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.1.0

Particle-Field Feynman Rules of Non-Abelian Gauge Theories fermion colours i , j = 1 ,..., N c for N c colours gauge-field colour combinations a , b , c , d = 1 ,..., dim G ( N c )= N 2 c − 1 in SU ( N c ) . a , µ k ν : − i δ ab g µν : i g γ µ ( t a ) ji a → b gauge-field propagator: qg vertex: µ k 2 + i ε colour i colour j a , µ flow like for γ -matrices: left-to-right against fermion arrow k ↓ : g f abc � g µν ( k − p ) ρ + g νρ ( p − q ) µ + g ρµ ( q − k ) ν � 3g self-interaction vertex: տ q p ր c , ρ b , ν a , µ b , ν : − i g 2 � g µρ g νσ − g µσ g νρ � g µν g ρσ − g µσ g ρν � 4g self- f abe f cde � + f ace f bde � interaction : g µν g σρ − g µρ g σν �� + f ade f bce � vertex c , ρ d , σ Even pure Yang-Mills Theory (no fermions) is self-interacting! = ⇒ Chance of confinement? All use same coupling strength g : delicate balance of terms to ensure gauge invariance. A few more interactions with “ghost” fields only cure a technicality (gauge invariance in loop diagrams). PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.1.1

SU ( 3 ) : Gell-Mann Matrices λ a And All. . . SU c ( 3 ) generated by Hermitean, traceless 3 × 3 matrices t a = λ a , ortho-normalised: 2tr [ λ a λ b 2 ] = δ ab 2 2 Structure Constants [ λ a 2 , λ b 2 ] = i f abc λ c : totally antisymmetric in all indices. 2 √ f 147 = − f 156 = f 246 = f 257 = f 345 = − f 367 = 1 3 f 123 = 1 f 458 = f 678 = rest 0 . 2 2 Explicit form never necessary in actual calculations! Only matrix dimension N c (= 3 ) , norm 2tr [ t a t b ] = δ ab , SU ( N c ) Casimir operator: t a t a = N 2 c − 1 . 2 N c PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.1.2

(e) Historical Notes O’Raifeartaigh: Dawn of Gauge Theories, CUP , p. 183 PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.1.3

Where the Name “Quark” Comes From James Joyce: Finnegans Wake (1939) 3 quark flavours form meson/baryon octets ( 8 ), 3 colours = ⇒ Quark (n., German) a sour curd or cottage cheese. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.1.4

Next: 2. Perturbative QCD Familiarise yourself with: [PRSZR 8.1-3, 14; HM 2.15, 10.3-9, 11.4/6-7; Tho 10.7/8; Ryd 3, end of 9.6; HG 12.3; PS 16.7; Per 6.5; lots more. . . ] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.1.5