- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Nambu–Jona-Lasinio model description of weakly interacting Bose condensate and BEC-BCS crossover in dense QCD-like theories

Lianyi He*

  • Frankfurt Institute for Advanced Studies and Institute for Theoretical Physics, J. W. Goethe University, 60438 Frankfurt am Main, Germany

  • *lianyi@th.physik.uni-frankfurt.de
PDF

Phys. Rev. D 82, 096003 – Published 8 November, 2010

DOI: https://doi.org/10.1103/PhysRevD.82.096003

Abstract

QCD-like theories possess a positively definite fermion determinant at finite baryon chemical potential μB and the lattice simulation can be successfully performed. While the chiral perturbation theories are sufficient to describe the Bose condensate at low density, to describe the crossover from Bose-Einstein condensation (BEC) to BCS superfluidity at moderate density we should use some fermionic effective model of QCD, such as the Nambu–Jona-Lasinio model. In this paper, using two-color two-flavor QCD as an example, we examine how the Nambu–Jona-Lasinio model describes the weakly interacting Bose condensate at low density and the BEC-BCS crossover at moderate density. Near the quantum phase transition point μB=mπ (mπ is the mass of pion/diquark multiplet), the Ginzburg-Landau free energy at the mean-field level can be reduced to the Gross-Pitaevskii free energy describing a weakly repulsive Bose condensate with a diquark-diquark scattering length identical to that predicted by the chiral perturbation theories. The Goldstone mode recovers the Bogoliubov excitation in weakly interacting Bose condensates. The results of in-medium chiral and diquark condensates predicted by chiral perturbation theories are analytically recovered. The BEC-BCS crossover and meson Mott transition at moderate baryon chemical potential as well as the beyond-mean-field corrections are studied. Part of our results can also be applied to real QCD at finite baryon or isospin chemical potential.

View figure in article

Article Text

References (97)

