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The Chiral Magnetic Effect and Axial Anomalies

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Strongly Interacting Matter in Magnetic Fields

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

Abstract

We give an elementary derivation of the chiral magnetic effect based on a strong magnetic field lowest-Landau-level projection in conjunction with the well-known axial anomalies in two- and four-dimensional space-time. The argument is general, based on a Schur decomposition of the Dirac operator. In the dimensionally reduced theory, the chiral magnetic effect is directly related to the relativistic form of the Peierls instability, leading to a spiral form of the condensate, the chiral magnetic spiral. We then discuss the competition between spin projection, due to a strong magnetic field, and chirality projection, due to an instanton, for light fermions in QCD and QED. The resulting asymmetric distortion of the zero modes and near-zero modes is another aspect of the chiral magnetic effect.

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Notes

  1. 1.

    Spontaneous symmetry breaking in two dimensions is a delicate subject whose details are beyond the scope of our topic. It suffices to mention that in the limit of large number of flavors it can be realized, for example in the Gross-Neveu model [28].

References

  1. D.E. Kharzeev, Parity violation in hot QCD: why it can happen, and how to look for it. Phys. Lett. B 633, 260 (2006). arXiv:hep-ph/0406125

    Article  ADS  Google Scholar 

  2. D. Kharzeev, A. Zhitnitsky, Charge separation induced by P-odd bubbles in QCD matter. Nucl. Phys. A 797, 67 (2007). arXiv:0706.1026 [hep-ph]

    Article  ADS  Google Scholar 

  3. D.E. Kharzeev, L.D. McLerran, H.J. Warringa, The effects of topological charge change in heavy ion collisions: ‘Event by event P and CP violation’. Nucl. Phys. A 803, 227 (2008). arXiv:0711.0950 [hep-ph]

    Article  ADS  Google Scholar 

  4. K. Fukushima, D.E. Kharzeev, H.J. Warringa, The chiral magnetic effect. Phys. Rev. D 78, 074033 (2008). arXiv:0808.3382 [hep-ph]

    Article  ADS  Google Scholar 

  5. D.E. Kharzeev, Topologically induced local P and CP violation in QCD × QED. Ann. Phys. 325, 205 (2010). arXiv:0911.3715 [hep-ph]

    Article  ADS  MATH  Google Scholar 

  6. H.J. Warringa, Dynamics of the chiral magnetic effect in a weak magnetic field. arXiv:1205.5679 [hep-th]

  7. R. Jackiw, Topological investigations of quantized gauge theories, in Current Algebra and Anomalies, ed. by S.B. Treiman, R. Jackiw, B. Zumino, E. Witten (Princeton University Press, Princeton, 1985)

    Google Scholar 

  8. M.A. Shifman, in ITEP Lectures on Particle Physics and Field Theory, Vols. 1, 2. World Sci. Lect. Notes Phys., vol. 62 (1999)

    Google Scholar 

  9. V.P. Gusynin, V.A. Miransky, I.A. Shovkovy, Dimensional reduction and catalysis of dynamical symmetry breaking by a magnetic field. Nucl. Phys. B 462, 249 (1996). arXiv:hep-ph/9509320

    Article  ADS  Google Scholar 

  10. G. Strang, Linear Algebra and Its Applications (Harcourt, San Diego, 1988)

    Google Scholar 

  11. Y. Aharonov, A. Casher, Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field. Phys. Rev. A 19, 2461 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  12. S.P. Novikov, B.A. Dubrovin, Ground states of a two-dimensional electron in a periodic magnetic field. Zh. Èksp. Teor. Fiz. 79, 1006 (1980). [Sov. Phys. JETP 52, 511 (1980)]

    MathSciNet  Google Scholar 

  13. S.P. Novikov, B.A. Dubrovin, Ground states in a periodic field. Magnetic Bloch functions and vector bundles. Dokl. Akad. Nauk SSSR 253, 1293 (1980)

    MathSciNet  Google Scholar 

  14. L.S. Brown, R.D. Carlitz, C. Lee, Massless excitations in instanton fields. Phys. Rev. D 16, 417 (1977)

    Article  ADS  Google Scholar 

  15. R.D. Carlitz, C. Lee, Physical processes in pseudoparticle fields: the role of fermionic zero modes. Phys. Rev. D 17, 3238 (1978)

    Article  ADS  Google Scholar 

  16. J. Hur, C. Lee, H. Min, Some chirality-related properties of the 4-D massive Dirac propagator and determinant in an arbitrary gauge field. Phys. Rev. D 82, 085002 (2010). arXiv:1007.4616 [hep-th]

