The strange critical endpoint and isentropic trajectories in an extended PNJL model with eight Quark interactions

Abstract

In this work, we explore the possible existence of several critical endpoints in the phase diagram of strongly interacting matter using an extended PNJL model with ’t Hooft determinant and eight quark interactions in the up, down and strange sectors. Besides, we also study the isentropic trajectories crossing both (light and strange) chiral phase transitions and around the critical endpoint in both the crossover and first-order transition regions.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The manuscript has no associated data, as it is a theoretical paper.]

Notes

  1. 1.

    \(\mathcal {P}\) is the path ordering operator.

  2. 2.

    For pure glue theory, the Polyakov loop is an exact order parameter. In the confined phase, the boundary conditions of QCD are respected by the \(Z(N_c)\) symmetry while in the deconfined phase it is broken.

  3. 3.

    Defined using the inflection point in the Polyakov loop.

  4. 4.

    This formula can be proved by induction.

References

  1. 1.

    A.M. Halasz, A. Jackson, R. Shrock, M.A. Stephanov, J. Verbaarschot, Phys. Rev. D 58, 096007 (1998). https://doi.org/10.1103/PhysRevD.58.096007

    ADS  Article  Google Scholar 

  2. 2.

    N. Brambilla et al., Eur. Phys. J. C 74(10), 2981 (2014). https://doi.org/10.1140/epjc/s10052-014-2981-5

    Article  Google Scholar 

  3. 3.

    Y. Aoki, G. Endrodi, Z. Fodor, S.D. Katz, K.K. Szabo, Nature 443, 675 (2006). https://doi.org/10.1038/nature05120

    ADS  Article  Google Scholar 

  4. 4.

    B. Friman, C. Hohne, J. Knoll, S. Leupold, J. Randrup, R. Rapp, P. Senger, Lect. Notes Phys. 814, 681 (2011)

    ADS  Google Scholar 

  5. 5.

    G. Endrodi, Z. Fodor, S. Katz, K. Szabo, JHEP 04, 001 (2011). https://doi.org/10.1007/JHEP04(2011)001

    ADS  Article  Google Scholar 

  6. 6.

    C.S. Fischer, Prog. Part. Nucl. Phys. 105, 1 (2019). https://doi.org/10.1016/j.ppnp.2019.01.002

    ADS  Article  Google Scholar 

  7. 7.

    P. Isserstedt, M. Buballa, C.S. Fischer, P.J. Gunkel, Phys. Rev. D 100(7), 074011 (2019). https://doi.org/10.1103/PhysRevD.100.074011

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345 (1961). https://doi.org/10.1103/PhysRev.122.345

    ADS  Article  Google Scholar 

  9. 9.

    Y. Nambu, G. Jona-Lasinio, Phys. Rev. 124, 246 (1961). https://doi.org/10.1103/PhysRev.124.246

    ADS  Article  Google Scholar 

  10. 10.

    G. ’t Hooft, Phys. Rev. D14, 3432 (1976). https://doi.org/10.1103/PhysRevD.14.3432

  11. 11.

    G.t. Hooft, Phys. Rev. D 18(6), 2199 (1978). https://doi.org/10.1103/PhysRevD.18.2199.3

  12. 12.

    T. Kunihiro, T. Hatsuda, Phys. Lett. B 206, 385 (1988). https://doi.org/10.1016/0370-2693(88)91596-1

    ADS  Article  Google Scholar 

  13. 13.

    V. Bernard, R.L. Jaffe, U.G. Meissner, Nucl. Phys. B 308, 753 (1988). https://doi.org/10.1016/0550-3213(88)90127-7

    ADS  Article  Google Scholar 

  14. 14.

    H. Reinhardt, R. Alkofer, Phys. Lett. B 207, 482 (1988). https://doi.org/10.1016/0370-2693(88)90687-9

    ADS  Article  Google Scholar 

  15. 15.

    A.A. Osipov, B. Hiller, J. da Providencia, Phys. Lett. B 634, 48 (2006). https://doi.org/10.1016/j.physletb.2006.01.008

    ADS  Article  Google Scholar 

  16. 16.

    A.A. Osipov, B. Hiller, A.H. Blin, J. da Providencia, Annals Phys. 322, 2021 (2007). https://doi.org/10.1016/j.aop.2006.08.004

    ADS  Article  Google Scholar 

  17. 17.

    K. Fukushima, Phys. Lett. B 591, 277 (2004). https://doi.org/10.1016/j.physletb.2004.04.027

    ADS  Article  Google Scholar 

  18. 18.

