Chiral magnetic effect in the hadronic phase Shota Imaki PRD 101, - PowerPoint PPT Presentation

Chiral magnetic effect in the hadronic phase Shota Imaki PRD 101, 074024 (2020) 1 /12

Summary Summary : - Chiral magnetic e ff ect in the hadronic phase involves pseudoscalar mesons. - Form of the CME current in the hadronic phase is model-independent and higher-loop immune. - This involvement of pseudoscalar mesons may decrease the CME strength. μ 5 j = e 2 N c 2 π 2 μ 5 B tr ( Q 2 ) Chiral phase j B μ 5 2 π 2 μ 5 B tr ( Q 2 + 1 j = e 2 N c 6 [ Q , Σ ][ Q , Σ † ] ) : mesons Hadronic phase Σ j B 2 /12

Contents: 1. Chiral magnetic e ff ect in the chiral phase 2. Chiral magnetic e ff ect in the hadronic phase 3. Strength of the chiral magnetic current in the hadronic phase 4. Conclusion 3 /12

Chiral phase Derivation : - One derivation is through the e ff ective action. - Derivative expansion of the e ff ective action and corresponding triangle diagram (Fig.) are: [ iD μ = i ∂ μ − eQA μ − γ 5 a μ , a μ = ( μ 5 , 0 ) ] S e ff = − i log Det( iD − m ) = e 2 N c 4 π 2 ∫ d 4 x a μ A ν ˜ F μν tr( Q 2 ) + ⋯ (Fig.) Corresponding diagram δ A = e 2 N c j = δ S e ff 2 π 2 μ 5 B tr( Q 2 ) CME current in the chiral phase - Note : In order to tame the renormalization scheme dependence, one may impose a physical requirement that the e ff ective action generates the canonical anomalous divergence, ∂ μ j μ = e 2 N c F R μν − F L μν ˜ μν ˜ 16 π 2 ( F R F L μν ) . 4 /12

Contents: 1. Chiral magnetic e ff ect in the chiral phase 2. Chiral magnetic e ff ect in the hadronic phase - Derivation 1: via a chiral e ff ective model - Derivation 2: via Wess-Zumino-Witten action 3. Strength of the chiral magnetic current in the hadronic phase 4. Conclusion 5 /12

Hadronic phase Derivation 1 (via a chiral e ff ective model) : - We adopt a chiral e ff ective model: [ M = P R Σ + P L Σ † , Σ = exp( i π A λ A / f π ) ] ℒ = ¯ q ( iD − gM ) q . - Derivative expansion of the e ff ective action and the corresponding triangle diagrams (Fig.) are: S e ff = − i log Det( iD − gM ) Σ −∂ 2 − g 2 ) + ⋯ = − i Tr ( γ 5 a i ∂ + gM † −∂ 2 − g 2 eQA i ∂ + gM † −∂ 2 − g 2 eQA i ∂ + gM † = e 2 N c 4 π 2 ∫ d 4 x a μ A ν ˜ F μν tr ( Q 2 + 1 6 [ Q , Σ ][ Q , Σ † ] ) + ⋯ . (Fig.) Corresponding diagrams - The e ff ective action involves pseudoscalar mesons as the diagrams illustrate. - Note : In order to tame the renormalization scheme dependence, one may impose a physical requirement that the e ff ective action reduces to that in the chiral phase for . Σ = 1 6 /12

Hadronic phase Derivation 2 (via Wess-Zumino-Witten action) : - Wess-Zumino-Witten action gives rise to the same e ff ective action. Wess-Zumino-Witten action* S e ff = e 2 N c 4 π 2 ∫ d 4 x a μ A ν ˜ F μν tr ( Q 2 + 1 6 [ Q , Σ ][ Q , Σ † ] ) + ⋯ . - This derivation implies that the e ff ective action is independent to microscopic details and higher- loop corrections. 7 /12 * Source: Kawai-Tye 1984.

Hadronic phase Result : - The chiral magnetic current reads: δ A = e 2 N c 2 π 2 μ 5 B tr ( Q 2 + 1 6 [ Q , Σ ][ Q , Σ † ] ) j = δ S e ff CME current in the hadronic phase - The current involves the pseudoscalar mesons. - The functional form of the current is independent to microscopic details and higher-loop corrections. Note : - Taking the expectation value of the pseudoscalar mesons, the current reduces to the familiar form: ⟨ j ⟩ = κ e 2 N c tr( Q 2 ) ⟨ tr ( Q 2 + 1 6 [ Q , Σ ][ Q , Σ † ] ) ⟩ . 1 2 π 2 μ 5 B tr( Q 2 ) , κ ≡ - With a physical value adjusted, the form of the anomalous current is still protected. H = κ B 8 /12

Contents: 1. Chiral magnetic e ff ect in the chiral phase 2. Chiral magnetic e ff ect in the hadronic phase 3. Strength of the chiral magnetic current in the hadronic phase 4. Conclusion 9 /12

Strength Dielectric constant : - Now our interest is on the strength of the CME current: ⟨ j ⟩ = κ e 2 N c tr( Q 2 ) ⟨ tr ( Q 2 + 1 6 [ Q , Σ ][ Q , Σ † ] ) ⟩ 1 2 π 2 μ 5 B tr( Q 2 ) , κ ≡ - Here is “dielectric constant” incorporating interactions with pseudoscalar mesons. κ Strength : - The dielectric constant as a function of T is analytically calculable for two-flavor free pion gas: κ ( T ) = 1 5 (12 + 3 e − 2 G + 9 e − G − 18 e − 1 G ≡ f − 2 2 G ) , π ⟨ π A ( x ) π A ( x ) ⟩ . chiral restoration - Mesonic medium reduces the current strength (Fig.). - It is interesting to note that the beam energy scan programs 
 have reported reduced CME signals for low beam energies 
 for which the fireball may have a short lifetime until it hadronizes. 10 /12 (Fig.) T dependence of dielectric constant

Contents: 1. Chiral magnetic e ff ect in the chiral phase 2. Chiral magnetic e ff ect in the hadronic phase 3. Strength of the chiral magnetic current in the hadronic phase 4. Conclusion 11 /12

Conclusion Conclusion : - CME in the hadronic phase involves interaction with pseudoscalar mesons. - The functional form of the current is independent to microscopic details and higher-loop corrections. - CME current could be weakened by pseudoscalar mesons as the two-flavor analysis implied. Outlooks : - Large multi-pion correlations may much more influence the strength of CME. - Other chiral transports in the hadronic phase deserve further study. - E.g. Chiral separation e ff ect is also modified, and thus chiral magnetic wave could be modified. - E.g. Chiral vortical e ff ect may also be modified. 12 /12