  1. N. Itoh, Prog. Theor. Phys. 44, 291 (1970); E. Witten, Phys. Rev. D 30, 272 (1984).
  2. F. Karsch, Lect. Notes Phys. 583, 209 (2002).
  3. S. Muroya, A. Nakamura, C. Nonaka, and T. Takaishi, Prog. Theor. Phys. 110, 615 (2003).
  4. D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 86, 592 (2001).
  5. D. T. Son and M. A. Stephanov, Phys. At. Nucl. 64, 834 (2001).
  6. J. B. Kogut, M. A. Stephanov, and D. Toublan, Phys. Lett. B 464, 183 (1999).
  7. J. B. Kogut, M. A. Stephanov, D. Toublan, J. J. M. Verbaarschot, and A. Zhitnitsky, Nucl. Phys. B582, 477 (2000).
  8. K. Splittorff, D. T. Son, and M. A. Stephanov, Phys. Rev. D 64, 016003 (2001).
  9. J. T. Lenaghan, F. Sannino, and K. Splittorff, Phys. Rev. D 65, 054002 (2002).
  10. K. Splittorff, D. Toublan, and J. J. M. Verbaarschot, Nucl. Phys. B620, 290 (2002).
  11. K. Splittorff, D. Toublan, and J. J. M. Verbaarschot, Nucl. Phys. B639, 524 (2002).
  12. T. Zhang, T. Brauner, and D. H. Rischke, J. High Energy Phys. 06 (2010) 64.
  13. For review, see J. O. Andersen, Rev. Mod. Phys. 76, 599 (2004).
  14. N. N. Bogoliubov, J. Phys. USSR 11, 23 (1947).
  15. T. D. Lee, K. Huang, and C. N. Yang, Phys. Rev. 106, 1135 (1957).
  16. R. F. Sawyer, Phys. Rev. Lett. 29, 382 (1972).
  17. D. J. Scalapino, Phys. Rev. Lett. 29, 386 (1972).
  18. G. Baym, Phys. Rev. Lett. 30, 1340 (1973).
  19. D. K. Campbell, R. F. Dashen, and J. T. Manassah, Phys. Rev. D 12, 979 (1975); 12, 1010 (1975).
  20. D. T. Son, Phys. Rev. D 59, 094019 (1999).
  21. T. Schäfer and F. Wilczek, Phys. Rev. D 60, 114033 (1999).
  22. D. Pisarski and D. H. Rischke, Phys. Rev. D 61, 074017 (2000); 61, 051501 (2000).
  23. T. Kanazawa, T. Wettig, and N. Yamamoto, J. High Energy Phys. 08 (2009) 003.
  24. D. M. Eagles, Phys. Rev. 186, 456 (1969).
  25. A. J. Leggett, in Modern Trends in the Theory of Condensed Matter, edited by A. Pekalski and R. Przystawa (Springer-Verlag, Berlin, 1980).
  26. M. Greiner et al., Nature (London) 426, 537 (2003); S. Jochim et al., Science 302, 2101 (2003); M. W. Zwierlein et al., Nature (London) 435, 1047 (2005).
  27. S. Hands, I. Montvay, S. Morrison, M. Oevers, L. Scorzato, and J. Skullerud, Eur. Phys. J. C 17, 285 (2000).
  28. S. Hands, I. Montvay, L. Scorzato, and J. Skullerud, Eur. Phys. J. C 22, 451 (2001).
  29. J. B. Kogut, D. K. Sinclair, S. J. Hands, and S. E. Morrison, Phys. Rev. D 64, 094505 (2001).
  30. J. B. Kogut, D. Toublan, and D. K. Sinclair, Phys. Lett. B 514, 77 (2001).
  31. S. Hands, S. Kim, and J. Skullerud, Eur. Phys. J. C 48, 193 (2006).
  32. S. Hands, S. Kim, and J. Skullerud, Phys. Rev. D 81, 091502(R) (2010).
  33. J. B. Kogut and D. K. Sinclair, Phys. Rev. D 66, 034505 (2002).
  34. J. B. Kogut and D. K. Sinclair, Phys. Rev. D 66, 014508 (2002).
  35. J. B. Kogut and D. K. Sinclair, Phys. Rev. D 70, 094501 (2004).
  36. P. Forcrand, M. A. Stephanov, and U. Wenger, Proc. Sci., LAT2007 (2007) 237.
  37. M. Loewe and C. Villavicencio, Phys. Rev. D 67, 074034 (2003).
  38. J. O. Andersen, Phys. Rev. D 75, 065011 (2007).
  39. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961).
  40. U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27, 195 (1991).
  41. S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992).
  42. T. Hatsuda and T. Kunihiro, Phys. Rep. 247, 221 (1994).
  43. D. Toublan and J. B. Kogut, Phys. Lett. B 564, 212 (2003).
  44. M. Frank, M. Buballa, and M. Oertel, Phys. Lett. B 562, 221 (2003).
  45. A. Barducci, R. Casalbuoni, G. Pettini, and L. Ravagli, Phys. Rev. D 69, 096004 (2004).
  46. A. Barducci, R. Casalbuoni, G. Pettini, and L. Ravagli, Phys. Rev. D 71, 016011 (2005).
  47. L. He and P. Zhuang, Phys. Lett. B 615, 93 (2005).
  48. L. He, M. Jin, and P. Zhuang, Phys. Rev. D 71, 116001 (2005).
  49. H. J. Warringa, D. Boer, and J. O. Andersen, Phys. Rev. D 72, 014015 (2005).
  50. L. He, M. Jin, and P. Zhuang, Phys. Rev. D 74, 036005 (2006).
  51. Z. Zhang and Y. -X. Liu, Phys. Rev. C 75, 064910 (2007).
  52. C. Ratti and W. Weise, Phys. Rev. D 70, 054013 (2004).
  53. G. Sun, L. He, and P. Zhuang, Phys. Rev. D 75, 096004 (2007).
  54. T. Brauner, K. Fukushima, and Y. Hidaka, Phys. Rev. D 80, 074035 (2009).
  55. J. O. Andersen and T. Brauner, Phys. Rev. D 81, 096004 (2010).
  56. J. Xiong, M. Jin, and J. Li, J. Phys. G 36, 125005 (2009).
  57. H. Hu, X.-J. Liu, and P. D. Drummond, Europhys. Lett. 74, 574 (2006); Nature Phys. 3, 469 (2007).
  58. R. B. Diener, R. Sensarma, and M. Randeria, Phys. Rev. A 77, 023626 (2008).
  59. X. Leyronas and R. Combescot, Phys. Rev. Lett. 99, 170402 (2007).
  60. M. Huang, P. Zhuang, and W. Chao, Phys. Rev. D 65, 076012 (2002).