    E. Megias, E. Ruiz Arriola, L.L. Salcedo, Phys. Rev. D 69, 116003 (2004). https://doi.org/10.1103/PhysRevD.69.116003

    ADS  Article  Google Scholar 

  19. 19.

    E. Megias, E. Ruiz Arriola, L. Salcedo, Phys. Rev. D 74, 065005 (2006). https://doi.org/10.1103/PhysRevD.74.065005

    ADS  Article  Google Scholar 

  20. 20.

    S. Roessner, C. Ratti, W. Weise, Phys. Rev. D 75, 034007 (2007). https://doi.org/10.1103/PhysRevD.75.034007

    ADS  Article  Google Scholar 

  21. 21.

    T. Hatsuda, T. Kunihiro, Phys. Rept. 247, 221 (1994). https://doi.org/10.1016/0370-1573(94)90022-1

    ADS  Article  Google Scholar 

  22. 22.

    P. Zhuang, J. Hufner, S. Klevansky, Nucl. Phys. A 576, 525 (1994). https://doi.org/10.1016/0375-9474(94)90743-9

    ADS  Article  Google Scholar 

  23. 23.

    M. Buballa, Phys. Rept. 407, 205 (2005). https://doi.org/10.1016/j.physrep.2004.11.004

    ADS  Article  Google Scholar 

  24. 24.

    H. Hansen et al., Phys. Rev. D 75, 065004 (2007). https://doi.org/10.1103/PhysRevD.75.065004

    ADS  Article  Google Scholar 

  25. 25.

    B. Hiller, J. Moreira, A.A. Osipov, A.H. Blin, Phys. Rev. D 81, 116005 (2010). https://doi.org/10.1103/PhysRevD.81.116005

    ADS  Article  Google Scholar 

  26. 26.

    P. Costa, M. Ruivo, C. de Sousa, Phys. Rev. D 77, 096001 (2008). https://doi.org/10.1103/PhysRevD.77.096001

    ADS  Article  Google Scholar 

  27. 27.

    P. Costa, C. de Sousa, M. Ruivo, H. Hansen, EPL 86(3), 31001 (2009). https://doi.org/10.1209/0295-5075/86/31001

    ADS  Article  Google Scholar 

  28. 28.

    J.M. Torres-Rincon, J. Aichelin, Phys. Rev. C 96(4), 045205 (2017). https://doi.org/10.1103/PhysRevC.96.045205

    ADS  Article  Google Scholar 

  29. 29.

    S. Borsanyi, Z. Fodor, C. Hoelbling, S.D. Katz, S. Krieg, C. Ratti, K.K. Szabo, JHEP 09, 073 (2010). https://doi.org/10.1007/JHEP09(2010)073

    ADS  Article  Google Scholar 

  30. 30.

    A. Bazavov, T. Bhattacharya, M. Cheng, C. DeTar, H. Ding et al., Phys. Rev. D 85, 054503 (2012). https://doi.org/10.1103/PhysRevD.85.054503

    ADS  Article  Google Scholar 

  31. 31.

    R. Bellwied, S. Borsanyi, Z. Fodor, J. Günther, S. Katz, C. Ratti, K. Szabo, Phys. Lett. B 751, 559 (2015). https://doi.org/10.1016/j.physletb.2015.11.011

    ADS  Article  Google Scholar 

  32. 32.

    A. Bazavov et al., Phys. Lett. B 795, 15 (2019). https://doi.org/10.1016/j.physletb.2019.05.013

    ADS  MathSciNet  Article  Google Scholar 

  33. 33.

    A. Osipov, B. Hiller, A. Blin, Eur. Phys. J. A 49, 14 (2013). https://doi.org/10.1140/epja/i2013-13014-y

    ADS  Article  Google Scholar 

  34. 34.

    A. Osipov, B. Hiller, A. Blin, Phys. Rev. D 88(5), 054032 (2013). https://doi.org/10.1103/PhysRevD.88.054032

    ADS  Article  Google Scholar 

  35. 35.

    J. Moreira, J. Morais, B. Hiller, A.A. Osipov, A.H. Blin, Phys. Rev. D 91, 116003 (2015). https://doi.org/10.1103/PhysRevD.91.116003

    ADS  Article  Google Scholar 

  36. 36.

    A.A. Osipov, B. Hiller, Phys. Lett. B 515, 458 (2001). https://doi.org/10.1016/S0370-2693(01)00889-9

    ADS  Article  Google Scholar 

  37. 37.

    A.A. Osipov, B. Hiller, Phys. Rev. D 64, 087701 (2001). https://doi.org/10.1103/PhysRevD.64.087701

    ADS  Article  Google Scholar 

  38. 38.

