Holography of the Chiral Magnetic Effect and Background-Dependence Ingo Kirsch DESY Hamburg, Germany ` International Symposium Ahrenshoop on the Theory of Elementary Particles Recent Developments in String and Field Theory, August 27-31, 2012, Berlin-Schmöckwitz Kalaydzhyan, I.K., Phys. Rev. Lett. 106 (2011) , Gahramanov, Kalaydzhyan, I.K., PRD 85 (2012) Based on work with T. Kalaydzhyan, I. Gahramanov

Motivation: Observed charge asymmetry in HICs Heavy-ion collision: Excess of positive charge (above Y R ) Excess of negative charge (below Y R ) + - Holography of the Chiral Magnetic Effect International Symposium Ahrenshoop Ingo Kirsch 2

spin mom. sphaleron background Chiral magnetic effect: 1. in presence of a magnetic field B, momenta of the quarks align along B 2. topological charge induces chirality 3. positively/negatively charged quarks move up/down (charge separation!) 4. an electric current is induced along the magnetic field B CME is a candidate for explaining an observed charge asymmetry in HIC Does the CME hold at strong coupling? AdS/CFT International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect Chiral magnetic effect + - CME in HICs: Kharzeev, McLerran and Warringa (2008), Fukushima, Kharzeev and Warringa (2008) earlier papers: Vilenkin (1980), Giovannini and Shaposhnikov (1998), Alekseev, Cheianov and Fröhlich (1998) 3

Outline: I.CME in hydrodynamics II.Fluid/gravity model of the CME III.CME in anisotropic fluids (v 2 -dependence) Conclusions Overview ETH Zurich, 30 June 2010International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 4

Part I: CME in hydrodynamics 4 University of Chicago, 23 April 2007International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect

Hydrodynamics vs. fluid/gravity model ETH Zurich, 30 June 2010 n=2 dual International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 6 Hydrodynamics Fluid/gravity model Multiple-charge model Holographic n-charge model U(1) n plasma with triangle anomalies 5D AdS black hole geometry with Son & Surowka (2009), n U(1) charges Neiman & Oz (2010) Two-charge model Holographic two-charge model U(1) V x U(1) A plasma n-charge model reduced to two charges recover CME (and other effects) recover holographic CME, etc.

U(1) n plasma with triangle anomalies: stress-energy tensor T mn and U(1) currents j am : Hydrodynamical model with n anomalous U(1) charges ETH Zurich, 30 June 2010 “New” transport coefficients (not listed in Landau-Lifshitz) - vortical conductivities Erdmenger, Haack, Kaminski, Yarom (2008) - magnetic conductivities Son & Surowka (2009) first found in a holographic context (AdS/CFT) E- & B-fields, vorticity: Son & Surowka (2009) International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 7

U(1) currents: vortical and magn. conductivities: Conductivities are non-zero only in fluids with triangle anomalies! First-order transport coefficients ETH Zurich, 30 June 2010 chemical potentials Son & Surowka (2009), Neiman & Oz (2010) coffee with sugar (chiral molecules) International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 8

U(1) A : provides chemical potential  5 (chirality) U(1) V : measures the electric current Hydrodynamical equations: Identifications: axial gauge field switched off! Constitutive equations: C-parity allows for: Two charge case (n=2): U(1) A x U(1) V ETH Zurich, 30 June 2010 Sadofyev and Isachenkov (2010), Kalaydzhyan & I.K. (2011)+(2012) CME coefficient International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 9

constitutive equations: transport coefficients (conductivities): C=chiral since CVE, CME prop. to chiral chemical potential m 5 Q=quark since QVE, QME prop. to quark chemical potential m (QME also called chiral separation effect (CSE)) creates an effective magnetic field Kharzeev and Son (2010) (Chiral) magnetic and vortical effects ETH Zurich, 30 June 2010 CME CSE (QME) CVE QVE International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 10

Part II: Fluid/gravity model of the CME Kalaydzhyan & I.K., PRL 106 (2011) University of Chicago, 23 April 2007International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect

Hydrodynamics vs. fluid/gravity model ETH Zurich, 30 June 2010 n=2 dual International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 12 Hydrodynamics Fluid/gravity model Multiple-charge model Holographic n-charge model U(1) n plasma with triangle anomalies 5D AdS black hole geometry with Son & Surowka (2009) n U(1) charges Neiman & Oz (2010) Two-charge model Holographic two-charge model U(1) V x U(1) A plasma n-charge model reduced to two charges recover CME (and other effects) recover holographic CME, etc.

Strategy: quark-gluon plasma is strongly-coupled use AdS/CFT to compute the transport coefficients relevant for the anomalous effects (CME, etc.) - find a 5d charged AdS black hole solution with several U(1) charges - duality: - use fluid-gravity methods to holographically compute the transport coefficients (i.e. CME and other effects) Hawking temperature Gravity: Holographic computation ETH Zurich, 30 June 2010 Kalaydzhyan & I.K. PRL 106 (2011) d fluid (QGP) `lives’ on the boundary of the AdS space mass m U(1) charges q a International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 13

Five-dimensional U(1) n Einstein-Maxwell theory with cosmological term: AdS black hole solution with multiple U(1) charges ETH Zurich, 30 June 2010 The information of the anomalies is encoded in the Chern-Simons coefficients Son & Surowka (2009) Fields: -metric g MN (M, N = 0,..., 4) -n U(1) gauge fields A M a (a = 1, …, n) - cosmological constant  = -6 International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 14