    Article  ADS  Google Scholar 

  17. W.A. Bardeen, B. Zumino, Consistent and covariant anomalies in gauge and gravitational theories. Nucl. Phys. B 244, 421 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  18. G.V. Dunne, C.A. Trugenberger, Odd dimensional gauge theories and current algebra. Ann. Phys. 204, 281 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. E.V. Gorbar, V.A. Miransky, I.A. Shovkovy, Chiral asymmetry and axial anomaly in magnetized relativistic matter. Phys. Lett. B 695, 354 (2011). arXiv:1009.1656 [hep-ph]

    Article  ADS  Google Scholar 

  20. N. Sadooghi, A. Jafari Salim, Axial anomaly of QED in a strong magnetic field and noncommutative anomaly. Phys. Rev. D 74, 085032 (2006). hep-th/0608112

    Article  ADS  Google Scholar 

  21. J.H. Gao, Z.T. Liang, S. Pu, Q. Wang, X.N. Wang, Chiral anomaly and local polarization effect from quantum kinetic approach. Phys. Rev. Lett. 109, 232301 (2012). arXiv:1203.0725 [hep-ph]

    Article  ADS  Google Scholar 

  22. W. Heisenberg, H. Euler, Consequences of Dirac’s theory of positrons. Z. Phys. 98, 714 (1936)

    Article  ADS  Google Scholar 

  23. J. Schwinger, On gauge invariance and vacuum polarization. Phys. Rev. 82, 664 (1951)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Y. Kluger, E. Mottola, J.M. Eisenberg, The quantum Vlasov equation and its Markov limit. Phys. Rev. D 58, 125015 (1998). hep-ph/9803372

    Article  ADS  Google Scholar 

  25. J.S. Schwinger, Gauge invariance and mass. Phys. Rev. 125, 397 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. J.S. Schwinger, Gauge invariance and mass. 2. Phys. Rev. 128, 2425 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. G. Basar, G.V. Dunne, D.E. Kharzeev, Chiral magnetic spiral. Phys. Rev. Lett. 104, 232301 (2010). arXiv:1003.3464 [hep-ph]

    Article  ADS  Google Scholar 

  28. D.J. Gross, A. Neveu, Dynamical symmetry breaking in asymptotically free field theories. Phys. Rev. D 10, 3235 (1974)

    Article  ADS  Google Scholar 

  29. T. Kojo, Y. Hidaka, L. McLerran, R.D. Pisarski, Quarkyonic chiral spirals. Nucl. Phys. A 843, 37–58 (2010). arXiv:0912.3800 [hep-ph]

    Article  ADS  Google Scholar 

  30. R. Peierls, The Quantum Theory of Solids (Oxford University Press, Oxford, 1955)

    Google Scholar 

  31. R. Peierls, More Surprises in Theoretical Physics (Princeton University Press, Princeton, 1991)

    Google Scholar 

  32. G. Basar, G.V. Dunne, M. Thies, Inhomogeneous condensates in the thermodynamics of the chiral NJL2 model. Phys. Rev. D 79, 105012 (2009). arXiv:0903.1868 [hep-th]

    Article  ADS  Google Scholar 

  33. G. Basar, G.V. Dunne, A twisted kink crystal in the chiral Gross-Neveu model. Phys. Rev. D 78, 065022 (2008). arXiv:0806.2659 [hep-th]

    Article  ADS  Google Scholar 

  34. V. Schon, M. Thies, Emergence of Skyrme crystal in Gross-Neveu and ’t Hooft models at finite density. Phys. Rev. D 62, 096002 (2000). arXiv:hep-th/0003195

    Article  ADS  Google Scholar 

  35. V. Schon, M. Thies, 2D model field theories at finite temperature and density, in At the Frontier of Particle Physics: Handbook of QCD, vol. 3, ed. by M.A. Shifman (World Scientific, Singapore, 2000). arXiv:hep-th/0008175

    Google Scholar 

  36. M. Thies, From relativistic quantum fields to condensed matter and back again: updating the Gross-Neveu phase diagram. J. Phys. A 39, 12707 (2006). hep-th/0601049

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. A. Bzdak, V. Koch, J. Liao, Remarks on possible local parity violation in heavy ion collisions. Phys. Rev. C 81, 031901 (2010). arXiv:0912.5050 [nucl-th]