    A.A. Osipov, B. Hiller, Phys. Rev. D 63, 094009 (2001). https://doi.org/10.1103/PhysRevD.63.094009

    ADS  Article  Google Scholar 

  39. 39.

    P. Costa, M. Ferreira, D.P. Menezes, J. Moreira, C. Providência, Phys. Rev. D 92(3), 036012 (2015). https://doi.org/10.1103/PhysRevD.92.036012

    ADS  Article  Google Scholar 

  40. 40.

    M. Ferreira, P. Costa, C. Providência, Phys. Rev. D 97(1), 014014 (2018). https://doi.org/10.1103/PhysRevD.97.014014

    ADS  Article  Google Scholar 

  41. 41.

    K. Fukushima, Phys. Rev. D 77, 114028 (2008). https://doi.org/10.1103/PhysRevD.77.114028

    ADS  Article  Google Scholar 

  42. 42.

    P. Costa, Phys. Rev. D 93(11), 114035 (2016). https://doi.org/10.1103/PhysRevD.93.114035

    ADS  Article  Google Scholar 

  43. 43.

    M. Ferreira, P. Costa, C. Providência, Phys. Rev. D 98(3), 034006 (2018). https://doi.org/10.1103/PhysRevD.98.034006

    ADS  Article  Google Scholar 

  44. 44.

    M. Aggarwal, et al., (2010)

  45. 45.

    B. Abelev et al., Phys. Rev. C 81, 024911 (2010). https://doi.org/10.1103/PhysRevC.81.024911

    ADS  Article  Google Scholar 

  46. 46.

    L. Adamczyk et al., Phys. Rev. Lett. 112, 032302 (2014). https://doi.org/10.1103/PhysRevLett.112.032302

    ADS  Article  Google Scholar 

  47. 47.

    A. Aduszkiewicz et al., Eur. Phys. J. C 76(11), 635 (2016). https://doi.org/10.1140/epjc/s10052-016-4450-9

    ADS  Article  Google Scholar 

  48. 48.

    V. Vovchenko, L. Jiang, M.I. Gorenstein, H. Stoecker, Phys. Rev. C 98(2), 024910 (2018). https://doi.org/10.1103/PhysRevC.98.024910

    ADS  Article  Google Scholar 

  49. 49.

    L. Adamczyk et al., Phys. Rev. C 96(4), 044904 (2017). https://doi.org/10.1103/PhysRevC.96.044904

    ADS  MathSciNet  Article  Google Scholar 

  50. 50.

    W. Busza, K. Rajagopal, W. van der Schee, Ann. Rev. Nucl. Part. Sci. 68, 339 (2018). https://doi.org/10.1146/annurev-nucl-101917-020852

    ADS  Article  Google Scholar 

  51. 51.

    P. Costa, Eur. Phys. J. A 52(8), 228 (2016). https://doi.org/10.1140/epja/i2016-16228-5

    ADS  Article  Google Scholar 

  52. 52.

    P. Costa, R.C. Pereira, Symmetry 11(4), 507 (2019). https://doi.org/10.3390/sym11040507

    Article  Google Scholar 

  53. 53.

    M. Bluhm, B. Kampfer, R. Schulze, D. Seipt, U. Heinz, Phys. Rev. C 76, 034901 (2007). https://doi.org/10.1103/PhysRevC.76.034901

    ADS  Article  Google Scholar 

  54. 54.

    C. DeTar, L. Levkova, S. Gottlieb, U. Heller, J. Hetrick, R. Sugar, D. Toussaint, Phys. Rev. D 81, 114504 (2010). https://doi.org/10.1103/PhysRevD.81.114504

    ADS  Article  Google Scholar 

  55. 55.

    S. Borsanyi, G. Endrodi, Z. Fodor, S. Katz, S. Krieg, C. Ratti, K. Szabo, JHEP 08, 053 (2012). https://doi.org/10.1007/JHEP08(2012)053

    ADS  Article  Google Scholar 

  56. 56.

    A.A. Osipov, B. Hiller, V. Bernard, A.H. Blin, Annals Phys. 321, 2504 (2006). https://doi.org/10.1016/j.aop.2006.02.010

    ADS  Article  Google Scholar 

  57. 57.

    C. Ratti, M.A. Thaler, W. Weise, Phys. Rev. D 73, 014019 (2006). https://doi.org/10.1103/PhysRevD.73.014019

    ADS  Article  Google Scholar 

  58. 58.

    J. Moreira, B. Hiller, A. Osipov, A. Blin, Int. J. Mod. Phys. A 27, 1250060 (2012). https://doi.org/10.1142/S0217751X12500601

    ADS  Article  Google Scholar 

  59. 59.

