Universality of the Chern-Simons Diffusion Rate

PHYSICAL REVIEW D 98, 106023 (2018) Universality of the Chern-Simons diffusion rate † ‡ Francesco Bigazzi,1,* Aldo L. Cotrone,1,2, and Flavio Porri1,2, 1INFN, Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino (Firenze), Italy 2Dipartimento di Fisica e Astronomia, Universit`a di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino (Firenze), Italy (Received 20 August 2018; published 27 November 2018) We prove the universality of the Chern-Simons diffusion rate—a crucial observable for the chiral magnetic effect—in a large class of planar strongly correlated gauge theories with dual string description. When the effects of anomalies are suppressed, the diffusion rate is simply given in terms of temperature, entropy density and gauge coupling, with a universal numerical coefficient. We show that this result holds, in fact, for all the top-down holographic models where the calculation has been performed in the past, even in the presence of magnetic fields and anisotropy. We also extend the check to further well-known models for which the same computation was lacking. Finally we point out some subtleties related to the definition of the Chern-Simons diffusion rate in the presence of anomalies. In this case, the usual definition of the rate—a late time limit of the imaginary part of the retarded correlator of the topological charge density—would give an exactly vanishing result, due to its relation with a nonconserved charge correlator. We confirm this observation by explicit holographic computations on generic isotropic black hole backgrounds. Nevertheless, a nontrivial Chern- Simons relaxation time can in principle be extracted from a quasinormal mode calculation. DOI: 10.1103/PhysRevD.98.106023 I. INTRODUCTION techniques, as first performed in [3]. Thus, we consider the planar, strong coupling regime of quantum field theories The Chern-Simons diffusion rate Γ is a fundamental CS which can model (but always present some differences observable in the study of chiral effects in the Quark-Gluon Plasma (QGP). Its value, setting the rate of (local) change of from) real world QCD. Holography has proven to give the Chern-Simons number, determines, through the axial fruitful indications about the physics of transport coeffi- anomaly, the (local) change rate of chirality imbalance in cients in QCD. In particular, when some universality can be the plasma. Chirality imbalance is then responsible, in the found among different holographic models, the result for presence of a magnetic field, for the chiral magnetic effect [1], the observable under scrutiny can be used as a benchmark which can be in principle measured in current experiments. for the experiments, since clearly it does not depend on Being inherently nonperturbative, the Chern-Simons many of the details of the theory. The shear viscosity is the diffusion rate cannot be computed in QCD from first most notable example of such a situation [4,5]. principles. In the literature there are interesting effective We are going to prove that at leading order in the planar field theory calculations, setting for example the behavior strong coupling limit, the Chern-Simons diffusion rate is with N (the number of colors of the theory) to be OðN0Þ universal in a relevant class of top-down holographic N 4 [2]. But it is fair to say that we lack a solid estimate models which includes, among the others, ¼ SYM (analogous to what can be obtained from the lattice for [3], even in the presence of anisotropies [6,7] and magnetic Γ fields [8], the Witten-Sakai-Sugimoto model [9–11] and the other observables) for the value of CS in the realistic strong coupling regime of the QGP. Maldacena-Nuñez model [12,13], for which the computa- Γ In this paper we address the problem of the calculation of tion of CS was lacking. In all the cases the field theory the Chern-Simons diffusion rate by means of holographic comes either from D3-branes or wrapped Dp-branes with p>3. More specifically, at leading order in the large N limit *[email protected] † where axial anomalies can be neglected, the Chern-Simons [email protected] ‡ [email protected] diffusion rate is given, with a universal coefficient, in terms of the temperature T, entropy density s and the gauge Published by the American Physical Society under the terms of α 2 4π coupling s ¼ gYM=ð Þ of the field theory the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to 2 Γ α ðTÞ the author(s) and the published article’s title, journal citation, CS ¼ s : ð1:1Þ and DOI. Funded by SCOAP3. sT 23π3 2470-0010=2018=98(10)=106023(14) 106023-1 Published by the American Physical Society BIGAZZI, COTRONE, and