Advertisement

Sensitivity of predictions in an effective model: Application to the chiral critical end point position in the Nambu-Jona-Lasinio model

  • Alexandre BiguetEmail author
  • Hubert Hansen
  • Pedro Costa
  • Pierre Borgnat
  • Timothée Brugière
Regular Article - Theoretical Physics

Abstract

The measurement of the position of the chiral critical end point (CEP) in the QCD phase diagram is under debate. While it is possible to predict its position by using effective models specifically built to reproduce some of the features of the underlying theory (QCD), the quality of the predictions (e.g., the CEP position) obtained by such effective models, depends on whether solving the model equations constitute a well- or ill-posed inverse problem. Considering these predictions as being inverse problems provides tools to evaluate if the problem is ill-conditioned, meaning that infinitesimal variations of the inputs of the model can cause comparatively large variations of the predictions. If it is ill-conditioned, it has major consequences because of finite variations that could come from experimental and/or theoretical errors. In the following, we shall apply such a reasoning on the predictions of a particular Nambu-Jona-Lasinio model within the mean field + ring approximations, with special attention to the prediction of the chiral CEP position in the (T-μ) plane. We find that the problem is ill-conditioned (i.e. very sensitive to input variations) for the T-coordinate of the CEP, whereas, it is well-posed for the μ-coordinate of the CEP. As a consequence, when the chiral condensate varies in a 10MeV range, μ CEP varies far less. As an illustration to understand how problematic this could be, we show that the main consequence when taking into account finite variation of the inputs, is that the existence of the CEP itself cannot be predicted anymore: for a deviation as low as 0.6% with respect to vacuum phenomenology (well within the estimation of the first correction to the ring approximation) the CEP may or may not exist.

