Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 047, 14 pages      arXiv:0812.0176      https://doi.org/10.3842/SIGMA.2009.047
Contribution to the Proceedings of the VIIth Workshop ''Quantum Physics with Non-Hermitian Operators''

PT Symmetry and QCD: Finite Temperature and Density

Michael C. Ogilvie and Peter N. Meisinger
Department of Physics, Washington University, St. Louis, MO 63130, USA

Received November 15, 2008, in final form April 10, 2009; Published online April 17, 2009

Abstract
The relevance of PT symmetry to quantum chromodynamics (QCD), the gauge theory of the strong interactions, is explored in the context of finite temperature and density. Two significant problems in QCD are studied: the sign problem of finite-density QCD, and the problem of confinement. It is proven that the effective action for heavy quarks at finite density is PT-symmetric. For the case of 1+1 dimensions, the PT-symmetric Hamiltonian, although not Hermitian, has real eigenvalues for a range of values of the chemical potential μ, solving the sign problem for this model. The effective action for heavy quarks is part of a potentially large class of generalized sine-Gordon models which are non-Hermitian but are PT-symmetric. Generalized sine-Gordon models also occur naturally in gauge theories in which magnetic monopoles lead to confinement. We explore gauge theories where monopoles cause confinement at arbitrarily high temperatures. Several different classes of monopole gases exist, with each class leading to different string tension scaling laws. For one class of monopole gas models, the PT-symmetric affine Toda field theory emerges naturally as the effective theory. This in turn leads to sine-law scaling for string tensions, a behavior consistent with lattice simulations.

Key words: PT symmetry; QCD.

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References

  1. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
  2. Bender C.M., Introduction to PT-symmetric quantum theory, Contemp. Phys. 46 (2005), 277-292, quant-ph/0501052.
  3. Bender C.M., Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), 947-1018, hep-th/0703096.
  4. Stephanov M.A., QCD phase diagram: an overview, PoSLAT 2006 (2006), 024, 15 pages, hep-lat/0701002.
  5. Lombardo M.P., QCD at non-zero density: lattice results, J. Phys. G: Nucl. Part. Phys. 35 (2008), 104019, 8 pages, arXiv:0808.3101.
  6. Alford M.G., Schmitt A., Rajagopal K., Schafer T., Color superconductivity in dense quark matter, Rev. Modern Phys. 80 (2008), 1455-1515, arXiv:0709.4635.
  7. Greensite J., The confinement problem in lattice gauge theory, Progr. Part. Nucl. Phys. 51 (2003), 1-83, hep-lat/0301023.
  8. Gross D.J., Pisarski R.D., Yaffe L.G., QCD and instantons at finite temperature, Rev. Modern Phys. 53 (1981), 43-80.
  9. Weiss N., The effective potential for the order parameter of gauge theories at finite temperature, Phys. Rev. D 24 (1981), 475-480.
  10. Bender C.M., Jones H.F., Rivers R.J., Dual PT-symmetric quantum field theories, Phys. Lett. B 625 (2005), 333-340, hep-th/0508105.
  11. Hands S., Kogut J.B., Lombardo M.P., Morrison S.E., Symmetries and spectrum of SU(2) lattice gauge theory at finite chemical potential, Nuclear Phys. B 558 (1999), 327-346, hep-lat/9902034.
  12. Kogut J.B., Sinclair D.K., Hands S.J., Morrison S.E., Two-colour QCD at non-zero quark-number density, Phys. Rev. D 64 (2001), 094505, 9 pages, hep-lat/0105026.
  13. Edwards S.F., Lenard A., Exact statistical mechanics of a one-dimensional system with Coulomb forces. II. The method of functional integration, J. Math. Phys. 3 (1962), 778-792.
  14. Davies N.M., Hollowood T.J., Khoze V.V., Mattis M.P., Gluino condensate and magnetic monopoles in supersymmetric gluodynamics, Nuclear Phys. B 559 (1999), 123-142, hep-th/9905015.
  15. Davies N.M., Hollowood T.J., Khoze V.V., Monopoles, affine algebras and the gluino condensate, J. Math. Phys. 44 (2003), 3640-3656, hep-th/0006011.
  16. Myers J.C., Ogilvie M.C., New phases of SU(3) and SU(4) at finite temperature, Phys. Rev. D 77 (2008), 125030, 10 pages, arXiv:0707.1869.
  17. Unsal M., Abelian duality, confinement, and chiral symmetry breaking in QCD(adj), Phys. Rev. Lett. 100 (2008), 032005, 4 pages, arXiv:0708.1772.
  18. Unsal M., Yaffe L.G., Center-stabilized Yang-Mills theory: confinement and large N volume independence, Phys. Rev. D 78 (2008), 065035, 17 pages, arXiv:0803.0344.
  19. Meisinger P.N., Ogilvie M.C., Complete high temperature expansions for one-loop finite temperature effects, Phys. Rev. D 65 (2002), 056013, 7 pages, hep-ph/0108026.
  20. Myers J.C., Ogilvie M.C., Exotic phases of finite temperature SU(N) gauge theories with massive fermions: F, Adj, A/S, arXiv:0809.3964.
  21. Meisinger P.N., Miller T.R., Ogilvie M.C., Phenomenological equations of state for the quark-gluon plasma, Phys. Rev. D 65 (2002), 034009, 10 pages, hep-ph/0108009.
  22. Hollowood T.J., Solitons in affine Toda field theories, Nuclear Phys. B 384 (1992), 523-540.
  23. Meisinger P.N., Ogilvie M.C., Polyakov loops, Z(N) symmetry, and sine-law scaling, Nuclear Phys. Proc. Suppl. 140 (2005), 650-652, hep-lat/0409136.
  24. Polyakov A.M., Quark confinement and topology of gauge groups, Nuclear Phys. B 120 (1977), 429-458.
  25. Weinberg E.J., Fundamental monopoles and multi-monopole solutions for arbitrary simple gauge groups, Nuclear Phys. B 167 (1980), 500-524.
  26. Lee K.M., Instantons and magnetic monopoles on R3 × S1 with arbitrary simple gauge groups, Phys. Lett. B 426 (1998), 323-328, hep-th/9802012.
  27. Kraan T.C., van Baal P., Exact T-duality between calorons and Taub-NUT spaces, Phys. Lett. B 428 (1998), 268-276, hep-th/9802049.
  28. Lee K.M., Lu C. H., SU(2) calorons and magnetic monopoles, Phys. Rev. D 58 (1998), 025011, 7 pages, hep-th/9802108.
  29. Kraan T.C., van Baal P., Periodic instantons with non-trivial holonomy, Nuclear Phys. B 533 (1998), 627-659, hep-th/9805168.
  30. Kraan T.C., van Baal P., Monopole constituents inside SU(n) calorons, Phys. Lett. B 435 (1998), 389-395, hep-th/9806034.
  31. Zarembo K., Monopole determinant in Yang-Mills theory at finite temperature, Nuclear Phys. B 463 (1996), 73-98, arXiv:hep-th/9510031.
  32. Giovannangeli P., Korthals Altes C.P., 't Hooft and Wilson loop ratios in the QCD plasma, Nuclear Phys. B 608 (2001), 203-234, hep-ph/0102022.
  33. Lucini B., Teper M., Wenger U., Glueballs and k-strings in SU(N) gauge theories: calculations with improved operators, J. High Energy Phys. 2004 (2004), no. 04, 012, 44 pages, hep-lat/0404008.
  34. Diakonov D., Petrov V., Confining ensemble of dyons, Phys. Rev. D 76 (2007), 056001, 22 pages, arXiv:0704.3181.

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