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

푑휓푑휓 푒푖푆1 퐴,휓,휓 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. Generating functional of connected Green functions of photons 4 푍 퐽 = 푑퐴푙 푑퐴 푑휓푑휓 푒푖 푑 푥ℒ 퐴,휓,휓

푑푡 may follow a closed time path to handle the non-equilibrium case. 훿 ln 푍 풜휇 푥 = −푖 훿퐽휇 푥 • Chiral magnetic effect (cont.): Quantum effective action:

4 Γ 풜 = −푖 ln 푍 퐽 − 푑 푥 퐽휇풜휇

4 훿Γ 푑 푞 푖푞∙푥 0 1 퐽휇 푥 = = 4 푒 퐽휇 푞 = 퐽휇 + 퐽휇 + ⋯ 훿풜휇 푥 2휋 0 1 퐽휇 = 푂 풜 , 퐽휇 = 푂 휇5풜 , … 푑4푘 퐽 1 푞 = −푖 Δ −푞, 푞 − 푘|0 휇 푘 풜 푞 − 푘 휇 2휋 4 4휇휈 5 휈

4 −푖푞∙푥 4 −푖푞∙푥 풜휇 푞 = 푑 푥 푒 풜휇 푥 휇5 푘 = 푑 푥 푒 휇5 푥 • Chiral magnetic effect (cont.): −푖휔푡 Consider 풜휇 푥 = 풜 푟 , 0 , 휇5 푥 = 휇5푒 4 3 휇5 푘 = 2휋 휇5훿 푘 훿 푘0 − 휔 퐽푖 푞 = −푖휇5Δ4푖푗 −푞, 푞 − 푘|0 풜푗 푞 − 푘 푘 = 0, 푖휔 푞 = 푞 , 푖휔 Anomalous Ward identity 푒2 푒2 푖휔∆ −푞, 푞 − 푘|0 = − 휖 푞 푞 − 푘 = 푖 휔휖 푞 4푖푗 2휋2 푖푗훼훽 훼 훽 2휋2 푖푗푘 푘 푒2 ∆ −푞, 푞 − 푘|0 = 휖 푞 4푖푗 2휋2 푖푗푘 푘 푒2휇 In the limit ω→0 퐽 푞 = −푖 5 휖 푞 풜 푞 푖 2휋2 푖푗푘 푘 푗 푒2휇 퐽 푟 = 5 퐵 푟 퐵 = ∇ × 풜 2휋2 III. Miscellaneous Topics: • Other chiral anomalies: Non-Abelian gauge anomalies:

4 푆1 퐴, 휓, 휓 = 푖 푑 푥 휓 퐿퐷퐿휓퐿 + 휓 푅퐷푅휓푅

푙 푙 푙 푙 퐷퐿 = 휕 − 푖퐴 푇퐿 퐷푅 = 휕 − 푖퐴 푇푅 퐽휇 = 푖 휓 퐿훾휇푇퐿휓퐿 + 휓 푅훾휇푇푅휓푅 푇퐿, 푇푅 = generators of gauge transformation of left and right fermions 푎 푎 푎푏푐 푏 푐 퐷휇퐽휇 ≡ 휕휇퐽휇 + 푓 퐴휇퐽휇 푎 Classical 퐷휇퐽휇 = 0 푎 푎푏푐 푏 푐 Quantum 퐷휇퐽휇 = 휅휖휇휈휌휆푑 퐹휇휈퐹휌휆 1 푑푎푏푐 = tr푇푎 푇푏, 푇푐 − tr푇푎 푇푏, 푇푐 2 푅 푅 푅 퐿 퐿 퐿 3rd Casmir • Other chiral anomalies (cont.): Chiral anomaly in curved space: 휇 푎 휇휈휌휆 푎푏푐 푏 푐 푎 훼 훽 퐷 퐽휇 = 휅휖 푑 퐹휇휈퐹휌휆 + 휅′푏 푅 훽휇휈푅 훼휌휆 1 퐷휇퐽푎 ≡ 휕 −푔퐽푎 + 푓푎푏푐퐴푏퐽푐 휇 −푔 휇 휇 휇 휇 훼 푅 훽휇휈 = Riemann tensor 1 푑푎푏푐 = tr푇푎 푇푏, 푇푐 − tr푇푎 푇푏, 푇푐 2 푅 푅 푅 퐿 퐿 퐿 푎 푎 푎 푏 = tr푇푅 − tr푇퐿 Gauge-gravity mixed anomaly Pure gravitational anomaly: 휇 퐷 푇휇휈 ≠ 0 Exists only in 푑 = 4푘 + 2 2,6,8,10, … dimensions • Covariant and consistent anomalies: One-loop effective action:

Γ 퐴 = −푖 ln 푑휓푑휓 푒푖푆1 퐴,휓,휓

An infinitesimal gauge transformation: 훿퐴휇 = 퐷휇휃

4 훿Γ 4 4 훿Γ 퐴 = 2tr 푑 푥 퐷휇휃 ∝ tr 푑 푥 휃퐷휇퐽휇 ∝ 푡푟 푑 푥휃풜 훿퐴휇 푥 Introduce the generator in functional space Anomaly 훿 𝒢 푥 ≡ −퐷휇 훿퐴휇 Lie algebra: 𝒢푎 푥 , 𝒢푏(푦) = 푖푓푎푏푐𝒢푐 푥 훿4 푥 − 푦 • Covariant and consistent anomalies (cont): Wess-Zumino consistent condition: 𝒢푎 푥 풜푏 푦 − 𝒢푏 푦 풜푎 푥 = 푖푓푎푏푐𝒢푐 푥 풜푐 푥 훿4 푥 − 푦 For non-Abelian chiral gauge theory ---- Covariant anomaly is not consistent; ---- Consistent anomaly is not covariant; ---- They share the same group theoretic factor. • Anomaly cancellation: Benign anomaly: Anomaly of global symmetry ⟹ Interesting physics, e.g., CME Bad anomaly: Anomaly of gauge symmetry ⟹ Jeopardize unitary and renormalizability ⟹ Has to be cancelled! Example 1: Electroweak theory, gauge group = 푆푈(2) × 푈(1) 1 푑푎푏푐 = tr푇푎 푇푏, 푇푐 − tr푇푎 푇푏, 푇푐 = 0 2 푅 푅 푅 퐿 퐿 퐿 푎 푎 푎 푏 = tr푇푅 − tr푇퐿 = 0 ⟹ Anomaly free! Example 2: Supersymmetric Yang-Mills in 푑 = 10 Anomalies to be cancelled: gauge, mixed and gravity Anomaly free gauge group: 푆푂(32) and 푬ퟖ × 푬ퟖ Superstring References: 1. S. Adler, Phys. Rev. 177 (5), p2426 (1969). 2. J. S. Bell and R. Jackiw, II Nuovo Cimento A, 60, p47 (1969). 3. J. Schwinger, Phys. Rev., 82, p914 (1951). 4. K. Fujikawa and Suzuki, Path integrals and quantum anomalies, Int. Series of Monographs on Physics, 122, Oxford Press, 2004. 5. K. Fukushima, D. E. Kharzeev and Warringa, Phys. Rev. D78, p074033 (2008). 6. M. B. Gree, J. H. Schwarz and Witten, Superstring theory, Cambridge University Press, Cambridge, 1987, vol. 2.