    J.I. Kapusta, Finite-temperature field theory (Cambridge University Press, Cambridge, 1989)

    Google Scholar 

  60. 60.

    V. Skokov, B. Stokic, B. Friman, K. Redlich, Phys. Rev. C 82, 015206 (2010). https://doi.org/10.1103/PhysRevC.82.015206

    ADS  Article  Google Scholar 

  61. 61.

    T.K. Herbst, J.M. Pawlowski, B.J. Schaefer, Phys. Lett. B 696, 58 (2011). https://doi.org/10.1016/j.physletb.2010.12.003

    ADS  Article  Google Scholar 

  62. 62.

    P. Costa, M. Ruivo, C. de Sousa, H. Hansen, Symmetry 2, 1338 (2010). https://doi.org/10.3390/sym2031338

    Article  Google Scholar 

  63. 63.

    G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lutgemeier, B. Petersson, Nucl. Phys. B 469, 419 (1996). https://doi.org/10.1016/0550-3213(96)00170-8

    ADS  Article  Google Scholar 

  64. 64.

    B.J. Schaefer, J.M. Pawlowski, J. Wambach, Phys. Rev. D 76, 074023 (2007). https://doi.org/10.1103/PhysRevD.76.074023

    ADS  Article  Google Scholar 

  65. 65.

    Y. Aoki et al., JHEP 06, 088 (2009). https://doi.org/10.1088/1126-6708/2009/06/088

  66. 66.

    W. Pauli, F. Villars, Rev. Mod. Phys. 21, 434 (1949). https://doi.org/10.1103/RevModPhys.21.434

    ADS  Article  Google Scholar 

  67. 67.

    M.K. Volkov, A.A. Osipov, Sov. J. Nucl. Phys. 41, 500 (1985)

    Google Scholar 

  68. 68.

    M.K. Volkov, A.A. Osipov, Yad. Fiz. 41, 785 (1985)

    Google Scholar 

  69. 69.

    S.P. Klevansky, Rev. Mod. Phys. 64, 649 (1992). https://doi.org/10.1103/RevModPhys.64.649

    ADS  MathSciNet  Article  Google Scholar 

  70. 70.

    M. Tanabashi et al., Phys. Rev. D 98(3), 030001 (2018). https://doi.org/10.1103/PhysRevD.98.030001

    ADS  Article  Google Scholar 

  71. 71.

    C.R. Allton, M. Doring, S. Ejiri, S.J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, K. Redlich, Phys. Rev. D 71, 054508 (2005). https://doi.org/10.1103/PhysRevD.71.054508

    ADS  Article  Google Scholar 

  72. 72.

    A. Bhattacharyya, P. Deb, S.K. Ghosh, R. Ray, Phys. Rev. D 82, 014021 (2010). https://doi.org/10.1103/PhysRevD.82.014021

    ADS  Article  Google Scholar 

  73. 73.

    H. Hansen, R. Stiele, P. Costa, Phys. Rev. D 101(9), 094001 (2020). https://doi.org/10.1103/PhysRevD.101.094001

    ADS  MathSciNet  Article  Google Scholar 

  74. 74.

    Y. Sakai, T. Sasaki, H. Kouno, M. Yahiro, Phys. Rev. D 82, 076003 (2010). https://doi.org/10.1103/PhysRevD.82.076003

    ADS  Article  Google Scholar 

  75. 75.

    P. Rehberg, S. Klevansky, J. Hufner, Phys. Rev. C 53, 410 (1996). https://doi.org/10.1103/PhysRevC.53.410

    ADS  Article  Google Scholar 

Download references

Acknowledgements

This work was supported by a research grant under project No. PTDC/FIS-NUC/29912/2017 (J.M.), funded by national funds through FCT (Fundação para a Ciência e a Tecnologia, I.P, Portugal)/ MCTES and co-financed by the European Regional Development Fund (ERDF) through the Portuguese Operational Program for Competitiveness and Internationalization, COMPETE 2020, by national funds from FCT under the IDPASC Ph.D. program (International Doctorate Network in Particle Physics, Astrophysics and Cosmology), with the Grant No. PD/BD/128234/2016 (R.C.P.), and under the Projects UID/FIS/04564/2019 and UID/FIS/04564/2020. The authors also acknowledge networking support by the COST Action CA15213 THOR (Theory of hot matter and relativistic heavy-ion collisions).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Renan Câmara Pereira.

Additional information

Communicated by Laura Tolos

Appendix A:The Mean Field approximation and Meson Masses

Appendix A:The Mean Field approximation and Meson Masses

We introduce the auxiliary scalar, \(s_a\), and pseudoscalar field variables, \(p_a\), written in terms of quark bilinear operators, \(s_a={\overline{q}} \lambda _a q\) and \(p_a={\overline{q}} i \gamma ^5 \lambda _a q\), with indices \(a=0,1,2,\ldots ,8\). Writing the Lagrangian density in terms of these new variables, yields:

(A.1)

Here, \(f_{abc}\) and \(d_{abc}\) are the totally antisymmetric and symmetric structure constants of the special unitary group SU(3), respectively. The constants \(A_{abc}\) are defined as:

$$\begin{aligned} A_{abc}&= \frac{2}{3} d_{abc} + \sqrt{ \frac{2}{3} } ( \delta _{a0} \delta _{b0} \delta _{c0} - \delta _{a0} \delta _{bc} \nonumber \\&\quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - \delta _{b0} \delta _{ca} - \delta _{c0} \delta _{ab} ) . \end{aligned}$$
(A.2)

In order to derive the thermodynamical potential of the model we consider the mean field approximation. In this approximation, all quark interactions are transformed into quadratic interactions by introducing auxiliary fields whose quantum fluctuations are neglected and only the classical configuration contributes to the path integral i.e., the functional integration is dominated by the stationary point. A quark bilinear operator, \(\hat{\mathcal {O}}\), can be written as its mean field value plus a small perturbation, . To linearize the product of \(N-\)operators, terms superior to \((\delta \hat{\mathcal {O}})^2\) must be neglected. Conveniently, the linear product between \(N=n+1\) operators can be written using the following formulaFootnote 4:

(A.3)

The Lagrangian density can then be trivially linearized, the quadratic fermion term can be exactly integrated out and the grand canonical potential of the model can be derived to yield Eq. (4).

The meson masses can be calculated by writing an effective Lagrangian, built by expanding the Lagrangian in Eq. (A.1) up to second order in the auxiliary fields, [75]. Following the linear expansion of the Lagrangian, to build the quadratic expansion, terms superior to \((\delta \hat{\mathcal {O}})^3\) must be neglected. More easily, the quadratic product between \(N=n+2\) operators, with \(n \ge 1\), can be written using the following formula4

(A.4)

Having the quadratic expansion of the Lagrangian, the pseudoscalar and scalar inverse propagators are defined as the coefficient of the second order terms in the auxiliary fields. The pseudoscalar and scalar meson propagators are then given by:

(A.5)
(A.6)

Here, the indices \(a,b=0,1,2,\ldots ,8\).

The pseudoscalar and scalar meson projectors, \(P_{ab}\) and \(S_{ab}\), with four, six and eight quark interactions, neglecting pseudoscalar condensates (), can be calculated to yield:

(A.7)
(A.8)

Using the diagonal matrices of SU\((3)_f\) and the identity, we can write the mean field values of the bilinear operators in the \(0-3-8\) basis. One can switch to the quark flavour basis, \(u-d-s\), doing a rotation as follows:

(A.9)

Here, the elements of the matrix \(T_{ai}\) are given by:

(A.10)

The polarization functions can be rotated between basis using,

$$\begin{aligned} \varPi _{ab} = T_{ai} T_{bj} \varPi _{ij}. \end{aligned}$$
(A.11)

The pseudoscalar and scalar polarization functions for two quarks with flavours i and j, are given by [75]:

Here,

(A.12)
(A.13)

Where \(\mathrm {p.v.}\) stands for the Cauchy principal value of the integral.

The mass of a given meson, \(M_M\), and its decay width, \(\varGamma _M\), can then be calculated by searching for the complex pole of its inverse propagator in the rest frame, i.e,

(A.14)

The correspondence between the auxiliary pseudoscalar fields and the physical pseudoscalar mesons can be performed using:

(A.15)

Where the pseudoscalar nonet was represented in the usual way. For the auxiliary scalar fields and the physical scalar fields, we use:

(A.16)

Using these correspondences the inverse propagator of a physical meson can be calculated using Eq. (A.14).

For the neutral mesons one must perform, as usual, a diagonalization of the quadratic contributions coming from the 0–3–8 channels. In the isotopic limit, one therefore obtains the straightforward extension of the results from [75] to include the eight quark contributions.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pereira, R.C., Moreira, J. & Costa, P. The strange critical endpoint and isentropic trajectories in an extended PNJL model with eight Quark interactions. Eur. Phys. J. A 56, 214 (2020). https://doi.org/10.1140/epja/s10050-020-00223-8

Download citation