    Article  ADS  Google Scholar 

  38. B.I. Abelev et al. (STAR Collaboration), Azimuthal charged-particle correlations and possible local strong parity violation. Phys. Rev. Lett. 103, 251601 (2009). arXiv:0909.1739 [nucl-ex]

    Article  ADS  Google Scholar 

  39. B.I. Abelev et al. (STAR Collaboration), Observation of charge-dependent azimuthal correlations and possible local strong parity violation in heavy ion collisions. Phys. Rev. C 81, 054908 (2010). arXiv:0909.1717 [nucl-ex]

    Article  ADS  Google Scholar 

  40. D. Kharzeev, R.D. Pisarski, M.H.G. Tytgat, Possibility of spontaneous parity violation in hot QCD. Phys. Rev. Lett. 81, 512 (1998). arXiv:hep-ph/9804221

    Article  ADS  Google Scholar 

  41. K.-Y. Kim, B. Sahoo, H.-U. Yee, Holographic chiral magnetic spiral. J. High Energy Phys. 1010, 005 (2010). arXiv:1007.1985 [hep-th]

    Article  ADS  Google Scholar 

  42. T. Sakai, S. Sugimoto, Low energy hadron physics in holographic QCD. Prog. Theor. Phys. 113, 843 (2005). hep-th/0412141

    Article  ADS  MATH  Google Scholar 

  43. T. Sakai, S. Sugimoto, More on a holographic dual of QCD. Prog. Theor. Phys. 114, 1083 (2005). hep-th/0507073

    Article  ADS  MATH  Google Scholar 

  44. G. Basar, G.V. Dunne, D.E. Kharzeev, Electric dipole moment induced by a QCD instanton in an external magnetic field. Phys. Rev. D 85, 045026 (2012). arXiv:1112.0532 [hep-th]

    Article  ADS  Google Scholar 

  45. V. Skokov, A.Y. Illarionov, V. Toneev, Estimate of the magnetic field strength in heavy-ion collisions. Int. J. Mod. Phys. A 24, 5925 (2009). arXiv:0907.1396 [nucl-th]

    Article  ADS  Google Scholar 

  46. A. Bzdak, V. Skokov, Event-by-event fluctuations of magnetic and electric fields in heavy ion collisions. Phys. Lett. B 710, 171 (2012). arXiv:1111.1949 [hep-ph]

    Article  ADS  Google Scholar 

  47. P.V. Buividovich, M.N. Chernodub, E.V. Luschevskaya, M.I. Polikarpov, Numerical study of chiral symmetry breaking in non-Abelian gauge theory with background magnetic field. Phys. Lett. B 682, 484–489 (2010). arXiv:0812.1740 [hep-lat]

    Article  ADS  Google Scholar 

  48. P.V. Buividovich, M.N. Chernodub, E.V. Luschevskaya, M.I. Polikarpov, Chiral magnetization of non-Abelian vacuum: a lattice study. Nucl. Phys. B 826, 313–327 (2010). arXiv:0906.0488 [hep-lat]

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. P.V. Buividovich, M.N. Chernodub, E.V. Luschevskaya, M.I. Polikarpov, Numerical evidence of chiral magnetic effect in lattice gauge theory. Phys. Rev. D 80, 054503 (2009). arXiv:0907.0494 [hep-lat]

    Article  ADS  Google Scholar 

  50. P.V. Buividovich, M.N. Chernodub, E.V. Luschevskaya, M.I. Polikarpov, Quark electric dipole moment induced by magnetic field. Phys. Rev. D 81, 036007 (2010). arXiv:0909.2350 [hep-ph]

    Article  ADS  Google Scholar 

  51. M. Abramczyk, T. Blum, G. Petropoulos, R. Zhou, Chiral magnetic effect in 2+1 flavor QCD+QED. PoS LAT2009, 181 (2009). arXiv:0911.1348 [hep-lat]

    Google Scholar 

  52. T. Blum, Talk at Workshop on P- and CP-odd Effects in Hot and Dense Matter, Brookhaven National Laboratory, April 2010

    Google Scholar 

  53. V.V. Braguta, P.V. Buividovich, T. Kalaydzhyan, S.V. Kuznetsov, M.I. Polikarpov, The chiral magnetic effect and chiral symmetry breaking in SU(3) quenched lattice gauge theory. PoS LAT2010, 190 (2010). arXiv:1011.3795 [hep-lat]

    Google Scholar 

  54. B.C. Tiburzi, Lattice QCD with classical and quantum electrodynamics. PoS LAT2011, 020 (2011). arXiv:1110.6842 [hep-lat]

    Google Scholar 

  55. L. Giusti, A. Gonzalez-Arroyo, C. Hoelbling, H. Neuberger, C. Rebbi, Fermions on tori in uniform Abelian fields. Phys. Rev. D 65, 074506 (2002). arXiv:hep-lat/0112017

    Article  ADS  Google Scholar 

  56. Y. Tenjinbayashi, H. Igarashi, T. Fujiwara, Dirac operator zero-modes on a torus. Ann. Phys. 322, 460 (2007). arXiv:hep-th/0506259

    Article  MathSciNet  ADS  MATH  Google Scholar 

  57. M.H. Al-Hashimi, U.J. Wiese, Discrete accidental symmetry for a particle in a constant magnetic field on a torus. Ann. Phys. 324, 343 (2009). arXiv:0807.0630 [quant-ph]

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. J. Zak, Magnetic translation group. Phys. Rev. 134, A1602 (1964)

    Article  MathSciNet  ADS  Google Scholar 

  59. G. ’t Hooft, Computation of the quantum effects due to a four-dimensional pseudoparticle. Phys. Rev. D 14, 3432 (1976)

    Article  ADS  Google Scholar 

  60. A.S. Schwarz, On regular solutions of Euclidean Yang-Mills equations. Phys. Lett. B 67, 172 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  61. J.E. Kiskis, Fermions in a pseudoparticle field. Phys. Rev. D 15, 2329 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  62. R. Jackiw, C. Rebbi, Spinor analysis of Yang-Mills theory. Phys. Rev. D 16, 1052 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  63. V.A. Rubakov, Classical Theory of Gauge Fields (Princeton University Press, Princeton, 2002)

    MATH  Google Scholar 

  64. R. Jackiw, C. Rebbi, Conformal properties of a Yang-Mills pseudoparticle. Phys. Rev. D 14, 517 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  65. S. Chadha, A. D’Adda, P. Di Vecchia, F. Nicodemi, Fermions in the background pseudoparticle field in an O(5) formulation. Phys. Lett. B 67, 103 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  66. M. Atiyah, V. Patodi, I. Singer, Spectral asymmetry and Riemannian geometry. Math. Proc. Camb. Philos. Soc. 77, 43 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  67. A.A. Belavin, A.M. Polyakov, A.S. Schwartz, Yu.S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations. Phys. Lett. B 59, 85 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  68. G. ’t Hooft, Some twisted selfdual solutions for the Yang-Mills equations on a hypertorus. Commun. Math. Phys. 81, 267 (1981)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  69. P. van Baal, Some results for SU(N) gauge fields on the hypertorus. Commun. Math. Phys. 85, 529 (1982)

    Article  ADS  MATH  Google Scholar 

  70. P. van Baal, SU(N) Yang-Mills solutions with constant field strength on T 4. Commun. Math. Phys. 94, 397 (1984)

    Article  ADS  Google Scholar 

  71. P. van Baal, Instanton moduli for \(T^{3}\times \mathbb{R}\). Nucl. Phys. B, Proc. Suppl. 49, 238 (1996). arXiv:hep-th/9512223

    Article  ADS  MATH  Google Scholar 

  72. M.A. Shifman, Wilson loop in vacuum fields. Nucl. Phys. B 173, 13 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  73. M.S. Dubovikov, A.V. Smilga, Analytical properties of the quark polarization operator in an external selfdual field. Nucl. Phys. B 185, 109–132 (1981)

    Article  ADS  Google Scholar 

  74. B.L. Ioffe, A.V. Smilga, Nucleon magnetic moments and magnetic properties of vacuum in QCD. Nucl. Phys. B 232, 109 (1984)

    Article  ADS  Google Scholar 

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Acknowledgements

We thank T. Blum, D. Kharzeev and H-U. Yee for helpful discussions. This work was supported by the US Department of Energy under grants DE-FG02-92ER40716 (GD) and DE-AC02-98CH10886, DE-FG-88ER41723 (GB).

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Authors and Affiliations

  1. Department of Physics, Stony Brook University, Stony Brook, USA

    Gökçe Başar

  2. Physics Department, University of Connecticut, Storrs, USA

    Gerald V. Dunne

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  1. Gökçe Başar
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  2. Gerald V. Dunne
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Correspondence to Gökçe Başar .

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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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Başar, G., Dunne, G.V. (2013). The Chiral Magnetic Effect and Axial Anomalies. 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_10

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