On the Universality of the Chern-Simons Diffusion Rate Aldo L. - PowerPoint PPT Presentation

On the Universality of the Chern-Simons Diffusion Rate Aldo L. Cotrone Florence University Supersymmetric Quantum Field Theories in the Non-perturbative Regime May 9, 2018 Work in collaboration with Francesco Bigazzi (INFN, Florence) and Flavio Porri (Florence University) arXiv:1804.09942 Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

The Chern-Simons diffusion rate Definition: 1 32 π 2 Tr F ˜ Q ( x ) = F Change in the Chern-Simons number � d 4 x Q ( x ) ∆ N CS = Chern-Simons diffusion rate Γ CS = � (∆ N CS ) 2 � � d 4 x � Q ( x ) Q (0) � = Vt V = volume , t = time Note: Minkowski correlator, real time physics. Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

The Chern-Simons diffusion rate On a state with temperature T (e.g. Quark-Gluon Plasma): Kubo formula: 2 T ω Im G R ( ω, � Γ CS = − lim k = 0) ω → 0 Thus: compute retarded correlator G R ( ω, � k = 0). Genesis: thermal fluctuations can excite sphalerons ⇒ sphalerons decay ⇒ ∆ N CS � = 0 (locally). Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

The Chern-Simons diffusion rate Why Γ CS interesting Baryogenesis in Standard Model: sphaleron transitions cause ∆( B + L ) � = 0. Many studies at weak coupling. Chiral magnetic effect in QGP. Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

The Chern-Simons diffusion rate Chiral magnetic effect [Fukushima-Kharzeev-Warringa 2008] Axial anomaly: ∂ µ J µ A = − 2 Q Then: ∆ N CS generates a ∆ chirality ⇒ µ A � = 0 (chemical potential). Non central collisions in QGP have large magnetic field � B . J em = σ CME � ∆ chirality + � B generate electric current � B , with σ CME = e 2 2 π µ A . Currently under experimental search at RHIC and LHC. Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

The Chern-Simons diffusion rate Magnitude of Γ CS in the QGP? Real time non-perturbative physics: no reliable computational methods in QCD. Effective theory result [Moore-Tassler 2010] : Γ CS ∼ c · λ 5 T 4 λ = g 2 YM N c ′ t Hooft coupling Notes: c is non-perturbative; result valid at α s ≪ 1. Γ CS ∼ O ( N 0 c ). Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

Holographic derivation The Chern-Simons diffusion rate in N = 4 SYM [Son-Starinets 2002] Background is BH − AdS 5 × S 5 , generated by N c D3-branes. D3-brane action contains � d 4 x C F ˜ F ⇒ gravity field dual to Q is RR-potential C . Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

Holographic derivation The Chern-Simons diffusion rate in N = 4 SYM [Son-Starinets 2002] Action for C in 5d: d 5 x √− g 5 � � − 1 � 2 ∂ M C ∂ M C Solve eq of motion for C ⇒ Retarded correlator G R . Use Kubo, result λ 2 256 π 3 T 4 Γ CS = Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

Holographic derivation Comments: N = 4 SYM “a bit different from QCD”. Other holographic results look different, eg: N = 4 SYM with magnetic field B [Basar-Kharzeev 2012] : � 2 � 1 λ Γ CS = Γ CS ( B = 0) · f ( B ) = sT s = entropy density 2 7 π 5 N c N = 4 SYM with anisotropy a [Bu 2014] : � 2 � 1 λ Γ CS = Γ CS ( a = 0) · g ( a ) = sT s = entropy density 2 7 π 5 N c Witten model of holographic Yang-Mills [Craps et al 2012] : � 2 � λ 3 KK T 6 = 1 1 1 λ Γ CS = sT s = entropy density 2 π 3 6 π 2 M 2 2 7 π 5 N c ... 1 Situation different from universal η s = [Kovtun-Son-Starinets 2004] . 4 π Or does it?! Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

Holographic derivation Comments: N = 4 SYM “a bit different from QCD”. Other holographic results look different, eg: N = 4 SYM with magnetic field B [Basar-Kharzeev 2012] : � 2 � 1 λ Γ CS = Γ CS ( B = 0) · f ( B ) = sT s = entropy density 2 7 π 5 N c N = 4 SYM with anisotropy a [Bu 2014] : � 2 � 1 λ Γ CS = Γ CS ( a = 0) · g ( a ) = sT s = entropy density 2 7 π 5 N c Witten model of holographic Yang-Mills [Craps et al 2012] : � 2 � λ 3 KK T 6 = 1 1 1 λ Γ CS = sT s = entropy density 2 π 3 6 π 2 M 2 2 7 π 5 N c ... 1 Situation different from universal η s = [Kovtun-Son-Starinets 2004] . 4 π Or does it?! Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

“Universality” of the result “Wrapped brane models” Wrap Dp-brane on ( p − 3)-cycle Ω p − 3 ⇓ 4d gauge theory in IR N = 4 SYM included. Some of the most interesting models included (Witten-Sakai-Sugimoto, Maldacena-Nu˜ nez, ...). All computations of Γ CS in the literature performed in this class of models. Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