5d AdS black hole solution (0 th order solution): with = four-velocity of the fluid Boosted AdS black hole solution ETH Zurich, 30 June 2010 n charges q a background gauge field added (needed to model external B-field)! International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 15

We use the standard procedure of Bhattacharyya et al. (2008) to holographically compute the transport coefficients (Torabian & Yee (2009) for n=3) and : 1. Vary 4-velocity and background fields (up to first order): The boosted black-brane solution (0 th order sol.) is no longer an exact solution, but receives higher-order corrections. Ansatz for first-order corrections: Need to determine first order corrections First-order transport coefficients ETH Zurich, 30 June 2010International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 16

2. Solve equations of motion (system of Einstein-Maxwell equations) and find the first-order corrections to the metric and gauge fields: (lengthy calculation, functions of ) 3. Read off energy-momentum tensor and U(1) currents from the near-boundary expansion of the first-order corrected background (e.g. Fefferman-Graham coordinates): First-order transport coefficients (cont.) ETH Zurich, 30 June 2010International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 17

4. Determine the vortical and magnetic conductivities and use identities (from zeroth order solution): transport coefficients: First-order transport coefficients (cont.) ETH Zurich, 30 June 2010 We recover the hydrodynamic result! International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 18

Using the same identifications as in hydrodynamics, but now for the holographically computed transport coefficients, we get Result: CME, CVE, etc. are realized in an n-charged AdS black hole model (plus background gauge field), when appropriately reduced to a two-charge model (n=2). Holographic magnetic and vortical effects ETH Zurich, 30 June 2010 CME QME CVE QVE Other AdS/QCD models: Lifschytz and Lippert (2009), Yee (2009), Rebhan, Schmitt and Stricker (2010), Gorsky, Kopnin and Zayakin (2010), Rubakov (2010), Gynther, Landsteiner, Pena- Benitez and Rebhan (2010), Hoyos, Nishioka, O’Bannon (2011) International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 19

Part III: CME in anisotropic fluids Gahramanov, Kalaydzhyan & I.K., PRD 85 (2012) [arXiv: ] 4 University of Chicago, 23 April 2007International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect

Event-by-event anisotropy (v 2 obs ) dependence (low p T ) Investigate the charge asymmetry as a function of the anisotropy v 2 obs of the measured particles in mid-central 20–40% centrality collisions ( ). Consider (rare) events with different v 2 obs. Possible dependence of the charge asymmetry on v 2 ETH Zurich, 30 June 2010 Quan Wang (2012) Observations: same-sign particles are emitted more likely in UD direction the larger v 2 obs same-sign particles are emitted less likely in LR direction the larger v 2 obs (the dependence is significantly weaker for opposite-sign particles) => strong v 2 obs dependence of the difference between UD and LR => charge separation depends approx. linearly on v 2 obs (apparently in contradiction with the CME) International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 21

Central question: In anisotropic fluids, does the chiral conductivity depend on v 2 ? Sketch of the time-evolution of the momentum anisotropy : Our model describes a state after thermalization with unequal pressures. We do not model the full evolution of. Build-up of the elliptic flow and momentum anisotropy ETH Zurich, 30 June 2010 at freeze-out: sketch based on Huovinen, Petreczky (2010) International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 22

Anisotropic fluid with n anomalous U(1) charges stress-energy tensor and U(1) currents : orthogonality and normalization: local rest frame: thermodynamic identity: Hydrodynamics of an anisotropic fluid ETH Zurich, 30 June 2010 direction of anisotropy Florkowski (various papers) International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 23

Anisotropic fluid with one axial and one vector U(1) Repeating the hydrodynamic computation of Son & Surowka (for n=2), we find the following result for the chiral magnetic effect: or, for small, multiple charge case (n arb.): Chiral magnetic and vortical effects ETH Zurich, 30 June 2010International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 24

5d AdS black hole solution (ansatz): asymptotic solution (close to the boundary): no analytic solution numerical solution Boosted anisotropic AdS black hole solution (w/ n U(1)’s) ETH Zurich, 30 June 2010 anisotropy parameter International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 25

Shooting techniques provide numerical plots for the functions f(r), A 0 (r), w T (r), w L (r) for  =10: outer horizon: Numerical solution for the AdS black hole background ETH Zurich, 30 June 2010 we need International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 26

The holographic computation of the transport coefficients is very similar to that in the isotropic case. Magnetic conductivities (result): find agreement with hydrodynamics if the orange factors agree (needs to be shown numerically) First-order transport coefficients ETH Zurich, 30 June 2010International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 27

Numerical plot w L (r + ) as a function of the anisotropy: Numerical agreement with hydrodynamics ETH Zurich, 30 June 2010 agreement with hydrodynamics since International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 28

I presented two descriptions of the CME (and related effects) in a) isotropic plasmas (P=P T =P L ): i) hydrodynamic model: U(1) A x U(1) V fluid with triangle anomaly ii) holographic fluid-gravity model: 5d AdS-Reissner-Nordstrom-like solution with two U(1) charges Agreement was found between both models. b) anisotropic plasmas (P T ≠ P L ): -experimental data suggests possible v 2 – dependence of the charge separation -Does the chiral magn. conductivity depend on v 2 ? -we constructed anisotropic versions of the above U(1) A x U(1) V models and found Is the observed charge asymmetry a combined effect of the CME (1 st term in ) and the dynamics of the system (2 nd term)? Conclusions International Symposium Ahrenshoop Ingo Kirsch Holography of the Chiral Magnetic Effect 29