References

  1. 1.
    M. Asakawa, K. Yazaki, Nucl. Phys. A 504, 668 (1989).CrossRefGoogle Scholar
  2. 2.
    B.I. Abelev et al., Phys. Rev. C 81, 024911 (2010).CrossRefADSGoogle Scholar
  3. 3.
    M.M. Aggarwal, An Experimental Exploration of the QCD Phase Diagram: The Search for the Critical Point and the Onset of De-confinement, arXiv:1007.2613 [nucl-ex].
  4. 4.
    Terence J. Tarnowsky, J. Phys. G 38, 124054 (2011).CrossRefADSGoogle Scholar
  5. 5.
    Roy A. Lacey, Phys. Rev. Lett. 114, 142301 (2015).CrossRefADSGoogle Scholar
  6. 6.
    Yasuyuki Akiba, Aaron Angerami, Helen Caines, Anthony Frawley, Ulrich Heinz, The Hot QCD White Paper: Exploring the Phases of QCD at RHIC and the LHC, arXiv:1502.02730 [nucl-ex] (2013).
  7. 7.
    In 8th International Workshop on Critical Point and Onset of Deconfinement, 2013.Google Scholar
  8. 8.
    Xiaofeng Luo, Ming Shao, Cheng Li, Hongfang Chen, Phys. Lett. B 673, 268 (2009).CrossRefADSGoogle Scholar
  9. 9.
    Rajiv V. Gavai, Acta Phys. Pol. B 43, 723 (2012).CrossRefGoogle Scholar
  10. 10.
    Marek Gazdzicki, J. Phys. G 38, 124024 (2011).CrossRefADSGoogle Scholar
  11. 11.
    D. Blaschke, Searching for for a QCD Mixed Phase at the Nuclotron-Based Ion Collider Facility (NICA White Paper) (Dubna, 2013).Google Scholar
  12. 12.
    Z. Fodor, S.D. Katz, JHEP 04, 050 (2004).CrossRefADSGoogle Scholar
  13. 13.
    Szabolcs Borsanyi, Gergely Endrodi, Zoltan Fodor, Antal Jakovac, Sandor D. Katz et al., JHEP 11, 077 (2010).CrossRefGoogle Scholar
  14. 14.
    A. Bazavov, T. Bhattacharya, M. Cheng, C. DeTar, H.T. Ding et al., Phys. Rev. D 85, 054503 (2012).CrossRefADSGoogle Scholar
  15. 15.
    K. Fukushima, Phys. Lett. B 591, 277 (2004).CrossRefADSGoogle Scholar
  16. 16.
    C. Ratti, M.A. Thaler, W. Weise, Phys. Rev. D 73, 014019 (2006).CrossRefADSGoogle Scholar
  17. 17.
    Pedro Costa, C.A. de Sousa, M.C. Ruivo, Yu.L. Kalinovsky, Phys. Lett. B 647, 431 (2007).CrossRefADSGoogle Scholar
  18. 18.
    K. Fukushima, Phys. Rev. D 77, 114028 (2008).CrossRefADSGoogle Scholar
  19. 19.
    K. Kashiwa, H. Kouno, M. Matsuzaki, M. Yahiro, Phys. Lett. B 662, 26 (2008).CrossRefADSGoogle Scholar
  20. 20.
    S. Rossner, T. Hell, C. Ratti, W. Weise, Nucl. Phys. A 814, 118 (2008).CrossRefADSGoogle Scholar
  21. 21.
    Pedro Costa, C.A. de Sousa, M.C. Ruivo, H. Hansen, EPL 86, 31001 (2009).CrossRefADSGoogle Scholar
  22. 22.
    P. Costa, H. Hansen, M.C. Ruivo, C.A. de Sousa, Phys. Rev. D 81, 016007 (2010).CrossRefADSGoogle Scholar
  23. 23.
    Pedro Costa, M.C. Ruivo, C.A. de Sousa, H. Hansen, Symmetry 2, 1338 (2010).CrossRefGoogle Scholar
  24. 24.
    Bernd-Jochen Schaefer, Jan M. Pawlowski, Jochen Wambach, Phys. Rev. D 76, 074023 (2007).CrossRefADSGoogle Scholar
  25. 25.
    Tina Katharina Herbst, Jan M. Pawlowski, Bernd-Jochen Schaefer, Phys. Lett. B 696, 58 (2011).CrossRefADSGoogle Scholar
  26. 26.
    Kenji Fukushima, Phys. Rev. D 77, 114028 (2008).CrossRefGoogle Scholar
  27. 27.
    Nino M. Bratovic, Tetsuo Hatsuda, Wolfram Weise, Phys. Lett. B 719, 131 (2013).CrossRefADSGoogle Scholar
  28. 28.
    G.A. Contrera, A.G. Grunfeld, D.B. Blaschke, Phys. Part. Nucl. Lett. 11, 342 (2014).CrossRefGoogle Scholar
  29. 29.
    Thomas Hell, Kouji Kashiwa, Wolfram Weise, J. Mod. Phys. 4, 644 (2013).CrossRefGoogle Scholar
  30. 30.
    O. Kaczmarek, F. Karsch, E. Laermann, C. Miao, S. Mukherjee et al., Phys. Rev. D 83, 014504 (2011).CrossRefADSGoogle Scholar
  31. 31.
    David Blaschke, David E. Alvarez-Castillo, Sanjin Benic, PoS CPOD2013, 063 (2013).Google Scholar
  32. 32.
    Albert Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (Society for Industrial and Applied Mathematics, 2005).Google Scholar
  33. 33.
    A.G. Ramm, Inverse problems, tomography, and image processing (Springer, 1997).Google Scholar
  34. 34.
    J. Dobaczewski, W. Nazarewicz, P.-G. Reinhard, J. Phys. G 41, 074001 (2014).CrossRefADSGoogle Scholar
  35. 35.
    J. Toivanen, J. Dobaczewski, M. Kortelainen, K. Mizuyama, Phys. Rev. C 78, 034306 (2008).CrossRefADSGoogle Scholar
  36. 36.
    P.G. Reinhard, W. Nazarewicz, Phys. Rev. C 81, 051303 (2010).CrossRefADSGoogle Scholar
  37. 37.
    F.J. Fattoyev, J. Piekarewicz, Phys. Rev. C 84, 064302 (2010).CrossRefADSGoogle Scholar
  38. 38.
    M. Kortelainen, T. Lesinski, J. More, W. Nazarewicz, J. Sarich et al., Phys. Rev. C 82, 024313 (2010).CrossRefADSGoogle Scholar
  39. 39.
    Mario Wschebor Felipe Cucker, Numer. Math. 94, 419 (2003).zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    J.W. Demmel, Numer. Math. 51, 251 (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    S.P. Klevansky, Rev. Mod. Phys. 64, 649 (1992).MathSciNetCrossRefADSGoogle Scholar
  42. 42.
    U. Vogl, W. Weise, Progr. Part. Nucl. Phys. 27, 195 (1991).CrossRefADSGoogle Scholar
  43. 43.
    Michael Buballa, Phys. Rep. 407, 205 (2005).CrossRefADSGoogle Scholar
  44. 44.
    Tetsuo Hatsuda, Teiji Kunihiro, Phys. Rep. 247, 221 (1994).CrossRefADSGoogle Scholar
  45. 45.
    H.G. Dosch, S. Narison, Phys. Lett. B 417, 173 (1998).CrossRefADSGoogle Scholar
  46. 46.
    J. Bordes, C.A. Dominguez, P. Moodley, J. Peñarrocha, K. Schilcher, JHEP 05, 064 (2010).CrossRefADSGoogle Scholar
  47. 47.
    Sinya Aoki, Yasumichi Aoki, Claude Bernard, Tom Blum, Gilberto Colangelo, Review of lattice results concerning low energy particle physics, arXiv:1310.8555 (2013).
  48. 48.
    O. Lourenco, M. Dutra, T. Frederico, A. Delfino, M. Malheiro, Phys. Rev. D 85, 097504 (2012).CrossRefADSGoogle Scholar
  49. 49.
    Micaela Oertel, Investigation of meson loop effects in the Nambu-Jona-Lasinio model, doctoral dissertation arXiv: hep-ph/0012224 (2000).
  50. 50.
    L.S. Celenza, Shun-fu Gao, Bo Huang, Huangsheng Wang, C.M. Shakin, Phys. Rev. C 61, 035201 (2000).CrossRefADSGoogle Scholar
  51. 51.
    L.S. Celenza, Huangsheng Wang, C.M. Shakin, Phys. Rev. C 63, 025209 (2001).CrossRefADSGoogle Scholar
  52. 52.
    Jenq-Neng Hwang, S.-R. Lay, A. Lippman, Nonparametric multivariate density estimation: a comparative study, in IEEE Transactions on Signal Processing, Vol. 42 (1994) pp. 2795-2810.Google Scholar

Copyright information

© SIF, Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Alexandre Biguet
    • 1
    Email author
  • Hubert Hansen
    • 1
  • Pedro Costa
    • 2
  • Pierre Borgnat
    • 3
  • Timothée Brugière
    • 1
  1. 1.Institut de Physique Nucléaire de Lyon, CNRS/IN2P3Université Claude Bernard de LyonVilleurbanne CedexFrance
  2. 2.Centro de Física Computacional, Departamento de FísicaUniversidade de CoimbraCoimbraPortugal
  3. 3.Laboratoire de PhysiqueCNRS, l’École normale supérieure de LyonLyon Cedex 07France

Personalised recommendations