PORRI PHYS. REV. D 98, 106023 (2018) Z Δ 2 At present there are no counterexamples to this formula in hð NCSÞ i 4 Γ ¼ ¼ d xhQðxÞQð0Þi; ð1:2Þ top-down holographic models. So, it is not excluded that its CS Vt validity goes beyond the class of models considered in this where the variation in the Chern-Simons number is paper. Thus, the holographic behavior of this observable Z Z can be used with a certain degree of confidence as a 1 Δ 4 4 ˜ benchmark for the QGP. For example, a very rough NCS ¼ d xQðxÞ¼ d x 16π2 TrFF: ð1:3Þ estimate of the critical temperature values αsðTcÞ ∼ 3 Here F is the Yang-Mills field strength, Q the topological 1=2;sðTcÞ ∼ 10Tc (lattice results without magnetic field Γ 4 ∼ 0 01 charge density operator [14,15]) gives the value CS=Tc . This would have quite a small effect in the QGP. 1 Q FF;˜ : From a technical point of view, the derivation of the ¼ 16π2 Tr ð1 4Þ result (1.1) relies on the fact that the quadratic part of a b the five dimensional bulk action for the field dual to the and we normalize the SUðNÞ generators so that Tr½t t ¼ topological charge density operator has a universal form in δab=2 for the fundamental representation. this class of models. It is the action of a massless scalar. Its The two-point function in formula (1.2) is the sym- kinetic term depends on a function which, in the models metrized Wightman one—everything is defined in real Γ at hand, is the dual of the coupling constant [Eq. (2.13)]. time. In a state at thermal equilibrium at temperature T, CS The universality of this result can be useful beyond the can be given by a Kubo formula determination of the Chern-Simons diffusion rate. 2 The most interesting correction to the holographic result Γ − T ω ⃗ 0 CS ¼ lim ImGRð ; k ¼ Þ: ð1:5Þ for the Chern-Simons diffusion rate in the planar expansion ω→0 ω comes from possible anomalies proportional to the topo- Thus its computation boils down to the determination of the logical charge density operator. In the bulk, these are small frequency, zero momentum retarded correlator described by a Stueckelberg action for the scalar mentioned ω ⃗ 0 above and a vector field dual to the anomalous current [16]. GRð ; k ¼ Þ of the operator Q. The retarded correlator We are going to show that, in this case, the usual holo- is precisely what can be directly computed in holography Γ from the standard prescriptions [3], once the bulk field dual graphic prescription for the calculation of CS gives an identically vanishing result.1 This fact is very general for to the operator Q is identified. holographic field theories and does not rely on the details of The paper is organized as follows. In Sec. II we prove the the dual gravity backgrounds. universal formulas (1.1), (2.13) for the Chern-Simons dif- As we are going to discuss, this feature is not related to the fusion rate and the gravity action of the field dual to the holographic limit, but it simply relies on the fact that the usual corresponding operator in a relevant class of top-down Γ holographic models in absence of anomalies. Then, in definition of CS, as the late time behavior of the (imaginary part of the) retarded correlator of the topological charge Sec. III we discuss the effects of anomalies confirming, density, gives zero if the latter is proportional, as it happens through a holographic computation on generic isotropic black due to the anomaly relation, to the retarded correlator of a hole backgrounds, that the standard prescription for the Γ nonconserved (axial) charge. The relaxation time of the latter calculation of CS gives an identically vanishing result. In can, instead, be consistently defined and expressed in terms turn, we discuss how to extract the Chern-Simons relaxation of the Chern-Simons diffusion rate computed with a cutoff in time from the quasinormal modes of the dual black hole time, i.e., effectively turning off the anomaly. The rate backgrounds. The Appendices include the analysis of the Γ computed with this prescription is not automatically vanish- direct calculation of CS in specific examples, including, for ing. Some considerations along this line can be already found the first time, the Maldacena-Nuñez model. Moreover they in [2,18] where the same issue has been pointed out for QCD. contain a sketch of the derivation of a linear response formula – Chiral effects in the presence of anomalies can be inves- for the axial relaxation time as well as specific anti de Sitter tigated, from a holographic point of view, through the (AdS) examples of the generic holographic results discussed analysis of quasinormal modes of the dual black hole in Sec.

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