  61. To be consistent with the convention used in [57, 58, 62], here ϕ†(Q) is defined as [ϕ(Q)]†.

  62. J. R. Engelbrecht, M. Randeria, and C. A. R. Sá de Melo, Phys. Rev. B 55, 15153 (1997).
  63. N. Nagaosa, Quantum Field Theory in Condensed Matter Physics (Springer, Heidelberg, Germany, 1999).
  64. L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University Press, Oxford, England, 2003).
  65. C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, England, 2002).
  66. H. J. Schulze, J. Phys. G 21, 185 (1995).
  67. S. Weinberg, Phys. Rev. Lett. 17, 616 (1966).
  68. L. He and P. Zhuang, Phys. Rev. D 75, 096003 (2007).
  69. T. Brauner, Phys. Rev. D 77, 096006 (2008).
  70. T. D. Cohen, R. J. Furnstahl, and D. K. Griegel, Phys. Rev. C 45, 1881 (1992).
  71. T. D. Cohen, R. J. Furnstahl, D. K. Griegel, and X. Jin, Prog. Part. Nucl. Phys. 35, 221 (1995).
  72. L. He, Y. Jiang, and P. Zhuang, Phys. Rev. C 79, 045205 (2009).
  73. D. T. Son and M. A. Stephanov, Phys. Rev. A 74, 013614 (2006).
  74. M. Kitazawa, D. H. Rischke, and I. A. Shovkovy, Phys. Lett. B 663, 228 (2008).
  75. P. Nozières and S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985).
  76. C. A. R. Sá de Melo, M. Randeria, and J. R. Engelbrecht, Phys. Rev. Lett. 71, 3202 (1993).
  77. Y. Nishida and H. Abuki, Phys. Rev. D 72, 096004 (2005).
  78. H. Abuki, Nucl. Phys. A791, 117 (2007).
  79. L. He and P. Zhuang, Phys. Rev. D 76, 056003 (2007).

  80. In nonrelativistic systems, the BCS and BEC states are distinguished by μ>0 and μ<0, respectively, where μ is the fermion chemical potential.

  81. D. Toublan and A. R. Zhitnitsky, Phys. Rev. D 73, 034009 (2006).

  82. This is only possible for our two-flavor case. For Nf>2, the chiral symmetry is spontaneously broken at high density [23].

  83. For a careful explanation of the meson spectra in chiral perturbation theory, see T. Brauner, Mod. Phys. Lett. A 21, 559 (2006).
  84. D. Blaschke, F. Reinholz, G. Ropke, and D. Kremp, Phys. Lett. 151B, 439 (1985).
  85. T. Hatsuda and T. Kunihiro, Phys. Rev. Lett. 55, 158 (1985); Phys. Lett. B 185, 304 (1987).

  86. The mesons we called here are the collective excitations which have the same quantum numbers as the pions and the sigma meson in the vacuum. They are also called “pions” and “sigma meson” in the chiral restored regime.

  87. For the studies of meson properties in color-superconducting quark matter, see D. Ebert, K. G. Klimenko, and V. L. Yudichev, Phys. Rev. C 72, 015201 (2005); D. Zablocki, D. Blaschke, and R. Anglani, AIP Conf. Proc. 1038, 159 (2008). However, in these papers the meson properties are investigated only at zero momentum.
  88. J. Hüfner, S. P. Klevansky, P. Zhuang, and H. Voss, Ann. Phys. (N.Y.) 234, 225 (1994).
  89. P. Zhuang, J. Hüfner, and S. P. Klevansky, Nucl. Phys. A576, 525 (1994).
  90. J. B. Kogut and D. Toublan, Phys. Rev. D 64, 034007 (2001).
  91. R. Rapp, T. Schaefer, E. V. Shuryak, and M. Velkovsky, Phys. Rev. Lett. 81, 53 (1998).
  92. M. Alford, K. Rajagopal, and F. Wilczek, Phys. Lett. B 422, 247 (1998).
  93. M. Alford, K. Rajagopal, T. Schäfer, and A. Schmitt, Rev. Mod. Phys. 80, 1455 (2008); K. Rajagopal and F. Wilczek, arXiv:hep-ph/0011333; D. K. Hong, Acta Phys. Pol. B 32, 1253 (2001); M. Alford, Annu. Rev. Nucl. Part. Sci. 51, 131 (2001); T. Schäfer, arXiv:hep-ph/0304281; D. H. Rischke, Prog. Part. Nucl. Phys. 52, 197 (2004); M. Buballa, Phys. Rep. 407, 205 (2005); H. -C. Ren, arXiv:hep-ph/0404074; M. Huang, Int. J. Mod. Phys. E 14, 675 (2005); I. A. Shovkovy, Found. Phys. 35, 1309 (2005); Q. Wang, Progress in Physics 30, 173 (2010).
  94. A. Perali, P. Pieri, G. C. Strinati, and C. Castellani, Phys. Rev. B 66, 024510 (2002); P. Pieri, L. Pisani, and G. C. Strinati, 70, 094508 (2004).
  95. R. Haussmann, W. Rantner, S. Cerrito, and W. Zwerger, Phys. Rev. A 75, 023610 (2007).
  96. Q. Chen, J. Stajic, S. Tan, and K. Levin, Phys. Rep. 412, 1 (2005).
  97. D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Phys. Rev. Lett. 93, 090404 (2004).

Outline

Information

Phys. Rev. D 82, 096003– Published 8 November, 2010
  • Received 25 July 2010

© 2010 The American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation