Chiral Anomalies I. Adler-Bell-Jackiw Anomaly: • Triangle diagram: 푆 퐴, 휓, 휓 Consider QED action 푓 ∥ 1 푆 퐴, 휓, 휓 = − 푑4푥퐹 퐹 − 푑4푥휓 훾 휕 − 푖푒퐴 + 푚 휓 4 휇휈 휇휈 휇 휇 휇 Path integral 푖 푑퐴 푑휓푑휓 푒푖푆 퐴,휓,휓 = [푑퐴] exp − 푑4푥퐹 퐹 + 푖Γ 퐴 4 휇휈 휇휈 푖Γ 퐴 푖푆푓 퐴,휓,휓 푒 = 푑휓푑휓 푒 = det 훾휇 휕휇 − 푖푒퐴휇 + 푚 Γ 퐴 = sum of diagrams with one fermion loop decorated by external photon vertices • Triangle diagram:(cont.) Classical symmetry in the chiral limit 푚 = 0: −푖훼 푖훼 푈푉 1 : 휓 → 푒 휓 휓 → 휓 푒 −푖훼훾5 −푖훼훾5 푈퐴 1 : 휓 → 푒 휓 휓 → 휓 푒 푆 퐴, 휓, 휓 → 푆 퐴, 휓, 휓 Currents: 퐽휇 = 푖휓 훾휇휓 퐽5휇 = 푖휓 훾휇훾5휓 휕휇퐽휇 = 0 ⟹ electric charge conservation 휕휇퐽5휇 = 0 ⟹ axial charge conservation Quantum mechanical: UV divergence demands regulators. 푈푉 1 has to be preserved because of gauge invariance. 푈퐴 1 is explicitly broken by a gauge invariant regulator; is not recovered when the regulator mass 푀 → ∞ ⟹ anomalous! 휕휇퐽휇 = 0 but 휕휇퐽5휇 = anomaly ≠ 0 • Triangle diagram:(cont.) Beyond chiral limit: 푚 ≠ 0: Classical: 휕휇퐽휇 = 0 휕휇퐽5휇 = 2푖푚휓 훾5휓 Quantum: 휕휇퐽휇 = 0 휕휇퐽5휇 = 2푖푚휓 훾5휓 + anomaly Axial current in an external EM field 푖푆 퐴,휓,휓 푑휓푑휓 푒 1 퐽 푥 퐴 5휇 < 퐽5휇 푥 > ≡ 푖푆1 퐴,휓,휓 4 −푖푘∙푥 푑휓푑휓 푒 퐴휇 푘 ≡ 푑 푥 푒 퐴휇 푥 4 4 1 푑 푘1 푑 푘2 = 푒푖 푘1+푘2 ∙푥∆ 푘 , 푘 퐴 푘 퐴 푘 + 푂 퐴3 2 2휋 4 2휋 4 휇휈휌 1 2 휈 1 휌 2 퐴 휕휇 < 퐽5휇 푥 > = 4 4 푖 푑 푘1 푑 푘2 푒푖 푘1+푘2 ∙푥 푘 + 푘 ∆ 푘 , 푘 퐴 푘 퐴 푘 + 푂 퐴3 2 2휋 4 2휋 4 1 2 휇 휇휈휌 1 2 휈 1 휌 2 • Triangle diagram:(cont.) 휇 휇 휈 푝 휌 휌 푝 휈 4 2 푑 푝 1 1 1 ∆휇휈휌 푘1, 푘2|푚 = −푒 4 tr훾5훾휇 훾휌 훾휈 2휋 푝 + 푘2 − 푚 푝 − 푚 푝 − 푘1 − 푚 1 1 1 + 훾휈 훾휌 푝 + 푘1 − 푚 푝 − 푚 푝 − 푘2 − 푚 • Triangle diagram:(cont.) 푑4푝 푘 + 푘 Δ 푘 , 푘 |푚 = −푖푒2 tr훾 푘 + 푘 × 1 2 휇 휇휈휌 1 2 2휋 4 5 1 2 1 1 1 1 1 1 훾휌 훾휈 + 훾휈 훾휌 푝 + 푘2 − 푚 푝 − 푚 푝 − 푘1 − 푚 푝 + 푘1 − 푚 푝 − 푚 푝 − 푘2 − 푚 Using the identity 1 1 1 1 훾 푞 = −훾 − 훾 푝 + 푞 − 푚 5 푝 − 푚 5 푝 − 푚 푝 + 푞 − 푚 5 1 1 −2푚 훾 푝 + 푞 − 푚 5 푝 − 푚 We find 푘1 + 푘2 휇Δ휇휈휌 푘1, 푘2|푚 = 퐼휈휌 푚 + 2푖푚Δ휈휌 푚 • Triangle diagram:(cont.) 4 2 푑 푝 1 1 1 1 퐼휈휌 푚 = 푖푒 4 tr훾5 −훾휈 훾휌 + 훾휈 훾휌 2휋 푝 + 푘2 − 푚 푝 − 푚 푝 − 푚 푝 − 푘2 − 푚 1 1 1 1 −훾휌 훾휈 + 훾휌 훾휈 푝 − 푚 푝 − 푘1 − 푚 푝 + 푘1 − 푚 푝 − 푚 The 1st (3nd) term differ from 2nd (4th) term by a shift of integration momentum, but each integral is linearly divergent ⟹ 퐼휈휌 푚 ≠ 0. 4 2 푑 푝 1 1 1 Δ휈휌 푚 = 푒 4 tr훾5 훾휌 훾휈 2휋 푝 + 푘2 − 푚 푝 − 푚 푝 − 푘1 − 푚 1 1 1 + 훾휈 훾휌 푝 + 푘1 − 푚 푝 − 푚 푝 − 푘2 − 푚 Convergent integral • Triangle diagram:(cont.) Pauli-Villars regularization: Preserve the vector current conservation 푅 Δ휇휈휌 푘1, 푘2|푚 ≡ lim Δ휇휈휌 푘1, 푘2|푚 − Δ휇휈휌 푘1, 푘2|푀 푀→∞ = lim 퐼휈휌 푚 − 퐼휈휌 푀 + 2푖 lim 푚Δ휈휌 푚 − 푀Δ휈휌 푀 푀→∞ 푀→∞ Shift integration momentum becomes legitimate! 푑4푝 lim 퐼휈휌 푚 − 퐼휈휌 푀 = lim ℐ휈휌 푝|푚 − ℐ휈휌 푝|푀 = 0 푀→∞ 푀→∞ 2휋 4 푅 푘1 + 푘2 휇Δ휇휈휌 푘1, 푘2|푚 = 2푖푚Δ휈휌 푚 − 2푖 lim 푀Δ휈휌 푀 푀→∞ Naïve Ward identity Anomalous Ward identity 푒2 푘 + 푘 Δ푅 푘 , 푘 |푚 = 2푖푚Δ 푚 + 휖 푘 푘 1 2 휇 휇휈휌 1 2 휈휌 2휋2 휈휌훼훽 1훼 2훽 Anomaly! • Triangle diagram:(cont.) 4 푝 + 푘 + 푀 훾 푝 + 푀 훾 푝 − 푘 + 푀 2 푑 푝 2 휌 휈 1 −2푖푀∆휈휌 푀 = 2푖푒 푀 4 tr훾5 2 2 2 2 2 2 2휋 푝 + 푘2 + 푀 푝 + 푀 푝 − 푘1 + 푀 + 푘1 ↔ 푘2, 휈 ↔ 휌 1 1−푥 4 푝 + 푘 + 푀 훾 푝 + 푀 훾 푝 − 푘 + 푀 2 푑 푝 2 휌 휈 1 = 4푖푒 푀 푑푥 푑푦 4 tr훾5 2 2 2 2 3 0 0 2휋 푝 + 푘2 푦 + 푝 − 푘1 푥 + 푝 1 − 푥 − 푦 + 푀 + 푘1 ↔ 푘2, 휈 ↔ 휌 1 1−푥 4 푁 푙, 푘 , 푘 + 푁 푙, 푘 , 푘 2 2 푑 푙 휌휈 1 2 휈휌 2 1 = 4푖푒 푀 푑푥 푑푦 4 2 2 2 2 2 3 0 0 2휋 푙 + 푀 + 푘1 푥 + 푘2푦 − 푘1푥 − 푘2푦 푁휌휈 푙, 푘1, 푘2 ≡ tr훾5훾휌 푙 + 푘1푥 − 푘2푦 훾휈 푙 − 푘1 1 − 푥 − 푘2푦 Working out