“Universality” of the result Expanding DBI+WZ action at low energies get 1 Tr F 2 − θ YM 32 π 2 Tr F ˜ L = − F 4 g 2 YM with 1 � p − 7 τ p (2 πα ′ ) 2 d p − 3 x e 4 φ � = det ( g E ) g 2 Ω p − 3 YM � τ p (2 πα ′ ) 2 θ YM = C p − 3 Ω p − 3 Thus: gravity field dual to Q is � C ≡ τ p (2 π ) 2 (2 πα ′ ) 2 C p − 3 Ω p − 3 Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

“Universality” of the result Derivation of the 5 d action of C Action of F ( p − 2) = dC ( p − 3) in 10 d d 10 x √− g 10 e � � 1 � − 1 7 − p 2 F 2 2 φ ( p − 2) 2 κ 2 10 Reduction ansatz 10 = e f ds 2 ds 2 5 + ds 2 int Reduction of F ( p − 2) C ∂ M ˜ ( p − 2) = ∂ M ˜ F 2 p − 3 )] − 1 e − f C [ det ( g Ω ′ where C = τ p (2 π ) 2 (2 πα ′ ) 2 Vol (Ω p − 3 ) ˜ C Ω ′ p − 3 has unit volume. Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

“Universality” of the result Final result: d 5 x √− g 5 H 1 � � − 1 � 2 ∂ M C ∂ M C 2 κ 2 5 with 2   g 4 1 1 YM H = = 4 φ √ det g E   p − 7 (2 π ) 4 (8 π 2 ) 2 τ p (2 πα ′ ) 2 � Ω p − 3 e ⇓ 1 Chern-Simons diffusion rate has “universal” form Γ CS = α 2 s ( T ) (2 π ) 3 sT 1[Son-Starinets 2002, Gursoy-Iatrakis-Kiritsis-Nitti-O’Bannon 2013] Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

“Universality” of the result Comments: Checked also in N = 4 SYM with flavors and N = 1 models. Can calculate first 1 /λ correction: Γ CS decreases [Bu 2014] . Is holographic result an upper bound on Γ CS ? Problem in extending result to other models: identification of coupling λ and gravity field dual to Q . Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

Inclusion of the anomaly Anomaly ∂ µ J µ A = − qQ holographically reproduced by Stuckelberg action [Klebanov et al 2002] d 5 x √− g 5 � � 1 � − 1 − 1 � ∂ M C + qA M � 4 F MN F MN 2 ( ∂ M C + qA M ) 2 κ 2 5 A M : gravity field dual to J µ A ; q : anomaly coefficient. From dimensional reduction of main holographic models: Klebanov-Strassler, N = 4 with flavors, Maldacena-Nu˜ nez, Witten-Sakai-Sugimoto. Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

Inclusion of the anomaly Define B = ( dC + qA ) ⇒ dB = qF ≡ F B ⇓ action for a massive vector (mass ∼ q ) d 5 x √− g 5 1 � − 1 − 1 � � 2 q 2 B 2 4 F B , MN F MN B 2 κ 2 5 q 2 Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

Inclusion of the anomaly Calculation of G R on generic BH background (mild assumptions): Near horizon � r � ∆ � � − i ω 1 − r T � � 1 − r � � b (0) + b (1) B t ∼ + · · · h h r h r h r h (4∆(∆ − 1) = q 2 ) Get 1 ω Im G R ∼ α · | b (0) h | 2 with α independent of ω . For ω → 0 � q 2 � b (1) · b (0) = i ω + regular in ω h h ⇒ Two possibilities: q = 0 (no anomaly), or b (0) ∼ ω a with a ≥ 1 for ω → 0 ⇒ Γ CS ( q � = 0) = 0. h Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

Inclusion of the anomaly Numeric result on AdS -BH: ( 0 ) 2 | b h 1.5 1.0 0.5 ω 0.2 0.4 0.6 0.8 1.0 Black: q = 0. Blue: q = 0 . 04. Red: q = 0 . 44. Green: q = 3. Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

Inclusion of the anomaly Why expected: Anomaly ( ∂ µ J µ A = − qQ ) ⇒ Γ CS ∼ � QQ � ∼ � ∂ J A ∂ J A � d 3 x J t � Q A = A not conserved (anomaly), thus Γ CS ∼ �Q A ( t → ∞ ) Q A (0) � R = 0 In fact, with only gapped modes expect �Q A ( t ) Q A (0) � R ∼ e − t τ τ : relaxation time. Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate

Inclusion of the anomaly Definition of Γ CS makes sense if there is separation of time scales: [Moore-Tassler 2010] ∆ t < t ∗ ≪ τ ∆ t = ( microscopic ) time scale of CS number fluctuations t ∗ = cut − off τ = relaxation time Thus can define � t ∗ � d 3 x � Q ( t , x ) Q (0) � Γ CS = dt Note: Can remove cut-off if τ → ∞ . Large N c : τ ∼ N 2 c / T ≫ 1 / T ∼ microscopic time scale. Aldo L. Cotrone On the Universality of the Chern-Simons Diffusion Rate