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Chiral Magnetic Effect in Hydrodynamic Approximation

  • Chapter
Strongly Interacting Matter in Magnetic Fields

Part of the book series: Lecture Notes in Physics ((LNP,volume 871))

Abstract

We review derivations of the chiral magnetic effect (ChME) in hydrodynamic approximation. The reader is assumed to be familiar with the basics of the effect. The main challenge now is to account for the strong interactions between the constituents of the fluid. The main result is that the ChME is not renormalized: in the hydrodynamic approximation it remains the same as for non-interacting chiral fermions moving in an external magnetic field. The key ingredients in the proof are general laws of thermodynamics and the Adler-Bardeen theorem for the chiral anomaly in external electromagnetic fields. The chiral magnetic effect in hydrodynamics represents a macroscopic manifestation of a quantum phenomenon (chiral anomaly). Moreover, one can argue that the current induced by the magnetic field is dissipation free and talk about a kind of “chiral superconductivity”. More precise description is a quantum ballistic transport along magnetic field taking place in equilibrium and in absence of a driving force. The basic limitation is the exact chiral limit while temperature—excitingly enough—does not seemingly matter. What is still lacking, is a detailed quantum microscopic picture for the ChME in hydrodynamics. Probably, the chiral currents propagate through lower-dimensional defects, like vortices in superfluid. In case of superfluid, the prediction for the chiral magnetic effect remains unmodified although the emerging dynamical picture differs from the standard one.

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Notes

  1. 1.

    The review is prepared for a volume of the Springer Lecture Notes in Physics “Strongly interacting matter in magnetic fields” edited by D. Kharzeev, K. Landsteiner, A. Schmitt, H.-U. Yee.

  2. 2.

    In the realistic QCD case the singlet axial current is anomalous and is not conserved. Therefore, introduction of the chemical potential μ 5 is rather a subtle issue. In the bulk of the text we ignore this problem concentrating mostly on academic issues. One could have in mind, for example, that the chemical potential μ 5≠0 is associated in fact with the axial current with isospin ΔI=1 which is conserved in the limit of vanishing quark masses. Another possible line of reasoning is to invoke large-N c limit of Yang-Mills theories. The contribution of the gluon anomaly is then suppressed by large N c and the chemical potential μ 5 can be introduced consistently for the singlet current as well.

  3. 3.

    To compare (11.9) and (11.7) one should keep in mind that in the notations of Ref. [24] the current j in (11.9) is the current of right-handed fermions alone and, thus, constitutes one half of the chiral current entering (11.7).

  4. 4.

    Note that in the underlying fundamental field theory there are no anomalies associated with a non-vanishing chemical potential μ. This observation is in no contradiction with the fact that such anomalies do arise in the language of the effective theory.

  5. 5.

    Alternatively, this counting can be considered as a proof of the possibility to introduce the Landau gauge (11.5).

  6. 6.

    We could have defined anomaly in such a way that it does not contribute to ∂ μ j μ. However, in the presence of both chemical potentials μ and μ 5 there is no physical motivation for such a regularization.

  7. 7.

    The author is thankful to L. Stodolsky for a detailed discussion of the subject.

  8. 8.

    This subsection is of rather technical nature and can be considered as an appendix. Moreover, the presentation is close to that of Ref. [23], with a substitution A i →∂ i ϕ.

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Acknowledgements

The author is grateful to T. Kalaydzhyan, D.E. Kharzeev, V.P Kirilin, A.V. Sadofyev, V.I. Shevchenko, L. Stodolsky and H. Verschelde for illuminating discussions. The work of the author was partially supported by grants PICS-12-02-91052, FEBR-11-02-01227-a and by the Federal Special-Purpose Program “Cadres” of the Russian Ministry of Science and Education.

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Authors and Affiliations

  1. Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya, 25, Moscow, Russia

    Valentin I. Zakharov

  2. Max-Planck Institut fuer Physik, Werner-Heisenberg Institut, Foehringer Ring, 6, Munich, Germany

    Valentin I. Zakharov

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  1. Valentin I. Zakharov
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Correspondence to Valentin I. Zakharov .

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Editors and Affiliations

  1. Department of Physics, Stony Brook University, Nicolls Rd 100, Stony Brook, 11794, New York, USA

    Dmitri Kharzeev

  2. Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, Nicolas Cabrera 13-15, Madrid, 28049, Spain

    Karl Landsteiner

  3. Institut für Theoretische Physik, Technische Universität Wien, Wiedner Haupstr. 8, Wien, 1040, Austria

    Andreas Schmitt

  4. Physics Department (MC 273), University of Illinois at Chicago, West Taylor Street 845Rm 2236, Chicago, 60607, Illinois, USA

    Ho-Ung Yee

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Zakharov, V.I. (2013). Chiral Magnetic Effect in Hydrodynamic Approximation. In: Kharzeev, D., Landsteiner, K., Schmitt, A., Yee, HU. (eds) Strongly Interacting Matter in Magnetic Fields. Lecture Notes in Physics, vol 871. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37305-3_11

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