the trace 4푖푒2푀2 1 1−푥 1 −2푖푀∆휈휌 푀 = 2 휖휈휌훼훽푘1훼푘2훽 푑푥 푑푦 2 2 2 2 휋 0 0 푀 + 푘1 푥 + 푘2 푦 − 푘1푥 − 푘2푦 푒2 ⟶ 휖 푘 푘 as 푀 → ∞ 2휋2 휈휌훼훽 1훼 2훽 • Triangle diagram:(cont.) Coordinate space: 푖푒2 휕 퐽 = 2푖푚휓 훾 휓 + 휖 퐹 퐹 휇 5휇 5 16휋2 휇휈휌휆 휇휈 휌휆 Generalization to QCD+QED 2 2 푁푓푔 푖휂푒 휕 퐽 = 2푖푚휓 훾 휓 + 푖 휖 퐹푙 퐹푙 + 휖 퐹 퐹 휇 5휇 5 32휋2 휇휈휌휆 휇휈 휌휆 16휋2 휇휈휌휆 휇휈 휌휆 where 퐹휇휈 = 휕휇퐴휈 − 휕휈퐴휇 푙 푙 푙 푙푚푛 푚 푛 퐹휇휈 = 휕휇퐴휈 − 휕휈퐴휇 + 푔푓 퐴휇 퐴휈 color-flavor factor 2 휂 = 푁푐 푞푓 푓 UV divergence ⟹ Anomaly ⟹ Anomaly is independent of temperature and chemical potential. • Other approches: Point-splitting: Schwinger 퐽5휇 푥 = lim 퐽5휇 푥, 훿 훿→0 퐽5휇 푥, 훿 ≡ 푖푈 푥+, 푥− 휓 푥+ 훾휇훾5휓 푥− 푥+ 훿 푈 푥 , 푥 = exp 푖푒 푑휉 퐴 휉 푥 = 푥 ± + − 휌 휌 ± 2 푥− Maintain gauge invariance 퐴 < 퐽5휇 푥, 훿 > = −푖푈 푥+, 푥− tr훾휇훾5푆퐴 푥−, 푥+ where 푆퐴 푥−, 푥+ = Dirac propagator in an external EM field 4 −훾휇 휕휇 − 푖푒퐴휇 푆퐴 푥−, 푥+ = 푖훿 푥 − 푦 푚 = 0 1 푆퐴 푥−, 푥+ = −푖 < 푥 푦 > 훾휇 휕휇 − 푖푒퐴휇 • Other approaches (cont.): 2 휕휇푈 푥+, 푥− = 푖푒훿휌휕휇퐴휌 푥 + 푂 훿 3 −휕휇푆퐴 푥−, 푥+ = 푖푒훿휌휕휌퐴휇 푥 + 푂 훿 퐴 휕휇 < 퐽5휇 푥, 훿 > = 푒퐹휇휌 푥 훿휌tr훾휇훾5푆퐴 푥−, 푥+ 2 4 = −푒 퐹휇휌 푥 훿휌 푑 푦 tr훾휇훾5푆퐹 푥− − 푦 훾휈퐴휈 푦 푆퐹 푦 − 푥+ 2 푒 2 4 휕 1 1 = − 2 푒 퐹휇휌 푥 훿휌휖휇훼휈훽 푑 푦 2 2 퐴훽 푦 16휋 휕푥−훼 푥1 − 푦 푦 − 푥+ 푖푒2 푖푒2 = 퐹 퐹 휖 훿 훿 = 휖 퐹 퐹 4휋2훿2 휇휌 훽휈 휇훼휈훽 휌 훼 16휋2 휇휈휌휆 휇휈 휌휆 where Schouten identity 휖휇훼휈훽훿휌 + 휖휌휇훼휈훿훽 + 휖훽휌휇훼훿휈 + 휖휈훽휌휇훿훼 + 휖훼휈훽휌훿휇 = 0 has been employed. • Other approaches (cont.): Change of the path integral measure: Fujikawa 4 푆1 퐴, 휓, 휓 = 푖 푑 푥휓 퐷휓 퐷 = −푖훾휇 휕휇 − 푖푒퐴휇 푍 퐴 = 푑휓푑휓 푒푖푆1 퐴,휓,휓 휓 푥 → 푒−푖훼 푥 훾5 휓 푥 휓 푥 → 휓 푥 푒−푖훼 푥 훾5 훿휓훼 푥 −푖훼 푥 훾5 퐴 = 훿훼훽푒 훿 푥 − 푥′ ≡ 풰 훼훽 푥, 푥′ 훿휓훽 푥′ 훿휓 훼 푥 −푖훼 푥 훾5 퐴 = 훿훼훽푒 훿 푥 − 푥′ ≡ 풰 훼훽 푥, 푥′ 훿휓 훽 푥′ • Other approaches (cont.): 4 푆1 퐴, 휓, 휓 → 푆1 퐴, 휓, 휓 − 푑 푥훼 휕휇퐽5휇 4 푑휓푑휓 → 푑휓푑휓 det풰퐴 2 ≡ 푑휓푑휓 푒푖 푑 푥훼푑 4 푑 = 푖훿 0 tr훾5 = ∞ × 0 Need regularization! 2 −2 2 푖푒 푑 ≡ 푖 lim Tr훾5푓 Λ 퐷 = 휖휇휈휌휆퐹휇휈퐹휌휆 Λ→∞ 16휋2 4 푍 퐴 → 푑휓푑휓 푒푖푆1 퐴,휓,휓 + 푑 푥훼 푑−휕휇퐽5휇 = 푍 퐴 푖푒2 ⟹ 휕 < 퐽 푥 >퐴= 휖 퐹 퐹 휇 5휇 16휋2 휇휈휌휆 휇휈 휌휆 −푖훼 푖훼 In contrast: 휓 → 푒 휓 휓 → 휓 푒 for 푈푉 1 푑휓푑휓 → 푑휓푑휓 ⟹ No anomaly • Non-renormalization Multi-loop diagrams can be regularized by Pauli-Villars scheme applied to internal Bose lines without offsetting gauge invariance. Therefore 휇 휇 휈 푝 휌 휌 푝 휈 푒2 푘 + 푘 Δ푅 푘 , 푘 |푚 = 2푖Δ 푚 + 휖 푘 푘 1 2 휇 휇휈휌 1 2 휈휌 2휋2 휈휌훼훽 1훼 2훽 from one-loop only II. Applications: 푘1, 휀1 • 휋0 ⟶ 2γ Decay amplitude 푞 풯 = lim ℱ 푞2 2 2 푞 →−푚휋 푘2, 휀2 2 2 2 2 4 4 푖(푘1∙푥+푘2∙푦) ℱ 푞 = 푞 + 푚휋 푒 휀1휇휀2휈 푑 푥푑 푦 < 0 푇퐽휇 푥 퐽휈 푦 휋 0 0 > 푒 2 Naïve PCAC relation 휕휇퐽5휇 = 푚휋푓휋휋 2 2 2 푞 + 푚휋 ℱ 푞 = 2 휀1휇휀2휈푞휌푇휇휈휌 푓휋푚휋 2 4 4 푖(푘1∙푥+푘2∙푦) 푇휇휈휌 ≡ −푖푒 푑 푥푑 푦 < 0 푇퐽휇 푥 퐽휈 푦 퐽5휌 0 0 > 푒 • 휋0 ⟶ 2γ (cont.): Electric current conservation + Bose symmetry 푞2퐹 푞2 푞휌푇휇휈휌 = 휖휇휈훼훽푘1훼푘2훽 2 2 푞 + 푚휋 where we assumed that 휋0 is the only low-lying pole 2 2 2 푞 + 푚휋 1 2 2 ℱ 푞 = 2 휀1휇휀2휈푞휌푇휇휈휌 = 2 휀1휇휀2휈휖휇휈훼훽푘1훼푘2훽푞 퐹 푞 푓휋푚휋 푓휋푚휋 ⟹ ℱ 0 = 0 0 2 Low 휋 mass ℱ 푚휋 ≅ ℱ 0 = 0 disagree with experiments. Modified PCAC relation 푖푒2 휕 퐽 = 푚2 푓 휋 + 휖 퐹 퐹 휇 5휇 휋 휋 32휋2 휇휈휌휆 휇휈 휌휆 ⟹ Decay width ≅ 7.63 eV Experimental value = 7.31 ± 1.5 eV • Chiral magnetic effect: QCD+QED Lagrangian with ordinary and chiral chemical potentials 1 1 ℒ 퐴, 휓, 휓 = − 퐹푙 퐹푙 − 퐹 퐹 − 휓 훾 휕 − 푖푔퐴푙 푇푙 − 푖푒퐴 휓 4 휇휈 휇휈 4 휇휈 휇휈 휇 휇 휇 휇 † † +휇휓 휓 + 휇5휓 훾5휓 + 퐽휇퐴휇 +gauge fixing terms and counter terms Both 휇 and 휇5 can be functions of space and time.